Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies 9783540226963, 3540226966

Structure-Borne Sound" is a thorough introduction to structural vibrations with emphasis on audio frequencies and t

122 62 11MB

English Pages 620 [618] Year 2005

Report DMCA / Copyright

DOWNLOAD PDF FILE

Recommend Papers

Structure-Borne Sound: Structural Vibrations and Sound Radiation at Audio Frequencies
 9783540226963, 3540226966

  • 0 0 0
  • Like this paper and download? You can publish your own PDF file online for free in a few minutes! Sign Up
File loading please wait...
Citation preview

L. Cremer M. Heckl B.A.T. Petersson Structure-Borne Sound Structural Vibrations and Sound Radiation at Audio Frequencies 3rd edition

L. Cremer M. Heckl B.A.T. Petersson

Structure-Borne Sound Structural Vibrations and Sound Radiation at Audio Frequencies 3rd edition With 215 figures and 13 tables

L. Cremer M. Heckl Prof. Dr. B.A.T. Petersson Technische Universität Berlin FG Technische Akustik Einsteinufer 25 10587 Berlin Germany

Library of Congress Control Number: 2004116521 ISBN 3-540-22696-6 Springer Berlin Heidelberg New York This work subject to copyright. All rights are reserved, whether the whole or part of the material is Library ofisCongress Control Number: 2004116521 concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or inBerlin otherHeidelberg ways, and storage in data banks. Duplication of this publication or ISBN 3-540-22696-6 Springer New York parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its work current version,to and permission for useare must alwayswhether be obtained from Springer-Verlag. Violations This is subject copyright. All rights reserved, the whole or part of the material is are liable to prosecution under concerned, specifically the German rights of Copyright translation,Law. reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in other ways, and storage in data banks. Duplication of this publication or Springer is a part of Springer Media parts thereof is permitted onlyScience+Business under the provisions of the German Copyright Law of September 9, 1965, springeronline.com in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable to prosecution under German Copyright Law. © Springer-Verlag Berlin Heidelberg 2005 Printed in Germany Springer is a part of Springer Science+Business Media springeronline.com The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absenceBerlin of a specific statement, © Springer-Verlag Heidelberg 2005 that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Printed in Germany Typesetting: Data conversion the author. The use of general descriptivebynames, registered names, trademarks, etc. in this publication does not imply, Final by PTP-Berlin Protago-T GmbH,are Germany evenprocessing in the absence of a specific statement, that such names exempt from the relevant protective laws EX-Production Cover-Design: deblik, Berlin free for general use. and regulations and therefore Printed on acid-free paper 62/3020/Yu – 5 4 3 2 1 0 Typesetting: Data conversion by the author. Final processing by PTP-Berlin Protago-TEX-Production GmbH, Germany Cover-Design: deblik, Berlin Printed on acid-free paper 62/3020/Yu – 5 4 3 2 1 0

3UHIDFH

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|SIHU,PHOPDQQ!IRU!WKHLU!UHDG\!HIIRUWV!LQ!SURFHVVLQJ WKH VRPHWLPHV ZHDULQJ WH[W HTXDWLRQV DQG ILJXUHV )LQDOO\!UHVWLQJ!RQ!WKH!VKRXOGHUV!RI!KLV!GLVWLQJXLVKHG!SUHGHFHVVRUV!LW!LV WKH DXWKRU V KRSH WKDW WKLV HGLWLRQ ZLOO SURYH DV XVHIXO DV WKH IRUPHU iÀˆ˜] -Փ“iÀ Óää{ÊÊÊÊÊÊÊÊ

ŸÀ˜ °/° *iÌiÀÃܘ

7DEOHRI&RQWHQWV

,QWURGXFWLRQ  $/LWWOH'\QDPLFV   6LQJOH'HJUHHRI)UHHGRP6\VWHPV  /DJUDQJH¶V(TXDWLRQV   5HFLSURFLW\DQG0XWXDO(QHUJ\  0RGDO6\QWKHVLV   (QHUJ\&RQVLGHUDWLRQV  0LQLPL]DWLRQRIWKH$YHUDJH(QHUJ\'LIIHUHQFH +DPLOWRQ¶V 3ULQFLSOH   7KH5D\OHLJK4XRWLHQW  5HIHUHQFHV   6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV  /RQJLWXGLQDO:DYHV  3XUH/RQJLWXGLQDO:DYHV   4XDVL/RQJLWXGLQDO:DYHVRQ%HDPVDQG3ODWHV   7UDQVYHUVH:DYHV  7UDQVYHUVH3ODQH:DYHV  7RUVLRQDO:DYHV   %HQGLQJ:DYHV  3XUH%HQGLQJ:DYHV   (QHUJ\ 5HODWLRQV  :DYH0RWLRQVRQ%HDPVRI)LQLWH/HQJWK  /RQJLWXGLQDO1DWXUDO9LEUDWLRQV  1DWXUDO9LEUDWLRQVLQ%HQGLQJ  7KH*HQHUDO)LHOG(TXDWLRQV   :DYH)LHOGDWD)UHH6XUIDFH  5HIOHFWLRQRI3ODQH:DYHV   ([FLWDWLRQRIDQ(ODVWLF+DOI6SDFH  6XUIDFH:DYHV   )UHH3ODWH:DYHV   %RXQGDU\&RQGLWLRQVDQG7\SHVRI6ROXWLRQV   :DYHVZLWK'LVSODFHPHQWVRQO\3DUDOOHOWRWKH6XUIDFH   :DYHVZLWK'LVSODFHPHQWV3HUSHQGLFXODUWRWKH6XUIDFH  

9,,,

7DEOHRI&RQWHQWV

 (TXDWLRQV RI 0RWLRQ IRU 7KLQ 3ODWHV IURP WKH *HQHUDO )LHOG (TXDWLRQV    +DPLOWRQ¶V 3ULQFLSOH IRU WKH 'HULYDWLRQ RI WKH (TXDWLRQV RI 0RWLRQ   )XQGDPHQWDOV   )ODW 3ODWH ZLWK 6KHDU 6WLIIQHVV 7KH &RUUHFWHG %HQGLQJ :DYH    &\OLQGULFDO 6KHOOV    6WUXFWXUH%RUQH 6RXQG ,QWHQVLW\    )XQGDPHQWDO (TXDWLRQV    ,QWHQVLW\ LQ WKLV 3ODWHV   3RZHU 7UDQVPLVVLRQ LQ 7KLQ:DOOHG &\OLQGULFDO 6KHOOV   5HIHUHQFHV    'DPSLQJ   'DPSLQJ 0HFKDQLVPV DQG WKHLU 0DWKHPDWLFDO 'HVFULSWLRQ   &RPSOH[ 0RGXOXV DQG :DYHQXPEHUV   5HVRQDQW 9LEUDWLRQV RI 'DPSHG %HDPV   4XDVL/RQJLWXGLQDO :DYHV DQG 7RUVLRQDO :DYHV    %HQGLQJ :DYHV   0HDVXUHPHQW RI &RPSOH[ 0RGXOL   0HDVXUHPHQWV RQ 6PDOO 6DPSOHV    0HDVXUHPHQWV RQ %HDPV    0HDVXUHPHQWV RQ 2WKHU WKDQ %HDP/LNH 6DPSOHV    ([SHULPHQWDO 'DWD    0HWDOV    3ODVWLFV   %XLOGLQJ 0DWHULDOV    3ODWHV ZLWK $WWDFKHG /D\HUV    3ODWHV ZLWK 6LPSOH ([WHQVLRQDOO\ /RDGHG /D\HUV    3ODWHV ZLWK 0XOWL/D\HU 7UHDWPHQWV    (TXDWLRQV RI 0RWLRQ RI /D\HUHG 3ODWHV    'DPSLQJ E\ PHDQV RI 5HVRQDQW 6\VWHPV   'DPSLQJ E\ 7KLFN /D\HUV %DOODVW    'DPSLQJ DW -RLQWV    'DPSLQJ E\ 5HODWLYH 0RWLRQ 1RUPDO WR WKH ,QWHUIDFH    'DPSLQJ E\ 5HODWLYH 0RWLRQ 7DQJHQWLDO WR WKH ,QWHUIDFH   5HIHUHQFHV    ,PSHGDQFH DQG 0RELOLW\    'HILQLWLRQV    0HDVXUHPHQW RI 0RELOLWLHV ,PSHGDQFHV  

7DEOHRI&RQWHQWV ,;

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

;7DEOHRI&RQWHQWV

 $WWHQXDWLRQ RI %HQGLQJ :DYHV   %ORFNLQJ 0DVVHV   $WWHQXDWLRQ RI /RQJLWXGLQDO :DYHV   $WWHQXDWLRQ RI %HQGLQJ :DYHV ± 6\PPHWULF %ORFNLQJ 0DVVHV    $WWHQXDWLRQ RI %HQGLQJ :DYHV ± (FFHQWULF %ORFNLQJ 0DVVHV    3HULRGLF 6WUXFWXUHV   3HULRGLF 0DVV6SULQJ 6\VWHPV    $WWHQXDWLRQ RI /RQJLWXGLQDO :DYHV    3HULRGLF %HQGLQJ :DYH*XLGH   +DPLOWRQ¶V 3ULQFLSOH IRU 7UDQVPLVVLRQ 3UREOHPV   3URFHGXUH   $Q ,QWURGXFWRU\ ([DPSOH    %HQGLQJ DQG /RQJLWXGLQDO :DYHV DW DQ (FFHQWULF %ORFNLQJ 0DVV   2EOLTXH ,QFLGHQFH   *HQHUDO &RQVLGHUDWLRQV    *HQHUDO &RQVHTXHQFHV RI WKH %RXQGDU\ &RQGLWLRQV    ([DPSOHV    $SSOLFDWLRQ RI +DPLOWRQ¶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

7DEOHRI&RQWHQWV ;,

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

;,,! 7DEOH RI &RQWHQWV

 &RPELQHG 4XDQWLWLHV   5HIHUHQFHV   /LVW RI 6\PEROV DQG 1RWDWLRQ  ,QGH[ 

,QWURGXFWLRQ

$PDMRUSRUWLRQRIWKHVRXQGWKDWUHDFKHVRXUHDUV±EHLWDFKRUGIURP D JXLWDU WKH VTXHHO IURP EUDNHV RU D OLYHO\ GLVFXVVLRQ DW WKH QHLJKERXUV ± RULJLQDWHV IURP RU LV WUDQVPLWWHG WKURXJK YLEUDWLQJ VROLG VWUXFWXUHV 7KLV ILHOGRISK\VLFVZKLFKHQFRPSDVVHVJHQHUDWLRQWUDQVPLVVLRQSURSDJDWLRQ DQG UDGLDWLRQ RI ZDYH PRWLRQV LQ VROLG VWUXFWXUHV LV WHUPHG 6WUXFWXUH %RUQH6RXQG,QWKHQDPHWKHZRUG³VRXQG´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

,QWURGXFWLRQ

WKH DFFHOHUDWLRQ FDQ EH H[SHFWHG WR EH LQ WKH UDQJH IURP D PPV WR NPV+HUHLQDUHWUHDWHGHVVHQWLDOO\OLQHDUSURFHVVHVVRWKDWDFRQVWDQW FKDQJHLQIRUFHUHVXOWVLQDFRUUHVSRQGLQJFRQVWDQWFKDQJHLQPRWLRQ,ILW LVDFFRXQWHGIRUHYHQWXDOFKDQJHVLQPDWHULDOSDUDPHWHUVGXHWRHJVWDWLF ORDGLQJWKHQ PRVWDSSOLFDWLRQVLQSUDFWLFHLQYROYHOLQHDUSURFHVVHV:LWK WKHSURFHVVHVEHLQJOLQHDUWKHDSSOLFDELOLW\ LVZDUUDQWHGRIWKHVXSHUSRVL WLRQSULQFLSOH

*HQHUDWLRQ

7UDQVPLVVLRQ

3URSDJDWLRQ

5DGLDWLRQ

)LJ6WUXFWXUDODFRXVWLFSURFHVV

7KHVWUXFWXUDODFRXVWLFSURFHVVFDQEHVXEGLYLGHGLQWRIRXUPDLQVWDJHV DVVKRZQLQ)LJ7KHILUVWVWDJH±JHQHUDWLRQ±FRPSULVHVWKHRULJLQRI DQ RVFLOODWLRQ LH WKH PHFKDQLVP EHKLQG LW 7KH VHFRQG ± WUDQVPLVVLRQ ± FRYHUV WKH WUDQVIHU RI RVFLOODWRU\ HQHUJ\ IURP WKH PHFKDQLVPV RI JHQHUD WLRQ WR D SDVVLYH  VWUXFWXUH ,Q WKH VWUXFWXUH EH LW WKH SDVVLYHSDUW RIWKH VRXUFHRUDQDWWDFKHGUHFHLYLQJVWUXFWXUHWKHWKLUGVWDJH±SURSDJDWLRQ±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x DPDFKLQH VRXUFH LQVWDOOHGLQDSDUWLFXODUHQYLURQPHQW UHFHLYHU  x DPDFKLQHSDUW VRXUFH LQVWDOOHGLQDPDFKLQHU\ UHFHLYHU  x DVHQVLWLYHSLHFHRIHTXLSPHQW UHFHLYHU DWWDFKHGWRDYLEUDWLQJV\VWHP VRXUFH  ,QWKHILUVWVLWXDWLRQWKHVRXUFHLV FDXVLQJXQZDQWHGVRXQGRUYLEUDWLRQV DWRUUHPRWHIURPWKHVRXUFHORFDWLRQHJDFRPSUHVVRULQDEXLOGLQJRUD VKLS7KHVHFRQGW\SHRIVLWXDWLRQUHODWHVWRHJHOHFWULFPRWRUVSXPSVRU

,QWURGXFWLRQ

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

$/LWWOH'\QDPLFV

6LQJOH'HJUHHRI)UHHGRP6\VWHPV 3UHFLVHO\DVIRUWKHWUHDWPHQWRIIOXLGERUQHVRXQGWKHNLQHPDWLFDQGG\ QDPLFILHOGYDULDEOHVFRQVWLWXWHWKHSULPDU\LQJUHGLHQWVLQWKHWUHDWPHQWRI VWUXFWXUHERUQH VRXQG $Q LPSRUWDQW GLIIHUHQFH LV KRZHYHU WKDW LQ PRVW PHDVXUHPHQWV RI IOXLGERUQH VRXQG WKH SULPDU\ TXDQWLW\ LV G\QDPLF LH WKHVRXQGSUHVVXUHZKHUHDVLWLVDNLQHPDWLFLQPHDVXUHPHQWVRIVWUXFWXUH ERUQH VRXQG LHWKHGLVSODFHPHQWYHORFLW\RUDFFHOHUDWLRQ 7KHG\QDPLF TXDQWLWLHVVXFKDVIRUFHVRUVWUHVVHVFDQEHREWDLQHGYLDWKHFRQVWLWXWLYHUH ODWLRQVIRUWKHPDWHULDOLQWKHODWWHUFDVH7KHUHDVRQIRUWKLVGLIIHUHQFHLV WKDWWKHSUHVVXUHLVDVFDODUTXDQWLW\ DQGKHQFHPRVWPDQDJHDEOHZKLOVWWKH YHFWRULDOPRWLRQTXDQWLWLHVGHVSLWHWKHLUGLUHFWLRQGHSHQGHQFHDUHVLPSOHU WRPHDVXUHIRUVWUXFWXUHV $VLQFODVVLFDOG\QDPLFVDFRUQHUVWRQHIRUVWUXFWXUHERUQHVRXQGLVWKH VLQJOHGHJUHHRIIUHHGRP V\VWHP RIWHQ FDOOHG WKH PDVVVSULQJGDPSHU V\VWHP)RUVXFKDV\VWHPWKHHTXDWLRQRIPRWLRQFDQEHZULWWHQDV P

G [ G[ U  V[  GW GW

)

 

ZKHUH[LVWKHGLVSODFHPHQWIURPHTXLOLEULXP

)LJ6LQJOHGHJUHHRIIUHHGRPV\VWHPZLWKH[FLWDWLRQ)DQGGLVSODFHPHQW[

6LQJOH'HJUHHRI)UHHGRP6\VWHPV

,Q WKLV V\VWHP GHSLFWHG LQ )LJ  WKH PDVV LV GHQRWHG P WKH YLVFRXV GDPSLQJUDQGWKHVWLIIQHVVV,IWKHUHLVQRH[FLWDWLRQVXFKWKDW) WKH JHQHUDOVROXWLRQLV [ W

&HDW  & HD  W 

+HUH WKH WZR FRQVWDQWV & DQG & DUH WR EH GHWHUPLQHG IURP WKH LQLWLDO FRQGLWLRQVDWWLPHW DDQGDDUHWKHHLJHQYDOXHVLHWKHWZRURRWVRI WKHFKDUDFWHULVWLFHTXDWLRQ PD   FD  V



:LWKDQH[WHUQDOH[FLWDWLRQ ) z WKHSDUWLFXODUVROXWLRQPXVWEHDGGHG VXFKWKDW [ W

ª º ª º   J W » HDW  «&  J  W » HD W  «&  D  D  D   D ¬ ¼ ¬ ¼

ZKHUHJDQGJUHSUHVHQWWKH LQGHILQLWH LQWHJUDOV J

³

J

³

) W P ) W P

H DW GW

H D W GW 

DQG & DQG & DV EHIRUH DUH WKH LQWHJUDWLRQ FRQVWDQWV ZKLFK VDWLVI\ WKH LQLWLDOFRQGLWLRQV $VVXPLQJQRZWKDWWKHV\VWHPLVVXEMHFWWRDVWHDG\VWDWHKDUPRQLFH[ FLWDWLRQDWWKHDQJXODUIUHTXHQF\ ZVXFKWKDWWKHH[FLWDWLRQLVJLYHQLQWKH IRUP ) W

Ö MZ W  )H

ZKHUH )Ö LVWKHIRUFHDPSOLWXGHDQG H MZ W

FRV Z W  M VLQ Z W

7KHXVHRIWKHFRPSOH[H[SRQHQWLDOHMZWWRGHVFULEHWKHWLPHGHSHQGHQFHRI WKHSK\VLFDOIRUFHPHDQVWKDWLPSOLFLWO\HLWKHULVXQGHUVWRRG ) W

Ö MZ W º 5H ª¬ )H ¼

)Ö FRV Z W 

) W

Ö MZ W º ,P ª¬ )H ¼

)Ö VLQ Z W

RUDOWHUQDWLYHO\

$/LWWOH'\QDPLFV

7KH VHOHFWLRQ RI WKH UHDO RU LPDJLQDU\ SDUW LV DUELWUDU\ DV ORQJ DV WKLV FKRLFHLVUHWDLQHGWKURXJKRXWWKHDQDO\VLVDQGWKHSK\VLFDOLQWHUSUHWDWLRQRI WKH UHVXOWV LV PDGH FRQVLVWHQWO\ +HUHLQ WKH ILUVW DOWHUQDWLYH ZLOO EH HP SOR\HG 1RZ RQFH WKH LQLWLDO WUDQVLHQWV KDYH IDGHG WKH VWHDG\VWDWH UHVSRQVH PXVWDOVREHKDUPRQLFRIWKHVDPHIUHTXHQF\DVWKDWRIWKHH[FLWDWLRQDQG RQHFDQDVVXPHWKDWLWFDQEHZULWWHQLQWKHIRUP [ W

Ö MZ W  ' Z )H

ZKHUH ' Z  LV WKH FRPSOH[ IUHTXHQF\ UHVSRQVH IXQFWLRQ RI WKH V\VWHP 8SRQVXEVWLWXWLQJWKLVLQWRWKHHTXDWLRQRIPRWLRQ  LWLVFOHDUWKDW

Z P  MUZ  V ' Z 



ZKLFKPHDQVWKDW   PZ  MUZ  V

' Z



,IWKHXQGDPSHGQDWXUDOIUHTXHQF\VTXDUHGLVGHILQHGE\ V  P

Z 

WRJHWKHUZLWKWKHGDPSLQJUDWLRDV ]

U  PZ 

WKHIUHTXHQF\UHVSRQVHIXQFWLRQFDQEHUHZULWWHQDV ' Z

  P ª¬ Z   ] MZZ   Z  º¼

)URP WKLV FRPSOH[YDOXHG IXQFWLRQ WKH DPSOLWXGH UDWLR EHWZHHQ WKH UH VSRQVHDQGH[FLWDWLRQFDQEHGHYHORSHGWR 

' PZ 

  ª§ Z · § Z · º «¨ ] ¸ » ¸  ¨  Z  ¹ © Z  ¹ » ¬«© ¼

DQGWKHDVVRFLDWHGSKDVHUHODWLRQ







6LQJOH'HJUHHRI)UHHGRP6\VWHPV

M'

ª Z « ] Z  DUFWDQ « «§ Z « ¨  ¬« © Z 

º » » ·» ¸» ¹ ¼»

6RIDUDOOZKDWKDVEHHQGLVFXVVHG IROORZVWKHFODVVLFDOSUHVHQWDWLRQRI PHFKDQLFDOYLEUDWLRQV)RUGLIIHUHQWUHDVRQVKRZHYHUXVHLVPDGHRIYH ORFLW\LQVWHDGRIGLVSODFHPHQWLQDFRXVWLFVDQGVWUXFWXUHERUQHVRXQG2QH VXFK UHDVRQ LV WKDW WKH VFDODU SURGXFW RI IRUFH DQG YHORFLW\ \LHOGV WKH SRZHUZKLFKLVDFRQYLQLHQWPHDVXUHIRUVWDWLRQDU\ SURFHVVHV7KHUHIRUH XSRQUHZULWLQJWKHHTXDWLRQRIPRWLRQIRUWKHPDVVVSULQJGDPSHUV\VWHP LQWHUPVRIWKHYHORFLW\ § V · ¨ MZ P  U  ¸Y M Z¹ ©

)

,Q WKLV FDVH WKH HPSOR\PHQW RI WKH IUHTXHQF\ UHVSRQVH IXQFWLRQ OHDGV WR WKHUHVSRQVH Ö MZ W  Y W < Z )H

VXFKWKDWE\PHDQVRIDVXEVWLWXWLRQLQWRWKHHTXDWLRQRIPRWLRQ < Z

 V MZ P  U  MZ



 

,Q )LJ  WKH PDJQLWXGH DQG SKDVH RI WKLV YHUVLRQ RI WKH IUHTXHQF\ UH VSRQVH IXQFWLRQ ± FRQYHQWLRQDOO\ WHUPHG WKH PRELOLW\ ±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

$/LWWOH'\QDPLFV

)LJ  0RELOLW\ RI PDVVVSULQJGDPSHU V\VWHP IRU WKUHH YDOXHV RI GDPSLQJ YHU VXV QRQGLPHQVLRQDO IUHTXHQF\

,Q!WKH!SKDVH!WKH!VWLIIQHVV!FRQWUROOHG!UHJLRQ!LV!DVVRFLDWHG!ZLWK!D!SRVLWLYH VLJQ!±!WKH!YHORFLW\!OHDGV!WKH!IRUFH!±!ZKHUHDV!IRU!WKH!PDVV!FRQWUROOHG!UDQJH WKH!SKDVH!LV!QHJDWLYH!±!WKH!YHORFLW\!ODJV!WKH!IRUFH!$V!FDQ!EH!H[SHFWHG WKHUH LQ EHWZHHQ LV D UHJLRQ LQ ZKLFK WKH SKDVH DSSURDFKHV ]HUR DQG WKH YH ORFLW\!LV!LQ!SKDVH!ZLWK!WKH!IRUFH!7KLV!LV!WKH!UHVRQDQWO\!RU!UHVLVWLYHO\!FRQ WUROOHG!UHJLRQ!ZKHUH!WKH!GDPSHU!HVVHQWLDOO\!GLFWDWHV!WKH!EHKDYLRU!RI!WKH V\VWHP )RU WKH XSVLGHGRZQ V\VWHP VHH )LJ  ZKHUH WKH VXSSRUWLQJ VWUXFWXUH UHDOL]HV D FRQVWDQW KDUPRQLF PRWLRQ WKH IUHH ERG\ GLDJUDP JLYHV

)LJ5HVLOLHQWO\VXVSHQGHGPDVVZLWKDPRYLQJEDVH

/DJUDQJH¶V(TXDWLRQV

YV  YP U  YV  YP MZ PYP

N MZ

)

   

)

8SRQHTXDWLQJWKHVHOLQHDUUHODWLRQVWKHYHORFLW\UDWLR YP YV

MUZ N Z  P MZ U   N N 

 



Z MZ    ]  Z Z

UHVXOWV GHPRQVWUDWLQJ WKDW IRU IUHTXHQFLHV EHORZ WKH XQGDPSHG QDWXUDO IUHTXHQF\ WKH YHORFLW\ RI WKH PDVV LV HVVHQWLDOO\ WKDW RI WKH VXSSRUWLQJ VWUXFWXUH)RUKLJKIUHTXHQFLHVRQWKHRWKHUKDQGWKHPDVVYHORFLW\ LVVXE VWDQWLDOO\VPDOOHUWKDQWKDWRIWKHVXSSRUWLQJVWUXFWXUH2QO\LQWKHYLFLQLW\ RI WKH XQGDPSHG QDWXUDO IUHTXHQF\ Z WKH PDVV YHORFLW\ H[FHHGV WKDW RI WKHVXSSRUWLQJVWUXFWXUH$JDLQWKHPD[LPXP LVVOLJKWO\ GLVSODFHGIURP WKH XQGDPSHGQDWXUDOIUHTXHQF\VHH)LJ

)LJ  7UDQVIHU IXQFWLRQ + Z WKUHHYDOXHVRIGDPSLQJ

YP YV IRU D UHVLOLHQWO\ VXVSHQGHG PDVV IRU

/DJUDQJH¶V(TXDWLRQV 1DWXUDOO\WKHDSSURDFKRIUDLVLQJWKHHTXDWLRQVRIPRWLRQYLDIRUFHHTXL OLEULXP FDQEHXVHGDOVRIRUPRUHFRPSOLFDWHGFRQILJXUDWLRQVWKDQWKHWZR

$/LWWOH'\QDPLFV

SUHYLRXVO\ GLVFXVVHG $QRWKHU SRVVLELOLW\ ZKLFK FDQ EH UDWKHU DGYDQWD JHRXVLVWKDWWHUPHG/DJUDQJH¶VHTXDWLRQVRIPRWLRQ7KHUHE\WKHH[SOLFLW QHHG RI WKH IRUFH LV UHPRYHG DQG VLQFH WKH WHUPV LQYROYHG DUH SULPDULO\ TXDGUDWLF WKH SUREOHP RI VLJQ HUURUV PDUNHGO\ UHGXFHG 7KH /DJUDQJH¶V HTXDWLRQV DUH DQ DOWHUQDWLYH HQHUJ\ EDVHG VWDWHPHQW RI 1HZWRQ¶V ODZ RI PRWLRQDQGDGHULYDWLRQFDQVWDUWHLWKHUIURP1HZWRQ¶VODZVRUDVZLOOEH VHHQLQVXEVHTXHQWO\IURP+DPLOWRQ¶VSULQFLSOH>@7KHPHWKRGIRUGH ULYLQJWKHHTXDWLRQVRIPRWLRQXVLQJ/DJUDQJH VHTXDWLRQVFRQVLVWVRIWZR VWHSV L  'HWHUPLQDWLRQ RI WKH WRWDO NLQHWLF DQG SRWHQWLDO HQHUJLHV RI WKH V\VWHP DQG LL GLIIHUHQWLDWLRQ RI WKH HQHUJLHV ZLWK UHVSHFW WR YHORFLWLHV DQG GLVSODFH PHQWV 7RGHPRQVWUDWHWKHPHWKRGWKHV\VWHPGHSLFWHGLQ)LJZLOOEHFRQ VLGHUHG7KHWRWDONLQHWLFHQHUJ\IRUWKHV\VWHPFDQEHZULWWHQDV   ¦ PQ YQ   Q

(NLQ

 

7KHSRWHQWLDOHQHUJ\LVFRQWDLQHGRQO\LQWKHVSULQJVLQWKLVFDVHDQGLW FDQEHREWDLQHGYLDWKHDVVXPSWLRQWKDWHDFKVSULQJLVFRPSUHVVHGLQVHY HUDOVPDOOVWHSVRIOHQJWK'O$IWHUWKHTWKFRPSUHVVLRQVWHSWKHWRWDOFRP SUHVVLRQLV OT T ˜ 'O DQGDWWKHHQGRIWKHFRPSUHVVLRQLHT 4WKHWRWDO FRPSUHVVLRQ LV 4'O )RU HDFK VWHS WKH FRPSUHVVLRQ IRUFH FDQ EH IRXQG IURP +RRNH VODZDV )T 'O ˜ V ZKHUH VLVWKHVSULQJVWLIIQHVVDQGDGGLQJ DOOWKHVHVWHSVWKHSRWHQWLDOHQHUJ\LVJLYHQE\ 4

( SRW

4

¦)

˜ OT

V 'O



T

4

V ¦ 'O ˜ OT

T 

T 

4 4   

|

V ¦ T 'O



V 'O

T 

  V 'O ˜ 4 



4

¦T T 

  V O4  

7KHSRWHQWLDOHQHUJ\ VWRUHGLQDVSULQJWKHUHIRUHLVSURSRUWLRQDOWRWKHWR WDOFRPSUHVVLRQVTXDUHG:KHQWKLVUHODWLRQLVDSSOLHGWRWKHV\VWHPLQ)LJ WKHSRWHQWLDOHQHUJ\LVIRXQGWREHJLYHQE\ ( SRW

    V [  [   V[   V [  [   V [  [  

^







`

V [   [  V [   [   V [   [ 

 

,W VKRXOG EH QRWHG WKDW WKH UHVXOWLQJHQHUJ\ LVLQGHSHQGHQW RI ZKHWKHU IRU H[DPSOH [ ± [  RU [ ± [  LV XVHG DQG WKH VLJQ GRHV QRW FDXVH D

/DJUDQJH¶V(TXDWLRQV

SUREOHP LQ WKLV FDVH ,Q WKH H[SUHVVLRQ DERYH DUH LQFOXGHG WKH GXPP\ VSULQJVDQGLWV ]HUR FRPSUHVVLRQ7KLVHVWDEOLVKHVDZD\WRDFFRXQWIRU DSRVVLEOHH[WHUQDOIRUFH

)LJ0XOWLGHJUHHRIIUHHGRPV\VWHP

$VLVUHDGLO\IRXQGIURPDWH[WERRNRQPHWKRGVRIWKHRUHWLFDOSK\VLFV WKH SURFHGXUH ZLWK WKH JHQHUDOL]HG /DJUDQJH¶VHTXDWLRQFRQVLVWVLQIRUP LQJWKHH[SUHVVLRQ G w ( SRW  (NLQ w ( SRW  (NLQ  GW wYL w[L



 

IRU L     « +HUH YL DQG [L DUH WKH XQNQRZQ YHORFLWLHV DQG GLV SODFHPHQWV UHVSHFWLYHO\ EXW QRW WKH SUHVFULEHG DQG WKXV NQRZQ GLVSODFH PHQW [,WPXVWEHSRLQWHGRXWWKDWLQWKHVHFDOFXODWLRQVYLDQG[LDUHUHDO WLPHGHSHQGHQWYDULDEOHVDQGQRSKDVRUV ,QDOOFDVHVWUHDWHGKHUHLQDVIRUPDQ\RWKHUSUREOHPVLQG\QDPLFVLWLV SRVVLEOH WR FKRRVH WKH FRRUGLQDWHV LQ VXFK D ZD\ WKDW WKH NLQHWLF HQHUJ\ GHSHQGVRQO\ RQWKHYHORFLWLHVZKLOVWWKHSRWHQWLDOHQHUJ\RQO\RQWKHGLV SODFHPHQWV(TXDWLRQ  FDQDFFRUGLQJO\EHUHZULWWHQDV G w(NLQ w( SRW  GW wYL w[ L

  L    

 

8SRQ VXEVWLWXWLQJ (TV   DQG   LQWR   WKH GLIIHUHQWLDWLRQV IXUQLVKVL[OLQHDUHTXDWLRQVHVWDEOLVKLQJWKHHTXDWLRQVRIPRWLRQ

$/LWWOH'\QDPLFV ª D  V  V  V   º « V D   V »    V » «    «  V  D    » « » «  V   D   » «  V   D  »   « » «¬  V    D »¼

ª[ º «[ » « » «[ » « » «[  » «[ » « » «¬[  »¼

ª V[  º « » «  » «  » « » «  » «  » « » «¬  »¼

LQZKLFKWKHDEEUHYLDWLRQV G G     V  V  V  V  V V V  D P      GW  GW  G G G D P   V  D P   V  D P   V GW GW GW G DQG D P   V GW D

P

KDYH EHHQ XVHG $Q H[WHUQDO H[FLWDWLRQ RI WKH V\VWHP FDQ UHDGLO\ EH DF FRXQWHGIRUE\VHOHFWLQJWKHVSULQJVWLIIQHVVVVPDOOVRWKDWV[ÆDQG[ VXFKWKDWWKHSURGXFWV[HTXDOVWKHH[WHUQDOIRUFH%\PHDQVRIDQLQYHU VLRQRIWKHPDWUL[WKHXQNQRZQGLVSODFHPHQWVFDQEHIRXQG $VDQH[DPSOHWKHPRELOLW\ [ )  [  @  >K)  K @   







 

,QWKLVH[SUHVVLRQVWRVDUHWKHVWLIIQHVVHVRIWKHLQGLYLGXDOVSULQJVDQGO WROWKHLUOHQJWKDWHTXLOLEULXPV[DQGV\ DUHWKHDX[LOLDU\VSULQJVIRUWKH H[WHUQDOH[FLWDWLRQ7KHH[SOLFLWFDOFXODWLRQLVSULQFLSDOO\TXLWHVLPSOHEXW VRPHZKDW WHGLRXV )DFLOLWDWLQJ LV D PDWKHPDWLFDO FRPSXWHU DSSOLFDWLRQ ZLWKV\PEROLFFDSDELOLWLHV7KHLQGLYLGXDOVWHSVDUH x 6XEVWLWXWLRQRI  LQWR  ZLWK[ [$\ \$[ [%\ \%DQG [ [)\ \)UHVSHFWLYHO\ x 6XEVWLWXWLRQRIWKHUHVXOWLQJWRWDOSRWHQWLDOHQHUJ\DVZHOODVWKHWRWDONL QHWLFHQHUJ\LQ  LQWR   x 'LIIHUHQWLDWLRQZLWKUHVSHFWWRYF[YF\ M [FKFDQGM x 7UDQVIRUPDWLRQWRSKDVRUQRWDWLRQ x /HWWLQJWKHVWLIIQHVVHVV[DQGV\DSSURDFK]HURZLWKDVLPXOWDQHRXVLQ FUHDVHRIWKHGLVSODFHPHQWVVXFKWKDWWKHSURGXFWV [  V [ DQG K V \ \LHOG WKHIRUFHV)[DQG)\UHVSHFWLYHO\ 7KHUHVXOWLQJVHWRIHTXDWLRQVFDQEHZULWWHQRQWKHIRUP ª $ $ $ º ­[ F ½ «$ $ $ » ° ° «    » ®KF ¾ «¬ $ $ $ »¼ ¯°M ¿°

­ ) [ ½ ° ° ® ) \ ¾ ° ° ¯ ) [  ) \ ¿

ZKHUHE\ $QPDUHWKHUDWKHUOHQJWK\H[SUHVVLRQVFRQWDLQLQJWKHJHRPHWULFDO LQIRUPDWLRQ ,Q)LJDUHH[HPSOLILHGWKHSRLQWPRELOLWLHVIRUWZRH[FLWDWLRQSRVL WLRQV2EVHUYHGLVWKHVWURQJGHSHQGHQFHRQH[FLWDWLRQSRVLWLRQVLQFHURWD WLRQV DUH HDVLO\ H[FLWHG IRU WKLV FRQILJXUDWLRQ 2EVHUYHG DUH DOVR WKH WZR µVLGHUHVRQDQFHV¶ZKLFKDSSHDURQHLWKHUVLGHRIWKHPDLQUHVRQDQFH$O

$/LWWOH'\QDPLFV

WKRXJKWKH\DUHFOHDUO\YLVLEOHIRUERWKH[FLWDWLRQSRVLWLRQVWKH\EHFRPH SDUWLFXODUO\ SURQRXQFHG ZKHQ WKH H[FLWDWLRQ LV UHPRWH IURP WKH FHQWUH RI JUDYLW\7KLVLVDFRQVHTXHQFHRIWKHIDFWWKDWWKHVSULQJVDUHQHLWKHUV\P PHWULFDOO\ DUUDQJHG QRU HTXDO 7KH FRXSOLQJ RI WKH GLIIHUHQW GHJUHHV RI IUHHGRP DQGWKHUHRIUHVXOWLQJPXOWLSOHUHVRQDQFH¶V PD[LPXPVL[ DUHRI WHQRYHUORRNHGE\PHDVXUHPHQWVRQTXDVLULJLGERGLHVDQGFDQEHWKHUHD VRQIRUHUURUV 7KHSURFHGXUHZLWKWKH/DJUDQJH¶VHTXDWLRQVLVQRWRQO\DYHU\ XVHIXO WRROWRHVWDEOLVKWKHHTXDWLRQVRIPRWLRQEXWLVDOVRPRVWVXLWDEOHIRUGHULY LQJLPSRUWDQWJHQHUDOWKHRUHPVIRUOLQHDUPXOWLSOHGHJUHHRIIUHHGRPV\V WHP2ZLQJWRWKHIDFWWKDWWKRVHWKHRUHPVDUHWKRURXJKO\ WUHDWHGLQWH[W ERRNV RQ PHFKDQLFDO YLEUDWLRQ WKH PRVW LPSRUWDQW WKHRUHPV DUH KHUHLQ JLYHQZLWKRXWSURRIV

)LJ3RLQWPRELOLWLHVDWGLIIHUHQWSRVLWLRQVRIDULJLGERG\ ±±±± DERYHFHQ WHURIJUDYLW\DQG  DWWKHHGJH3DUDPHWHUVV VV VV V U4 O4 PU4

5HFLSURFLW\DQG0XWXDO(QHUJ\ 2ZLQJWRWKHTXDGUDWLFIRUPVRIWKHLQWKLVFRQWH[WLQWHUHVWLQJHQHUJ\ UHOD WLRQVWKHV\VWHPVRIHTXDWLRQVDUHFRQVLVWHQWO\V\PPHWULF2QHRIWKHFRQ VHTXHQFHVLVWKHUHFLSURFLW\ SULQFLSOH7KLVSULQFLSOHHVWDEOLVKHVDUHODWLRQ EHWZHHQWKHILHOGYDULDEOHVVXFKWKDWWKHH[FLWDWLRQDQGUHVSRQVHSRVLWLRQV FDQEHLQWHUFKDQJHG7KHSUHUHTXLVLWHLV WKDWWKHSURGXFWRIWKHYDULDEOHVWR EHLQWHUFKDQJHG\LHOGVWKHSRZHURUHQHUJ\$JHQHUDOL]DWLRQRIWKHUHFLS

5HFLSURFLW\DQG0XWXDO(QHUJ\

)LJ5HFLSURFLW\DQGPXWXDOHQHUJLHV

URFLW\ SULQFLSOH LV WKH WKHRUHP RI WKH PXWXDO HQHUJLHV >@ ,Q )LJ  DUH H[HPSOLILHG WKH FRUUHVSRQGLQJ UHODWLRQV LQ SKDVRU QRWDWLRQ )RU!V\VWHPV!FRPSULVLQJ!UHFLSURFDO!WUDQVGXFHUV!WKH!PXWXDO!SURGXFW!RI FXUUHQW!DQG!YROWDJH!FDQ!EH!FRPSULVHG!LQ!WKH!UHFLSURFLW\!UHODWLRQV!7KH SURRI!FDQ!EH!REWDLQHG!YLD!+DPLOWRQ¶V!SULQFLSOH!ZKHQ!WKLV!LV!HPSOR\HG!RQ HOHFWULFDO HQHUJ\ UHVHUYRLUV VHH (T  

$/LWWOH'\QDPLFV

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

 

)

ZLWKWKHYHORFLW\ [ WKHSRZHUEDODQFH   U[   V[[ P[[

 G  G P[   V[   U[  GW  GW





)[

LVREWDLQHG%\UHZULWLQJLWDV G §     ·  ¨ P[  V[ ¸  U[ GW ©   ¹

) [

LWLVVHHQWKDWWKLVLVHTXLYDOHQWZLWK G (NLQ  ( SRW  :G GW

:

 

(QHUJ\&RQVLGHUDWLRQV

7KLVH[SUHVVLRQUHYHDOVWKDWWKHWUDQVPLWWHGSRZHUWRWKHV\VWHP : ) ˜ Y LVHLWKHUGLVVLSDWHGE\ WKHGDPSHURUVHUYHVDVWRFKDQJHWKHWRWDOHQHUJ\ FRPELQHG RI WKH NLQHWLF DQG SRWHQWLDO SDUWV ,Q VWDWLRQDU\ VWDWH WKH SDUW SUHFHGHG E\ GGW LH DQ\ FKDQJH LQ WRWDO HQHUJ\ YDQLVKHV SHU GHILQLWLRQ DQGWKHSRZHUWUDQVPLVVLRQHTXDOVWKDWGLVVLSDWHG,Q(TV  DQG   WKHLQVWDQWDQHRXVYDOXHVDUHXVHGZKLFKPHDQVWKDWIRUWKHHPSOR\PHQWRI SKDVRUQRWDWLRQLWKDVWREHREVHUYHGWKDWSRZHURUHQHUJ\ DUHSURGXFWVRI ILHOGYDULDEOHV7KHUHIRUH )[

:

5H ^ )H MZ W ` 5H ^YH MZ W ` 

) ˜Y

VLQFHWKHRSHUDWRU5H^`DQGWKHWLPHEDVHHMZWFDQQRWEHRPLWWHG /HWWLQJ) _)_HMMDQGY _Y_HM\UHVSHFWLYHO\WKHSRZHUFDQEHUHZULW WHQDV :

^

)

Y 5H H

M Z W  M

` 5H^H ` M Z W \

ZKLFKLVHTXLYDOHQWWR Y FRV Z W  M FRV Z W  \

:

)

 ) 

Y ª¬FRV Z W  M  \  FRV M  \ º¼ 

RU :

)RUKDUPRQLFYLEUDWLRQVDFFRUGLQJO\WKHSRZHUFRQVLVWVRIDSDUWZLWKWKH GRXEOHIUHTXHQF\DQGDSDUWWKDWLVWLPHLQYDULDQW7KLVPHDQVWKDWIRUWKH XVXDOO\PRVWLQWHUHVWLQJWHPSRUDODYHUDJH :

 ) 

 ) 

Y FRV M  \

^

Y 5H H M M \

`

 5H ^ ) H MM Y H  M\ `  

ZKLFKFOHDUO\LVHTXLYDOHQWWR :

 5H ^) ˜ Y `  

 

ZKHUH GHQRWHVWKHFRPSOH[FRQMXJDWH:LWKWKLVUHODWLRQDSSOLHGWR(T    U 5H [ [ 

^

`

U  [ 

U  Y 

 5H ^ ) ˜ Y `  

$Q H[WHQVLRQ WR PXOWLSOHGHJUHHRIIUHHGRP V\VWHPV LV UDWKHU VLPSOH VWDUWLQJIURP(T  DQGSHUIRUPLQJWKHVDPHVWHSVDIWHULQWURGXFWLRQRI WKHGDPSLQJWHUPVDQGH[WHUQDOIRUFHV

$/LWWOH'\QDPLFV

0LQLPL]DWLRQRIWKH$YHUDJH(QHUJ\'LIIHUHQFH +DPLOWRQ¶V3ULQFLSOH  2QHRIWKHPRVWLPSRUWDQWODZVLQSK\VLFVLV+DPLOWRQ¶VSULQFLSOH:KHQ DSSOLHGLQFODVVLFDOG\QDPLFVLWVWDWHVWKDWWKHPRWLRQRIWKHGLIIHUHQWFRP SRQHQWVRIDV\VWHPDOZD\VDGMXVWWKHPVHOYHVVRWKDWDQH[WUHPHYDOXHRI WKHWHPSRUDOO\DYHUDJHGGLIIHUHQFHEHWZHHQNLQHWLFDQGSRWHQWLDOHQHUJ\LV REWDLQHG )RU WKH FDVHV VWXGLHG LQ WKLV FRQWH[W WKH H[WUHPXP LV DOZD\V D PLQLPXP,WLVDVWRQLVKLQJWKDWVRVLPSO\VWDWHGDSULQFLSOHFDQUHSUHVHQW VXFKDPXOWLWXGHRISK\VLFDOSURFHVVHV)XUWKHUPRUHLWLVQRWHZRUWK\WKDW E\ HPSOR\LQJ +DPLOWRQ¶V SULQFLSOH DOO SK\VLFDO SURFHVVHV FDQ EH GH VFULEHG ZLWKRXW WKH FRQFHSW RI IRUFH LH ³ZLWKRXW SUHVVXUH RU HQIRUFH PHQWFRQVWUDLQW´ DOEHLW WUDQVFULSWLRQV PLJKW EH UHTXLUHG LQ VRPH FDVHV 6RPHLPSOLFLWSUREOHPVDVVRFLDWHGZLWK WKHIRUFHFRQFHSWDUHGLVFXVVHGLQ >@ 7KHPDWKHPDWLFDOUHSUHVHQWDWLRQRI+DPLOWRQ¶VSULQFLSOHFDQEHZULWWHQ DV  W  W

W

³ (

NLQ

 ( SRW GW

([WUHPXP 0LQLPXP 

 

W

6LQFH D PLQLPXP RU DQ H[WUHPXP RI D VPRRWK ZHOOEHKDYHG IXQFWLRQ LV IHDWXUHGE\ QRRUVPDOOHUWKDQOLQHDUYDULDWLRQVRIWKHRUGLQDWHLQLWVYLFLQ LW\WKHVWDWHPHQWLQ(T  LVHTXLYDOHQWWRDYDQLVKLQJYDULDWLRQRIWKH HQHUJ\GLIIHUHQFHLH G³

W

W

(

NLQ

 ( SRW GW

 

D 

7KH FRQVWDQW IDFWRUW±WWKXVGLVDSSHDUVG GHQRWHVWKHYDULDWLRQDQGLQ WKHQH[WH[DPSOHZLOOEHVKRZQKRZLWFDQEHREWDLQHGWKURXJKGLIIHUHQWLD WLRQV$JDLQDVLQJOHGHJUHHRIIUHHGRPV\VWHPLVFRQVLGHUHGH[FLWHGE\ DSUHVFULEHGGHIRUPDWLRQ[RIDVSULQJVDVVKRZQLQ)LJ&OHDUO\ WKHH[SUHVVLRQ W §    · G ³ ¨ P[   V[   V [   [  ¸ GW W   © ¹



 

LVWREHPDQLSXODWHG)RUWKHYDULDWLRQ WKHIDFWWKDWVPDOOFKDQJHV YDULD WLRQV  RI DQ DUELWUDU\ IXQFWLRQ J [ [ «  FDQ EH REWDLQHG IURP WKH FKDQJHVRIWKHDUJXPHQWVDV G ^ J [  [  `

wJ wJ G [  G [   w[ w[

(QHUJ\&RQVLGHUDWLRQV

+HUHLQG[DQGG[HWFDUHWKHYDULDWLRQVRIWKHDUJXPHQWV$SSOLFDWLRQRQ (T  \LHOGV W

   V[G[  V [ ³ P[G[ 

W



 

 [ G[ GW



)LJ0DVVVSULQJV\VWHPH[FLWHGE\PHDQVRIDSUHVFULEHGGLVSODFHPHQWYLD DVSULQJ

%\PHDQVRILQWHJUDWLRQE\SDUWVWKHILUVWWHUPFDQEHEURXJKWRQWRWKH LQWKLVFDVHPRUHXVHIXOIRUPRI W  GW P ³ [G[



W

W

W

G[ § G[ · G¨ ¸ GW GW © GW ¹

W



W G [ ª G[ º G[ GW  P « G[ »  P ³ W  GW  ¬ GW ¼ W

,I WKH LQWHJUDWLRQ OLPLWV DUH FKRVHQ VXFK WKDW WKH YHORFLW\ G[GW LV ]HUR DW WKRVH SRLQWV WKH WHUP LQ WKH EUDFNHW YDQLVKHV DQG KHQFH (T   EH FRPHV

³

W

W

§ · G [ ¨ P   V[  V[  V[  ¸ G[ GW GW © ¹

 

7KHH[SUHVVLRQDERYHPXVWEHHTXDOWR ]HURIRUDOOSRVVLEO\WLPHGHSHQG HQWYDULDWLRQVG[7KLVLVRQO\ SRVVLEOHLIWKHH[SUHVVLRQZLWKWKHSDUHQWKH VLVLVLGHQWLFDOO\]HURDQGDFFRUGLQJO\ P

G [  V[  V[ GW 

V[ 

ZKLFK LV WKH HTXDWLRQ RI PRWLRQVRXJKWIRUWKHV\VWHPLQ)LJ:LWK VHYHUDOPDVVHVDQGVSULQJVLQYROYHGWKHSURFHGXUHLVSULQFLSDOO\ WKHVDPH EXWVLQFHWKHFRUUHVSRQGLQJRSHUDWLRQVDUHLGHQWLFDOWRWKRVHGHVFULEHGSUH YLRXVO\LQFRQMXQFWLRQZLWKWKH/DJUDQJH¶VHTXDWLRQV (T  LWLVKHUH UHIUDLQHG IURP VXFK GHYHORSPHQWV 6XEVHTXHQWO\ +DPLOWRQ¶V SULQFLSOH ZLOOEHHPSOR\HGRQRWKHULQWHUHVWLQJFDVHV

$/LWWOH'\QDPLFV

7KH5D\OHLJK4XRWLHQW 6RIDUWKHLPSRUWDQWUHODWLRQVGLVFXVVHGKDYHEHHQGHULYHGIURPVXPVDQG GLIIHUHQFHV RI HQHUJLHV ,Q WKLV VHFWLRQ LV GHVFULEHG D SURFHGXUH EDVHG RQ HTXDOLW\LQHQHUJLHV>@ $ SUHUHTXLVLWH IRU WKH DSSOLFDWLRQ RI WKH 5D\OHLJK TXRWLHQW LV WKH DV VXPSWLRQWKDWDWDOOUHVRQDQFHV EXWQRWRXWVLGHUHVRQDQFHV DQGIUHHYLEUD WLRQV ( SRW

(NLQ

 

LVYDOLGZKHUHE\WKHRYHUEDUGHQRWHVDYHUDJLQJRYHURQHRUPRUHSHULRGV RIYLEUDWLRQ ,IIRUH[DPSOHDPXOWLGHJUHHRIIUHHGRPV\VWHPYLEUDWHVDWRQHRILWV HLJHQIUHTXHQFLHV ZQ WKHQ WKH GLVSODFHPHQW DQG WKH YHORFLW\ RI WKH QWK PDVVKDYHWKHWLPHGHSHQGHQFLHV [Q W [Q $ FRV Z Q W  MQ  YQ W

Z Q[Q $ VLQ Z Q W  M Y 

UHVSHFWLYHO\8SRQDYHUDJLQJRYHURQHRUPRUHSHULRGVLVREWDLQHG [Q W 

  [Q $  

YQ W 

   Z Q [Q $  

$FFRUGLQJO\WKHHLJHQIUHTXHQF\VTXDUHG HLJHQYDOXH RIDPXOWLGHJUHH RIIUHHGRPV\VWHPZKHUHWKHQWKPDVVKDVDGLVSODFHPHQWDPSOLWXGH[Y$ LVJLYHQE\ Z Q

( SRW   ¦ PQ [Q$  

 

2ZLQJWRWKHWHPSRUDODYHUDJHQRSKDVHLQIRUPDWLRQLVUHTXLUHGDQGWKH SRWHQWLDOHQHUJ\LVDOVRDIXQFWLRQRIWKHDPSOLWXGH[Y$RQO\ 7KHDGYDQWDJHVRIWKHH[SUHVVLRQLQ(T  IRUWKHHVWLPDWLRQRIHL JHQIUHTXHQFLHV UHVRQDQFHIUHTXHQFLHV DUHWKDW x VXUSULVLQJO\ JRRG DSSUR[LPDWLRQV DUH REWDLQHG IRU WKH ILUVW IHZ HLJHQ IUHTXHQFLHV ZKHQ UHDOLVWLF DVVXPSWLRQV IRU WKH DPSOLWXGHV DUH LQWUR GXFHGDQG x H[SUHVVLRQ   KDV WKH FKDUDFWHU RI D PLQLPXP VXFK WKDW ZKHQ JXHVVHGDPSOLWXGHVDUHHPSOR\HGDQXSSHUOLPLWLVIRXQGIRUWKHHLJHQ IUHTXHQF\,QGHHGWKLVPLQLPXPFKDUDFWHURIWKH5D\OHLJKTXRWLHQWFDQ EHXVHGWRILQGWKHWUXHDPSOLWXGHLWHUDWLYHO\E\PHDQVRIYDULDWLRQXQWLO WKHHLJHQYDOXH Z Q EHFRPHVDPLQLPXP>@

5HIHUHQFHV

7RKLJKOLJKWWKLVDVLPSOHPDVVVSULQJPDVVV\VWHP FDQEHFRQVLGHUHG ZKHUHWKHWZRPDVVHVKDYHWKHDPSOLWXGHV [$ DQG [  $ UHVSHFWLYHO\7KH WHPSRUDOO\DYHUDJHGNLQHWLFDQGSRWHQWLDOHQHUJLHVDUHWKXVJLYHQE\ (NLQ

  Z P[$  P[ $  

( SRW

 V [ $  [  $   

ZKLFKVXEVWLWXWHGLQWR(T  \LHOGV Z Q

V

[ $  [  $  P[$  P[ $

V

  D   P  PD 

LQZKLFK D [  $ [ $ LVLQWURGXFHGLQWKHODVWHTXDOLW\7RILQGWKHHLJHQ YDOXH Z Q  D PXVWEHFKRVHQVXFKWKDWDPLQLPXPLVHVWDEOLVKHG$VFDQEH UHDGLO\ GHPRQVWUDWHG IURP D YDQLVKLQJ ILUVW GHULYDWLYH WKHUH DUH WZR H[ WUHPHV7KRVHDUHREWDLQHGIRU D

 œ [ $

[ $

DQG D

 P P œ P[  $

P[ $ 

7KHDVVRFLDWHGHLJHQIUHTXHQFLHVDUHWKXVIRXQGWREHJLYHQE\ Z Q



ZQ

V

P  P PP

UHVSHFWLYHO\ERWKRIZKLFKDUHLQDJUHHPHQWZLWKFODVVLFDOUHVXOWV

5HIHUHQFHV >@0RUVH30DQG)HVKEDFK+0HWKRGVRIWKHRUHWLFDO SK\VLFV9RO  &K0F*UDZ+LOO%RVWRQ0$ >@ 7LPRVKHQNR 63 DQG :RLQRZVN\.ULHJHU 6  7KHRU\ RI SODWHV DQG VKHOOV QGHG &K0F*UDZ+LOO$XFNODQG >@-RKQVRQ./&RQWDFW0HFKDQLFV&K&DPEULGJH8QLYHUVLW\3UHVV >@+HDYLVLGH2(OHFWULFDO3DSHUV9ROS9ROS >@%OHYLQV5')RUPXODVIRUQDWXUDOIUHTXHQF\DQGPRGHVKDSH.ULHJHU 3XEOLVKLQJ&RPSDQ\0DODEDU)O >@&UDQGDOO6+DQG0F&DOOH\ -U5%6KRFNDQGYLEUDWLRQKDQGERRN HGV+DUULV&UHGH &K0DWUL[PHWKRGVRIDQDO\VLV0F*UDZ+LOO1HZ @ =XUPKO 5  3UDNWLVFKH 0DWKHPDWLN †† 6SULQJHU 9HUODJ %HUOLQ >@ 7HPSOH * DQG %LFNOH\ :*  5D\OHLJK¶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

)UHH3ODWH:DYHV

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

,UURWDWLRQDO3DUW /RQJLWXGLQDOYROXPHZDYHV

'LYHUJHQFH)UHH3DUW 7UDQVYHUVHYROXPHZDYHV

6XUIDFHZDYHV

7UDQVYHUVHYROXPHZDYHV

6XUIDFHZDYHV

6XUIDFHZDYHV

7KHGLVSHUVLRQSUREOHPFRQVLVWVRIGHWHUPLQLQJ YÖ/   YÖ/   YÖ7   YÖ7  IURPWKH ILUVWIRXUHTXDWLRQVLQ  ,QWKLVSXUVXLWLWLVFRQYHQLHQWWRUHSODFHWKH H[SRQHQWLDOV ZLWK WKH DVVRFLDWHG WULJRQRPHWULF IXQFWLRQV 7KHQ WKH ILUVW HTXDWLRQLVDGGHGWRDQGVXEWUDFWHGIURPWKHVHFRQGUHVSHFWLYHO\WKHWKLUG VLPLODUO\DGGHGWRDQGVXEWUDFWHGIURP WKHIRXUWK,QWKLVZD\LVREWDLQHG Z N[ SÖ   SÖ   * Z N \7 N \7 N \/ YÖ/   YÖ/  VLQ /  $ YÖ7   YÖ7  VLQ 7 SÖ [\   SÖ [\    M* Z N[  $ YÖ/   YÖ/  VLQ /  N [ YÖ7   YÖ7  VLQ 7 SÖ   SÖ    M* Z N \7 N \7 N \/ YÖ/   YÖ/  FRV /  $ YÖ7   YÖ7  FRV 7 SÖ [\   SÖ [\    M*  $ YÖ/   YÖ/  FRV /  N [ YÖ7   YÖ7  FRV 7

 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

+HUHE\WKHHTXDWLRQVDUHEURXJKWLQWRWZRLQGHSHQGHQWSDLUVZKLFKUHDG LO\ FDQ EH VROYHG ,Q WKLV FRQWH[W WKH VWXG\ LV FRQILQHG WR WKH IUHH SODWH ZDYHV ZKLFK PHDQV WKDW WKRVH ZDYHQXPEHUV DUH VRXJKW OHDGLQJ WR QRQ WULYLDO VROXWLRQV 0DWKHPDWLFDOO\ WKLV LV DJDLQ IRXQG IURP WKH YDQLVKLQJ GHWHUPLQDQWVWRWKHWZRSDLUVRIHTXDWLRQ  VLQ / VLQ 7 ª  N7  N [ FRW /  N [ N \/ N \7 FRW 7 º  «¬ »¼  'HW ,, FRV / FRV 7 ª  N7  N [ WDQ /  N [ N \/ N \7 WDQ 7 º  «¬ »¼

'HW ,

DE 

,QWKHH[SUHVVLRQVDERYH $ N7  N [ LVEDFNVXEVWLWXWHGEXWWKHDEEUHYLD WLRQV/ N\/ KDQG7 N\7 KDUHUHWDLQHG (TXDWLQJ DE WR]HURIXUQLVKHVWKHZDYHQXPEHUVRIWKHIUHHSODWH ZDYHVDQG WDQ 7 WDQ /



WDQ / WDQ 7



N [ N \/ N \7

N

 7

 N [





F 





G 

N [ N \/ N \7

N

 7

 N [

7KH FRVLQH DQGVLQHIDFWRUVLQ(TV DE UHVXOWV LQ QR DGGLWLRQDO URRWV RI WKH GHWHUPLQDQW (TXDWLRQV F G  WKHUHIRUH HVWDEOLVK DOO WKH URRWVDQGLWLVVXIILFLHQWWKDWRQHRIWKHWZRHTXDWLRQVYDQLVKHVIRUDIUHH SODWHZDYHWRH[LVWDQGSURSDJDWH,IIRUH[DPSOH(T F YDQLVKHV WKHQWKHWZRILUVWHTXDWLRQVLQ  KDYHQRQWULYLDOVROXWLRQVIRUDYDQ LVKLQJULJKWKDQGVLGH$WWKHVDPHWLPHWULYLDOVROXWLRQVFDQEHFKRVHQIRU WKHODWWHUWZRVXFKWKDW YÖ/  YÖ/  DQG YÖ7  YÖ7  UHVXOW7KLVPHDQVWKDWIRU WKRVHIUHHSODWHZDYHVJRYHUQHGE\(T F WKHPRWLRQLVJLYHQLQWKH IRUP Y [ \

 ª¬ YÖ/ FRV N \/ \  YÖ7 FRV N \7 \ º¼ H MN[ [ 

Y [ \

ª N \/ º N YÖ/ VLQ N \/ \  [ YÖ7 VLQ N \7 \ » H  MN [ [  M« N \7 ¬« N [ ¼»

 

7KHVHZDYHVUHSUHVHQWVWKHV\PPHWULFPRGHV ,I LQ FRQWUDVW G  YDQLVKHV WKH VDPH UHDVRQLQJ DSSOLHV EXW ZLWK FRVLQHDQGVLQHLQWHUFKDQJHGLQ  ,QVXFKDFDVHWKHDQWLV\PPHWULF PRGHVDUHSURPRWHG 7KH GLVSHUVLRQ HTXDWLRQ \LHOGLQJ WKH ZDYHQXPEHUV RI WKH IUHH SODWH ZDYHV FRXOG KDYH EHHQ GHYHORSHG DOVR HPSOR\LQJ WKH ZDYH WUDLQ FORVXUH

)UHH3ODWH:DYHV

SULQFLSOH)RUWKLVIRUZDUGVSURSDJDWLQJDQGUHIOHFWHGZDYHVDUHDVVXPHG DVLQ6HFWDQGWKHZDYHQXPEHUDQGDQJOHVUHVSHFWLYHO\ VRXJKWIRU ZKLFK WKH SKDVH FKDQJH GXH WR UHIOHFWLRQ DQG SURSDJDWLRQLVPXOWLSOHVRI S7KLVDSSURDFKLVKHUHQRWFKRVHQDVLWLVLQWHUHVWDOVRWRHQFRPSDVVWKH FDVHRIIRUFHGYLEUDWLRQ 7RIXUWKHUHYDOXDWH(TV FG XVXDOO\QXPHULFDOPHWKRGVDUHHP SOR\HG$QH[DPSOHWDNHQIURP>@LVSUHVHQWHGLQ)LJ

)LJ'LVSHUVLRQGLDJUDPIRUIUHHSODWHZDYHVFRPSULVLQJGLVSODFHPHQWVSHU SHQGLFXODUWRWKHVXUIDFHDIWHU>@ ±±± V\PPHWULFDQG  DQWLV\PPHWULF PRGHV

)RUVSHFLILFYDOXHVRIWKHWUDFHZDYHQXPEHULQWKH[GLUHFWLRQN[ ZF[LW LVHDV\WRILQGWKHFRUUHVSRQGLQJYDOXHVRIN7 ZF7)RUN[ IRUH[DP SOHDUHN\7 N7DQGN\/ N/7KHUHE\YDQLVKHVWKHULJKWKDQGVLGHRI(T F  DQG KHQFH WKH JRYHUQLQJ HTXDWLRQV IRU WKH IUHH ZDYHQXPEHUV UHDGWDQ7 DQGWDQ/ fIRUWKHV\PPHWULFPRGHVZKHUHDVWDQ/ DQG WDQ7  f IRU WKH DQWLV\PPHWULF 7KH IUHTXHQFLHV FRUUHVSRQGLQJ WR WKLV WUDFHZDYHQXPEHULHIRUDWUDFHVSHHGRIF[ fDUH

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV S F7 Q K S F/ S Q   Ÿ Z Q     K S F/ SQ Ÿ Z Q K S F7 S Q   Ÿ Z Q    K

N7 K  N/ K  N/ K  N7 K 

SQ Ÿ Z

D 

$ FORVHG IRUP DSSUR[LPDWLRQ LV DOVR SRVVLEOH IRU WKLQ SODWHV DQG ORZ IUHTXHQFLHV:LWKWKHWDQJHQWLDOIXQFWLRQVVHULHVH[SDQGHGWRILUVWRUGHU N \7 K



N \/ K

N [ N \7 N \/

N

 7

 N [



DQG

N

 7

 N [



N [ N /  N [

UHVSHFWLYHO\,WIROORZVWKDW N [

N7  N7  N /

E 

OHDGLQJWR F[ F7

N7 N[

   K

    P  N7 K  

,W LV HDV\ WR FRQYLQFH RQHVHOI WKDW WKLV PHDQV WKDW F/ ! F[ ! F7 DQG WKDW F[ F/,VHH(T  )RUWKLQSODWHVWKHV\PPHWULFPRGHLVKHQFHLGHQ WLFDO WR WKH WHQVLRQDO ZDYH ,W FRQVLVWV RI D URWDWLRQDO IUHH SDLU RI VXUIDFH ZDYHEHFDXVHRIN[!N/RUF[F/DQGRIWZRFURVVLQJWUDQVYHUVDOZDYHV VLQFHN[N7RUF[!F7 ,IDOVRWKHFXELFWHUPVDUHWDNHQLQWRDFFRXQWLQWKHVHULHVH[SDQVLRQRI WKHWDQJHQWLDOIXQFWLRQVWKHQ N [

RU

ª  § K ·    Q  º «  ¨ N7 ¸ »  N7  N / «¬  ©  ¹    Q  »¼ N7

)UHH3ODWH:DYHV

F[

ª  § N K · § P · º F/, «  ¨ / ¸ ¨ ¸ » ¬«  ©  ¹ ©   P ¹ ¼»

F 

UHVSHFWLYHO\:LWKDW\SLFDO3RLVVRQ¶VUDWLRRIP LWPHDQVWKDWDSKDVH VSHHGGHYLDWLRQRIDURXQGRIWKDWRIF/,ILUVWWKHQDULVHVZKHQO/,K 7RILQGDQDSSUR[LPDWLYHVROXWLRQIRUWKHDQWLV\PPHWULFPRGH G  DWORZIUHTXHQFLHVLWLVQHFHVVDU\WRH[SDQGWKHWDQJHQWLDOIXQFWLRQWRWKLUG RUGHUDOUHDG\IRUWKHVLPSOHVWDSSUR[LPDWLRQLH  § N \/ K ·  ¨ ¸ ©  ¹ WDQ / N \/ K   | WDQ 7 N \7 K    § N \7 K ·  ¨ ¸ ©  ¹



N [ N \/ N \7

N

 7

 N [

D 



,QWKLVFDVHLWIROORZVWKDW N7 

K   ª N / N7  N [ N7  N / N7  N [ N /  N7 º ¼  ¬



)RUWKHDSSUR[LPDWLRQRILQWHUHVWZKHUHN[!!N7LWVXIILFLHQWWRUHWDLQWKH KLJKHVWRUGHURIN[LQWKHEUDFNHWZKLFK\LHOGV N [



N7  K  N7  N /



N7   N7 K    Q 

E 

%\ XVLQJ WKH UHODWLRQV         DQG F  WKLV PHDQVWKDWWKHWUDFHVSHHGDSSURDFKHV F[

 Z  K*  U   P

 Z  K (  U   P 

%cZ  Pcc

F /,

Z  K  N7 K   

F 

+HUHLQ %c LVWKHEHQGLQJVWLIIQHVVRIWKHSODWHDQG Pcc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

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

,QWKLVKLJKIUHTXHQF\UDQJHWKHGLVSHUVLRQGLDJUDPPRUHDQGPRUHUH VHPEOHV )LJ  DQG RQH LV DOPRVW LQFOLQHG WR DVVXPH WKDW WKH W\SH 7 ZDYHVDSSURDFKF[ F7ZKLOVWWKHW\SH/ZDYHVF[ F/7KHODWWHULVKRZ HYHU QRW WKH FDVH ,QVWHDG LV UHYHDOHG E\ QXPHULFDO FDOFXODWLRQV WKDW DOO FXUYHV IRU WKH KLJK RUGHUV VWHSZLVH DSSURDFK F[  F/ ZKHUHE\RQHFXUYH VXFFHHGVWKHRWKHU7KHVHVWHSVQRWRQO\DSSURDFKWKHDV\PSWRWHF[ F/DW N7K Æ fEXWIDUHDUOLHU7KHDSSURDFKIROORZVDSSUR[LPDWHO\WKHUHODWLRQ F[

F/  VLQ -/

 

ZKLFKLVDQDORJRXVWRWKDWLQ(T F DQGZKLFKLVDQHFHVVDU\FRQGL WLRQIRUWKHIRUPDWLRQRIREOLTXHO\SURSDJDWLQJZDYHVDVGLVFXVVHGLQWKDW FRQWH[W 7KH ILUVW GLVSHUVLRQ FXUYHV RI WKLV NLQG LH 7 / DQG 7 REWDLQHG E\ *|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

F/ VLQ -/ 

 

$W YHU\ KLJKIUHTXHQFLHVDFFRUGLQJO\REOLTXHO\ SURSDJDWLQJ ORQJLWXGL QDOUD\VPD\EHFRQVWUXFWHGIURPSODWHZDYHV7KHVDPHWKLQJLVWUXHIRU WKHFXUYHVZKLFKDSSURDFKIURPEHORZWKHJHQXLQHDV\PSWRWHUHSUHVHQWHG E\ WKH WUDQVYHUVH ZDYH VSHHG F7 HYHQ LI LW LV WHPSRUDULO\ H[FHHGHG 7KH

)UHH3ODWH:DYHV

IDFW WKDW WKH TXDVLORQJLWXGLQDO DQG EHQGLQJ ZDYHV HQGV DW WKH VRPHZKDW ORZHU5D\OHLJKZDYHVSHHGZKLFKH[KLELWVQRGLVSHUVLRQLVVKDGRZHGLQ WKHILJXUHE\WKHFRQYHUJHQFHRIWKHPDQ\FXUYHEUDQFKHV ,QSUDFWLFHPRVWVWUXFWXUHERUQHVRXQGSUREOHPVRQO\FRQFHUQVWKHOLP LWLQJFDVHVRITXDVLORQJLWXGLQDODQGEHQGLQJZDYHVZKHQKO7LVVPDOO

)LJ'LVSHUVLRQGLDJUDPIRUWKHJURXSYHORFLWLHVFRUUHVSRQGLQJWR)LJ >@

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

w[ w]   DK c  D w[ w] w[ w]  D  D   K c  D * w[ w] V] w[ w] D  DK c  D   * w[ w] W [\ wK W \] wK W []   [c ] c * * * w[ w]

D  

D  w[ w]  w] w[

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

DQG Z  U[ Z  UK Z  U]

wV [ wW  W [\c  []  w[ w] wW [\ wW \]   V \c  w[ w] wW [] wV  W [\c  ]  w[ w]

E 

+HUHLQLVLQWURGXFHG P   P

D

F 

DVDQDEEUHYLDWLRQ:LWKWKHSKDVRUQRWDWLRQWKHWLPHGLIIHUHQWLDWLRQLVUH SODFHG E\ D PXOWLSOLFDWLRQ E\ MZ'LIIHUHQWLDWLRQZLWKUHVSHFWWR\PRUH RYHULVDEEUHYLDWHGE\DµSULPH¶RZLQJWRLWVVSHFLDOUROHDQGIUHTXHQF\LQ WKHVXEVHTXHQWGHYHORSPHQW 8SRQHPSOR\LQJWKHDERYHUHODWLRQVWRWKHFDVHRIDIODWSODWHRIWKLFN QHVVKDVGHSLFWHGLQ)LJWKHVWUHVVHVDWWKHVXUIDFHV\ r KFDQEH H[SDQGHGLQ7D\ORUVHULHVVXFKWKDW K K K V \ K   V \   V \c   V \cc   V \ccc    S      K K K V \ K   V \   V \c   V cc   V \ccc    S    

D 

+HUHLQ S DQG SGHQRWHWKHDSSOLHGQRUPDOVWUHVV%\PHDQVRIDGGLWLRQ DQGVXEWUDFWLRQRIWKHWZROLQHVLQ D RQHREWDLQV V \ 



S  S K  cc  V \c  V \     



S  S K  ccc V \     K

E 

,QWKHVDPHPDQQHUWKHVKHDUVWUHVVHVDFWLQJDWWKHVXUIDFHV\ r K\LHOG W \[  W \] 

S \[   S \[  K  cc S \[   S \[  K W \[c   W \[ccc   W [\     K   S S S S \]    K K \]  \]  \]  W \]c   W \]ccc    W \]cc     K 

F 

:LWKWKHDEEUHYLDWLRQVIRUWKHYDULRXVVWUHVVHVLQ)LJ S  S S \[   S \[ 

S'  S 

S  S

S% 

S \]   S \] 

S \[   S \[  S 

S 

S \]   S \] 

S 

 

)UHH3ODWH:DYHV

LQWURGXFHGLQ D WR F DQGWKH(T E VXEVWLWXWHGLVRE WDLQHG Z  U[ Z  UK Z  U]

wV [ wW [] K  ccc S   W [\   w[ w]  K K  wW [\cc K  ccc K  wW \]  wS S%  wS    V\      w[   w]  w[ K  w] wW [] wV ] K  ccc S   W \]   w[ w]  K

 

,QRUGHUWRHVWDEOLVKDVHWRIHTXDWLRQVRIPRWLRQIURP  LWLVVXLW DEOHWRUHZULWHWKHQRUPDODQGVKHDUVWUHVVHVVXFKWKDWDOOGLIIHUHQWLDOVZLWK UHVSHFWWR\DUHHOLPLQDWHGDWWKHILQDOVWDJH+HUHE\LWVKRXOGEHREVHUYHG WKDWLQWKHSODQH\ WKHUHFDQRQO\RFFXUGLVSODFHPHQWVDQGVWUHVVHVDV ZHOODVWKHLU[DQG]GHULYDWLYHV)RUWKHVDNHRIFRQYHQLHQFHWKHDUJXPHQW  DSSHDULQJLQ(TV EF LVRPLWWHGLQWKHIROORZLQJ :LWKWKHUHODWLRQ E LQWURGXFHGLQWKHVHFRQGHTXDWLRQRI D  LWIROORZVWKDW 

S K V \cc  ' * *

D

w[ w]   D   K c  D w[ w]

ZKLFKPHDQVWKDWDVKDSSURDFKHV]HUR Kc

D § w[ w]  ¨ D   © w[ w]

S' · ¸ ¹ * D  

 P § w[ w]  ¨   P © w[ w]

·    P S'  ¸ ¹ *   P

 

7KHUHE\ K c  FDQ EH HOLPLQDWHG IURP WKH UHVW RI WKH HTXDWLRQV LQ D  DQGDIWHUVRPHPDQLSXODWLRQLWLVIRXQGWKDW V[ *

 § w[ w] P ¨   P © w[ w]

V] *

P · ¸   P ¹  § w[ w] · P  ¨P ¸   P © w[ w] ¹   P

W [] *

§ w[ w] ·  ¨ ¸ © w] w[ ¹

S'  * S'  *

 

8SRQFRPELQLQJ(TV  DQG  WKHVLPSOHVWIRUPRIHTXDWLRQ RIPRWLRQIRUDIODWSODWHHPHUJHVZKHQDOOWHUPVRIRUGHUKRUKLJKHUDUH RPLWWHG7KLVPHDQVWKDW

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV Z  U [ *

 w [ w [   P w ] P  wS' S        P w[  w]    P w[w]   P * w[ *K

Z  U ] *

  P w [  w ] w ] P  wS' S          P w[w]   P w] w[   P * w] *K

D 

$UHDUUDQJHPHQWRIWKHVHH[SUHVVLRQV\LHOGV §  w ·   P w ] w    N7 ¸ [  ¨    P w[w] w] ©   P w[ ¹ ·   P w [ §  w  w ¨   N7 ¸ ]     P w[w] ©   P w] w[ ¹

wS' S º ª P  » « * «¬    P w[ K »¼ wS' S º ª P  «  » * «¬    P w] K »¼



E 

+HUHLQN7LVWKHVKHDURUWUDQVYHUVDOZDYHQXPEHUFI6HFW7KHIDFW WKDWVKHDUDQGH[WHQVLRQDOZDYHVFRPELQHFDQEHVHHQE\IRUH[DPSOHOHW WLQJ ww]   ZKLFK PHDQV WKDW DOO TXDQWLWLHV DUH FRQVWDQW LQ WKH ] GLUHFWLRQ7KLVOHDGVWR §  w ·  N7 ¸ [ ¨   P [  w © ¹ § w  · ¨   N7 ¸ ] © w[ ¹



wS' S º ª P  » « * ¬«    P w[ K ¼»

 

S   *K

7KHVHFRQGHTXDWLRQLVVHHQWREHWKHVKHDUZDYHHTXDWLRQZLWKWKHH[FLWD WLRQS*K2ZLQJWR N7

 P 

ZU

 P *

ZU

 P *   P

ZU   P  (

Z F/,

N /, 

D 

VHH(TV  DQG  WKHILUVWHTXDWLRQLQ  WXUQVLQWR § w  ¨   N /, [ w ©

· ¸[ ¹



P   P wS' (

w[



  P  S  ( *K

E 

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

)UHH3ODWH:DYHV [

[Ö H  MN[ [ H  MN ] ]  ]

]Ö H  MN[ [ H  MN ] ] 

 

LQ(T E LWLVIRXQGWKDW §   ·  P N [  N ]  N7 ¸ [Ö  N [ N ]]Ö ¨  P © P ¹ §   ·  P N [ N ][Ö  ¨ N ]  N [  N7 ¸ ]Ö  P ©  P ¹



  

7KHVHWZRHTXDWLRQVKDYHQRQWULYLDOVROXWLRQVRQO\ZKHQWKHGHWHUPLQDQW RIWKHFRHIILFLHQWVYDQLVKHV5HZULWLQJ P DV P  P WKH FKDUDFWHULVWLFHTXDWLRQLQN[DQGN]EHFRPHV §  ·  P  ·§   P  · §  P   N [ ¸ ¨ N [  N ]  N7  N] ¸  ¨ N[ N] ¸ ¨ N [  N ]  N7      P  P  P ¹ © ¹© ¹ ©



 

7KLV H[SUHVVLRQ LV RI WKH IRUP D  E  D  F  ± EF  D D  E  F    +HQFH WKH SDLU RI HTXDWLRQV LQ   KDV QRQWULYLDO VROXWLRQV LI HLWKHU D RU DEF  LH N [  N ]

N7 

N [  N ] 

 P  N[  N]  P

 N[  N]  P

N7 Ÿ N [  N ]

N /, 

D 

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rK  



r P K § w[ w] · K    P  S'  ¨ ¸    P © w[ w] ¹ *   P

 

%HQGLQJ:DYHVLQ)ODW,VRWURSLF3ODWHV 7RDQDO\VHWKHEHQGLQJPRWLRQRIDSODWHLWLVQHFHVVDU\WRGHWHUPLQHWKH GLVSODFHPHQWK RILWVQHXWUDOOD\HU$VLQGLFDWHGE\WKHPLGGOHH[SUHVVLRQ

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

LQ(T  WKHGHULYDWLYHVRIWKHQRUPDODQGWDQJHQWLDOVWUHVVHVDUHUH TXLUHG Z  U K



wW \]cc º S% K  ª wW [\cc   V ccc  « »  ¬ w[  w] ¼ K

 

)RUUHDVRQVRIEUHYLW\ LWLVKHUHDVVXPHGWKDWWKHQRUPDOVWUHVVHVDWWKHXS SHUDQGORZHUVXUIDFHVDUHHTXDOLQPDJQLWXGHDQGWKDWDOOVKHDULQJVWUHVVHV SSSDQGSYDQLVKHV )RUWXQDWHO\ QR SULQFLSDO GLIILFXOWLHV DULVH LQ WKH GHWHUPLQDWLRQ RI W \[cc HWF VLQFH WKHVH TXDQWLWLHV DUH PXOWLSOLHGE\ WKHIDFWRUKDQGDFFRUGLQJO\ WKH VWUHVVHV QHHG EH GHWHUPLQHG RQO\ XS WR WKH RUGHU K ,Q WKH IROORZLQJ GHYHORSPHQWRQHPD\IXUWKHUPRUHVHWWKHSODWHWKLFNQHVVKHTXDOWR]HUR WRUHWDLQWKHSUHFLVLRQGHVLUHGLH(T  WR  FDQEHXVHGWRGH WHUPLQH W \[cc ,QSULQFLSOHLWLVWKHQSRVVLEOHWRXVHWKHUHVXOWVUHFXUUHQWO\WR REWDLQDSSUR[LPDWLRQVXSWRRUGHUK7KHGUDZEDFNKRZHYHULVWKDWWKH H[SUHVVLRQVEHFRPHUDWKHUOHQJWK\DQGLQWUDFWDEOH 7KHILUVWGHULYDWLYHZLWKUHVSHFWWRWKH\GLUHFWLRQRIWKHVWUHVVHVVRXJKW LPPHGLDWHO\IROORZVIURP E  W [\c V \c W \]c

wV [ wW []   Z  U[  w[ w] wW [\ wW \]    Z  UK  w[ w] wW wV  []  ]  Z  U]  w[ w] 

 

,QRUGHUWRFDUU\RXWWKHQH[WGLIIHUHQWLDWLRQILUVWWKH [ cK c ] c V [cV ]c DQG W []c PXVWEHGHWHUPLQHG7KLVLVQRWGLIILFXOWVLQFHW\[ W\] WRWKHUH TXLUHGSUHFLVLRQZKLFKPHDQVWKDW [c



wK  ]c w[



wK w]

D 

IROORZV IURP WKH ODVW HTXDWLRQ LQ D  0RUHRYHU ZLWK S  S LH S' IROORZVIURP(T  WKDW Kc

 P § w[ w]  ¨   P © w[ w]

· ¸ ¹

7KHFRQGLWLRQS' LQ  IXUWKHUPRUHLPSOLHVWKDW

E 

)UHH3ODWH:DYHV V [c *

 § w K w K · V ] c  P ¨ ¸   P © w[  w]  ¹ *

W [\c

w K   w[w]

*

 § w K w K ·  ¸ ¨P   P © w[  w]  ¹

 

XVLQJ(T  8SRQLQWURGXFLQJ WKLVLQ  WRJHWKHUZLWK   RQHREWDLQV W [\cc

wK * w’ K  Z UK    P w[ w[

W ]\cc

* w’ K wK  ZU K    P w] w]

 

+HUHLQ ’

w w  w[  w] 

LVWKHWZRGLPHQVLRQDO/DSODFHRSHUDWRU :LWKDIXUWKHUVXEVWLWXWLRQRI  LQWRWKHPLGHTXDWLRQRI   WDNLQJLQWRDFFRXQWWKDW K cc

 P § w[ c w] c ·  ¨ ¸   P © w[ w] ¹

P ’ K   P

 

LWIROORZVIURP DE WKDWWKHWKLUGGHULYDWLYHRIWKHQRUPDOVWUHVVLQ WKH\GLUHFWLRQLV V \ccc

*  P ’ K Z  U’ K  P  P

 

)LQDOO\LVREWDLQHGWKHHTXDWLRQIRUEHQGLQJPRWLRQRIDSODWHE\ VXEVWL WXWLQJ  DQG  LQ   K *   K   P   K ’ K Z U’ K  Z  U K    P     P 

S%  

D 

RU ( K  K   P   K Z U’ K  Z  U KK ’     P   P  

S% 

E 

,QWKLVHTXDWLRQ %c (K> P @ LVWKHEHQGLQJVWLIIQHVVDQG Pcc  UK WKH PDVV SHU XQLW DUHD S%  S  S GHVFULEHV WKH H[WHUQDO IRUFHV H[HUWHG RQWRWKHSODWH7KLVPHDQVWKDW(T E FDQEHUHZULWWHQLQWKHIRUP

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV %c  ’ K  Z K Pcc

S  S S  FRUUHFWLRQ | %  Pcc Pcc

D 

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

[c



wK   E] w[

]c



wK w]

E 

IRU WKH VPDOO DPSOLWXGHV RI LQWHUHVW ,Q YLHZ RI   WKH PRPHQWV DUH DQDORJRXVO\ JLYHQE\WKHSURGXFWRIORFDOVWUHVVDQGGLVWDQFHIURPWKHQHX WUDOOD\HU K

0 []

³

K

V [ \G\

* § w K w K ·  ¨   P  ¸ ³ \ G\   P © w[ w] ¹  K  

V ] \G\

K * § w K w K ·  ¨ P    ¸ ³ \ G\   P © w[ w] ¹  K  

K   K

0 ][

³



K   K

0 [[

 0 ]]

³

§ w K w K ·  %c ¨   P  ¸  w] ¹ © w[  § w K w K ·  %c ¨ P    ¸  w] ¹ © w[

K

W [] \G\

K  

*

w K \  G\ w[w]  K³ 

 %c   P

F 

w K  w[w]

,QWKHDERYHH[SUHVVLRQV V [c  V ]c DQG W []c DUHLQWURGXFHGIURP  FRQ VLGHULQJ WKDW V [ V [c \ V ] V ] c \ W [] W []c \  DSSO\ LQ WKH DSSUR[LPDWLRQ XVHGKHUH)XUWKHUPRUHLVXVHGWKHIOH[XUDOVWLIIQHVVRIDSODWH %c

( K   P  

* K    P 

G 

)URPWKHPRPHQWVWKHFURVVVHFWLRQDOIRUFHVDUHREWDLQHGDV 4[ 4]

FI)LJ

w0 [] w0 ]]  w[ w] w0 ][ w0 ]]   w] w[ 

w’K  w[ w’K %c w] %c

H 

)UHH3ODWH:DYHV

,QVXPPDU\LWFDQEHVWDWHGWKDWWKHPRWLRQRIDWKLQSODWHLVZHOOGH VFULEHG E\ E  DQG DH  ZKHUHE\ E  \LHOGV WKH LQSODQH PRWLRQVZKLOVW D WKRVHRXWRISODQH7KHDFFXUDF\LVVDWLVIDFWRU\IRU PRVWHQJLQHHULQJDSSOLFDWLRQSURYLGHGWKHSODWHWKLFNQHVVLVVPDOOHUWKDQD VL[WKRIWKHZDYHOHQJWK

)LJ9HORFLWLHVPRPHQWVDQGIRUFHVIRUDSODWHLQEHQGLQJ

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

V [c ˜ \

V]

V ]c ˜ \

W []

W []c ˜ \

§ ([ w K w K ·   ( ¨ ¸ \ P   w]  ¹ ©   P w[ § (] w K · w K (   ¨ P  ¸ \ w[   P  w]  ¹ © *\

w K  w[w]

 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

([DQG(]DUHWKH@ %[ c

w K w K w K c c c  %  % %     Z  PccK * ] P w[  w[  w]  w] 





S [ ]  W 

 

+HUHLQ %[c LVWKHEHQGLQJVWLIIQHVVLQRQHGLUHFWLRQ %]c WKDWLQWKHSHUSHQ GLFXODU GLUHFWLRQ %Pc GHQRWHV WR WKH VWLIIQHVV UHODWLQJ WR (P ZKHUHDV %*c WDNHV WKH VKHDU VWLIIQHVV LQWR DFFRXQW )RU D SODWH RI WKLFNQHVV K DQG RI RUWKRWURSLFPDWHULDOZLWKWKHPRGXOL([(](PDQGVKHDUPRGXOXV*LWIRO ORZVIURP  WKDW %[ c

([ K  %] c   P  

(] K  %P c   P  

(P

K  %*c 

*

K  

D 

$VDQH[DPSOH FDQEHPHQWLRQHGWKDWIRUDWKUHHOD\HUSO\ZRRGODPLQDWH WKHIROORZLQJDSSUR[LPDWHYDOXHVDSSO\ ([ |  (] |  (P |  * |  *3D 

)RU FRUUXJDWHG RU VWLIIHQHG SODWHV RI KRPRJHQHRXV PDWHULDO ZLWK @ FDQEHXVHGDVGLVFXVVHGLQ WKHQH[WVHFWLRQ7KHUHVXOWJLYHQKHUHZLWKRXWSURRIUHDGV %’ K  7 c’ K  V ccK  Pcc

w K wW 

S [ ]  W

 

IRUWKHPRWLRQRXWRIWKHSODQHRIWKHSODWH7KHXQLWVRIWKHTXDQWLWLHVXVHG DUH %c  >1P@ 7 c  >1P@ V cc  >1P@ Pcc  >NJP@ K >P@ DQG S >1P@ S []W LVWKHH[WHUQDOO\DSSOLHGSUHVVXUH

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

,W LV FOHDU WKDW (T   FRPELQHV WKH SODWH (T LQ D  DQG D PHPEUDQHHTXDWLRQ,QFOXGHGLVDOVRDQDGGLWLRQDOUHDFWLRQIRUFH V cc K%\ LQWURGXFLQJWKHFRQYHQWLRQDOH[SRQHQWLDOVROXWLRQWKHGLVSHUVLRQHTXDWLRQ LVIRXQGIRUS []W  WREH 

%c N [  N ]  7 c N [  N ]  Z  Pcc  V cc

 

D 

UHYHDOLQJWKHSRVVLEOHIUHHZDYHQXPEHUV$FORVHUVFUXWLQ\ RIWKLVH[SUHV VLRQOHDGVWRWKHIROORZLQJILQGLQJV x 3URSDJDWLQJZDYHVDUHRQO\SRVVLEOHZKHQ Z !

V cc  7    %c  Pcc

ZKLFKIRUYDQLVKLQJSUHVWUHVVPHDQVWKDWWKHIUHTXHQF\PXVWEHKLJKHU WKDQ V cc  Pcc )RUORZHUIUHTXHQFLHVRQO\HYDQHVFHQWZDYHVH[LVW x $FRQVHTXHQFHRIWKHOLPLWLQJIUHTXHQF\ MXVWGHILQHGLVWKDWLQFRQWUDVW WRWKHSODWHZLWKRXWD:LQNOHUEHGWKHZDYHVSHHGGRHVQRWGHFUHDVHEH ORZ D FHUWDLQ OLPLWLQJ YDOXH DOVR IRU YHU\ ORZ IUHTXHQFLHV )RU SODWHV ZLWKRXW SUHVWUHVV 7 c   WKH OLPLWLQJ SKDVH VSHHG LV F%PLQ   Vcc%c  Pcc  ZKLFK LV UHDFKHG DW ZPLQ   Vcc  Pcc  )RU UDLOZD\ UDLOVWKHPLQLPXPSKDVHVSHHGLVW\SLFDOO\VRPHPVDQGWKHFXWRQ RFFXUVLQWKHUDQJHWR+] x :KHQWKHSODWHLVFRPSUHVVHG 7 c  LWDSSHDUVVRPHZKDWPRUHIOH[L EOH 7KHRUHWLFDOO\ D YDQLVKLQJ ZDYHQXPEHU LV SRVVLEOH EXW KDV WR EH UXOHGRXWRQSK\VLFDOJURXQGVVLQFHWKHSODWHZLOOEXFNOH x 8SRQ FRPELQLQJ D PHPEUDQH HTXDWLRQ FRPSULVLQJ WKH SUHVWUHVVHV 7[c DQG 7]c ZLWKWKHHTXDWLRQRIPRWLRQIRURUWKRWURSLFSODWHVLQ  WKH DSSOLFDELOLW\RI(T  FDQEHIXUWKHUH[WHQGHG

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH (TXDWLRQVRI0RWLRQ )XQGDPHQWDOV $VZDVDOUHDG\VWDWHGLQ6HFW+DPLOWRQ¶VSULQFLSOHUHDOL]HVRQHRI WKHPRVWLPSRUWDQWUHODWLRQVRISK\VLFV,WLVWKHUHIRUHWREHDQWLFLSDWHGWKDW LWLVDSSOLFDEOHDOVRLQVWUXFWXUHERUQHVRXQG$VIDUDVGLVFUHWL]DWLRQVRID VWUXFWXUHDUHFRQFHUQHGWKHXVHRI+DPLOWRQ¶VSULQFLSOHKDVH[SHULHQFHGD UHQDLVVDQFH LQ UHFHQW \HDUV VLQFH WKH DSSOLFDWLRQ RI WKH ILQLWH HOHPHQW PHWKRG )(0  FDQ EH VHHQ DV DQ HPSOR\PHQW RI WKH SULQFLSOH 'HVSLWH WKLV)(0ZLOOQRWEHDWRSLFLQWKLVERRNEHFDXVHDODUJHQXPEHURISXEOL

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

FDWLRQVLVGHYRWHGWRWKH PHWKRG,QVWHDGWKHZDYHHTXDWLRQVZLOOEHGH YHORSHGIURP+DPLOWRQ¶VSULQFLSOHLQ&KDSWHUVDQGZKHUHE\WKHSODWH DQG WKH VKHOO VHUYH DV H[DPSOHV FI6HFW,Q6HFWPRUHRYHUD IXUWKHUXVHZLOOEHGHPRQVWUDWHG 1DWXUDOO\DOVRWKHZDYHHTXDWLRQVHPSOR\HGLQSUHYLRXVVHFWLRQVFRXOG KDYH EHHQ GHULYHG IURP +DPLOWRQ¶V SULQFLSOH EXW EHVLGHV WKH SK\VLFV RI VWUXFWXUHERUQHVRXQGDOVRGLIIHUHQWPHWKRGVVKRXOGEHSUHVHQWHG ,QPHFKDQLFVWKH+DPLOWRQ¶VSULQFLSOHUHDGVFI6HFW W

W

W

W

G ³ (NLQ  ( SRW GW  ³ G :GW

 

 

+HUHLQ(NLQDQG(SRWDUHWKHWRWDONLQHWLFDQGSRWHQWLDOHQHUJLHVRIWKHV\V WHPUHVSHFWLYHO\ : LVWKHH[WHUQDOO\VXSSOLHGZRUN7KHV\PEROGPHDQV WKDWDYDULDWLRQLVXQGHUWDNHQ(TXDWLRQ  WKXVVWDWHVWKDWDPHFKDQL FDOV\VWHPDGMXVWVLWVHOIVXFKWKDWWKHYDULDWLRQYDQLVKHVRI (NLQ±(SRW DQG :UHVSHFWLYHO\,QSDUWLFXODU (NLQ±(SRW LVDPLQLPXPZKHQQRHQHUJ\LV LPSDUWHGRQWKHV\VWHPH[WHUQDOO\7KHWDVNFRQVLVWVRIH[SUHVVLQJ(NLQ(SRW DQG : LQ WHUPV RI WKH ILHOG YDULDEOHV HJ [ K DQG ] DQG WR SHUIRUP WKH YDULDWLRQ±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

[ 0  M [ \ K K  ]

7KHNLQHWLFHQHUJ\WKHUHE\EHFRPHV

] 0  M ] \ 

 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

(NLQ

U Y[  Y\  Y] G[G\G] ³   U ª¬[0  \ M [ º¼  K   ª¬]0  \ M ] º¼ G[G\G] ³  U  K [ 0  K   ]0  M [  M ] G[G] ³ 

D 

,Q WKH ODVW YHUVLRQ RI D  WKH LQWHJUDWLRQ DFURVV WKH FURVVVHFWLRQ LV XQGHUWDNHQIURP±KWRK

)LJ &RRUGLQDWHV\VWHP IRUWKHGHULYDWLRQRIWKHµFRUUHFWHGEHQGLQJZDYH¶ HTXDWLRQ

)RU WKH SRWHQWLDO HQHUJ\ RI D YROXPH HOHPHQW WKH GLIIHUHQW VWUHVVHV ZLWK WKH DFFRPSDQ\LQJ VWUDLQV H DQG J DUH WR EH PXOWLSOLHG DQG DGGHG FI  DQG  *HQHUDOO\ ( SRW

 ªV [ H [  V \ H \  V ] H ]  W [\J [\  W \]J \]  W []J [] º¼ G[G\G]  ³¬

E 

)RUWKLQSODWHVZLWKVWUHVVIUHHVXUIDFHVV\FDQEHVHWWR]HURIRUWKHFRP SOHWHFURVVVHFWLRQ7KHQFH E IROORZVIURP  ZKLFKUHVXEVWL WXWHGLQ  JLYHV V[

( § w[ w] P ¨ w]   P  © w[

· ¸ V ] ¹

( § w[ w]  ¨P   P  © w[ w]

· ¸  ¹

F 

5HJDUGLQJ WKH VKHDU VWUHVVHV WKH JHQHUDO H[SUHVVLRQV LQ (T   DUH XVHG7KHVWUDLQVDQGVKHDUDQJOHVUHDG

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

H[ J [\ J []

wM w[ w[ 0 w]  \ [  H] w[ w[ w[ w] w[ wK wK w]  J \] M[   w\ w[ w[ w\ wM w] w[ w] w[ 0  \ [  0 w] w[ w] w] w[

wM w] 0 \ ] w] w] wK wK  M]   w] w] wM \ ] w[

G 

:LWK WKHVH H[SUHVVLRQ LQVHUWHG LQ E  UHVXOWV DIWHU LQWHJUDWLRQ RYHU WKHFURVVVHFWLRQ ( SRW

 (K [ 0  [  P [ 0  [] 0  ]  ] 0  ] G[G]   P ³  *K  [ 0  ]  ] 0  [ G[G] ³    (K K  ª   P  M [  [  P M [  [M ]  ]  M ] ]  M [ ]  M ]  [ º» G[G]    P   ³ «¬  ¼   *K ª  M [  K [  M ]  K ] º¼» G[G]  ³ ¬«

H 

,QWKLVH[SUHVVLRQWKHQRWDWLRQ³´LVXVHGLQLQGLFHVWRGHQRWHGLIIHUHQWLD WLRQVZLWKUHVSHFWWRWKHVXEVHTXHQWFRRUGLQDWH 7KHILUVWDQGVHFRQGOLQHVLQ H UHODWHWRWKHSRWHQWLDOHQHUJ\ RI WKHLQSODQHZDYHV7KHWKLUGOLQHGHVFULEHVWKHSRWHQWLDOHQHUJ\RIWKHRU GLQDU\ .LUFKKRII¶V EHQGLQJ WKHRU\ ZKHUHDV WKH ODVW OLQH JLYHV WKH 7L PRVKHQNR0LQGOLQFRUUHFWLRQ>@VWHPPLQJIURPWKHILQLWHVKHDUVWLII QHVV 8SRQXQGHUWDNLQJWKHYDULDWLRQDIWHUVXEVWLWXWLRQLQ  DVHWRIH[ SUHVVLRQVUHVXOWVLQWKHIROORZLQJIRUP 

§ w[ · G ³ ¨ 0 ¸ G[G]GW © w[ ¹ § w[ 0 · G[ 0 G]GW w[ ¸¹

³  ¨©

§ w[ 0 · § w[ 0 · G G[G]GW w[ ¸¹ ¨© w[ ¸¹

³  ¨©

§ w [ 0 · G [ 0 G[G]GW  ³ ¨  ¸ [ © w[ ¹ [

§ w [ 0 · G[ 0 G[G]GW  ³ ¨  ¸ © w[ ¹

+HUHLQ ILUVW WKH YDULDWLRQ LV SHUIRUPHG DFFRUGLQJ WR WKH SURGXFW UXOH DQDORJRXV WR WKDW LQ D GLIIHUHQWLDWLRQ  7KHQ DQ LQWHJUDWLRQ E\ SDUWV LV PDGHDQGILQDOO\LWLVDVVXPHGWKDWWKHILHOGTXDQWLWLHVYDQLVKDWWKHOLPLWV RILQWHJUDWLRQHTXLYDOHQWWRDYDQLVKLQJYDULDWLRQ %\PHDQVRIWKHFRUUHVSRQGLQJSURFHGXUHLVREWDLQHG

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV w[ 0 w] 0 w [ 0 w ] 0 G] 0 G[G]GW  ³ G[G[GW  ³ G[ 0 G[G]GW  w[ w] w[w] w[w] wM wK wK G ³M[ G[G]GW  ³ [ GK G[G]GW  ³ GM [ G[G]GW  w[ w[ w[ G³

I 

DQGDVHWRIVLPLODUH[SUHVVLRQV 5HJDUGLQJWKHH[WHUQDOZRUN:LWLVDVVXPHGWKDWLWLVVROHO\GXHWRGLV WULEXWHG IRUFHV LH SUHVVXUH 6LQFH ZRUN LV WKH SURGXFW RI IRUFH DQG GLV SODFHPHQW :

³ S K  S [ %

 0

 S] 0 G[G]

 

$V LQ )LJ  DQG LQ (T   S% LV WKH H[WHUQDO SUHVVXUH LQ WKH \ GLUHFWLRQDQGSDQGSDUHWKHSUHVVXUHVLQWKH[DQG]GLUHFWLRQVUHVSHF WLYHO\ 8SRQ XQGHUWDNLQJ WKH VDPH PDQLSXODWLRQ DV LQ (T I  LW LV IRXQGWKDW W

³ G :GW ³ S GK  S G[ %



0

 SG] 0 G[G]GW

 

W

ZKHUHWKHSUHVVXUHQHHGQRWEHYDULHGVLQFHLWLVSUHVFULEHG %\FRPELQLQJDOOHTXDWLRQVEHJLQQLQJZLWK  LVREWDLQHGDQH[ SUHVVLRQLQWKHIRUP

³ > $ G[ 

0

 $GK  $G] 0  $GM [  $GM ]  S%GK  SG[ 0  SG] 0 @ G[G]GW



7KLV IRUPXOD UHYHDOV ZK\ DOO YDULDWLRQV LQ I  DUH WUDQVSRVHG WR WKH TXDQWLWLHV [0 K ]0 M[ M] 3LYRWDO LV QDPHO\ WKDW +DPLOWRQ¶V SULQFLSOH PXVWEHYDOLGIRUDQ\YDULDWLRQ)RULQVWDQFHDOVRIRUWKHYDULDWLRQG[0 z EXWGK G]0 GM[ GM] ,WIROORZVWKDW$SPXVWYDQLVKVLQFHG[0 LV DUELWUDU\ 7KLV DUJXPHQW DSSOLHG RQ GK HWF VKRZV WKDW WKH IDFWRUV LQ IURQW RI WKH YDULDWLRQV G[0 GK DQG WKH RWKHU PXVW YDQLVK LQGHSHQGHQWO\ :LWKWKHH[SUHVVLRQVIRU$WR$LQWURGXFHGIURP(T  WR   UHVXOW

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ ª ( w S w º   P w ] 0 * G[ 0  U[0  «  *  » [0    w] ¼   P w[w] K ¬  P w[    ª ( w S  P w [0 w º G] 0   *  U]0  «  *  »] 0     P w[w] K w[ ¼ ¬  P w] GK  U KK  *K’ K  *K

wM [ wM  *K ] w[ w]

S%

 

ª §   P w M ] wK w · w º %c  U , M[  «* ¨ K  ,  ¸  %c  » M [   w[w] w[ w] ¹ w[ ¼ ¬ ©    ª § wK   P w M [ w · w º %c GM ]  *K   U ,M]  «* ¨ K  ,  ¸  % c  » M ] w] w[w] w[ ¹ w] ¼  ¬ ©

GM [  *K

 

ZKHUH %c  (, ±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wM wM · § *K ¨ ’ K  [  ] ¸ U KK  S% w[ w] ¹ © § w M ] w M [ %c ª    «   P ’ M [    P ¨  «¬ © w[w] w[

·º wK · § ¸ »  *K ¨ M [  ¸ w[ ¹ © ¹ »¼ § w M [ w M ] · º %c ª wK · §    ¸ »  *K ¨ M ]  «   P ’ M ]    P ¨ ¸ [ ]  ¬« w] ¹ ¼» w] ¹ © ©w w

U , M[

D 

U , M]

ZKHUHXVHLVPDGHRIWKHUHODWLRQ*,  %c  P %\GLIIHUHQWLDWLQJWKH SHQXOWLPDWHHTXDWLRQZLWKUHVSHFWWR[ DQGWKHODVWZLWKUHVSHFWWR]DVXE VHTXHQWDGGLWLRQ\LHOGV *K ’ K  \

U KK  S% 

% c’ \  *K ’ K  \

ZKHUHWKHDEEUHYLDWLRQ

U ,\

E 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

\

wM [ wM ]   w[ w]

F 

LVLQWURGXFHGIRUFODULW\)URPDQHOLPLQDWLRQRI\LWLVIRXQGWKDW U § %cU ·  %c’ K  ¨  U , ¸ ’ K  U KK  U , K * © * ¹

S% 

%c  U,  ’ S%  S% *K *K

 

7KLV IRUPXOD ZKLFK LQ WKH RQHGLPHQVLRQDO FDVH GDWHV EDFN WR 7L PRVKHQNR >@DQGWR0LQGOLQ >@LQWKH WZRGLPHQVLRQDO IXUQLVKHV DQLPSURYHPHQWRIWKHVLPSOHEHQGLQJZDYH(TV  DQG D UH VSHFWLYHO\ ,W WDNHV WKH VKHDU VWLIIQHVV LQWR DFFRXQW YLD WHUPV LQYROYLQJ , 2IWKHWZRWKHODWWHUDOUHDG\GHULYHGE\5D\OHLJKLVWKHOHDVWLPSRUWDQW 2IWHQ WKH PRGLILHG VKHDU VWLIIQHVV *  FI 6HFW  LVLQWURGXFHG LQ (T  LQVWHDGRIWKHRUGLQDU\VKHDUPRGXOXV*,QWKLVZD\WKHIOH[ XUDOZDYHVSHHGDSSURDFKHVWKH5D\OHLJKZDYHVSHHGDWKLJKIUHTXHQFLHV 7RDVVHVVKRZPXFK  GHYLDWHVIURP  DQG D UHVSHF WLYHO\ D RQHGLPHQVLRQDO FDVH ZLWKRXW H[WHUQDO H[FLWDWLRQ FDQ EH FRQVLG HUHG8SRQOHWWLQJ K  H MZ W H  MN [ WKHGLVSHUVLRQHTXDWLRQUHVXOWVDV [

ª Z  º N&  Z  N& «    »  N %    F/,, F7 ¬ F7 F/,, ¼



D 

+HUHLQ N% LV WKHQRQFRUUHFWHGIOH[XUDOZDYHQXPEHUDFFRUGLQJWR   DQGN&WKHIUHHZDYHQXPEHURIIOH[XUDOZDYHVLQWKLFNSODWHV7KLVPHDQV WKDWWKHZDYHVSHHGRIWKHFRUUHFWHGIOH[XUDOZDYHLVDSSUR[LPDWHGE\ ª  §  ·º  F& | F% «  F% ¨    ¸ » | F% ª   K  O% º ¼ ¬ ¬«  © F7 F/,, ¹ »¼

E 

$  GHYLDWLRQ LV DFFRUGLQJO\ ILUVW WKHQ DSSURDFKHG ZKHQ WKH ZDYH OHQJWKO%EHFRPHVVPDOOHUWKDQVL[WLPHVWKHSODWHWKLFNQHVVO%K)XU WKHUDVSHFWVRISODWHVZLWKILQLWHVKHDUVWLIIQHVVDUHJLYHQLQ6HFW &\OLQGULFDO6KHOOV )XQGDPHQWDO(TXDWLRQV ,Q )LJV  DQG  DUH GHSLFWHG WKH SRODU FRRUGLQDWHV DQG WKH GLV SODFHPHQW YDULDEOHV ± [ WDQJHQWLDO K UDGLDO DQG ] D[LDO ± DSSURSULDWH IRU F\OLQGULFDOVKHOOV7KHFRQYHUVLRQWR&DUWHVLDQFRRUGLQDWHVIRUYHU\ODUJH UDGLLLVJLYHQE\ U- Æ [U Æ \] Æ ]

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

7R EH DEOH WR HPSOR\ +DPLOWRQ¶V SULQFLSOH WKH SRWHQWLDO HQHUJ\ LV UH TXLUHG JLYHQ E\ WKH VWUHVVHV DQG VWUDLQV 7KLV PHDQV WKDW WKH JHQHUDO VWUHVVVWUDLQ UHODWLRQV KDYH WR EH GHYHORSHG LQ SRODU FRRUGLQDWHV $V FDQ EHVHHQIURP)LJWKHVWUDLQLQUDGLDOGLUHFWLRQLV HU

wK · º  ª§ 'U ¸  K » ¨K  « 'U ¬© wU ¹ ¼

wK wU

D 

)LJ'HIRUPDWLRQ D RIDYROXPHHOHPHQWLQSRODUFRRUGLQDWHVDQGQRUPDO DQGVKHDUVWUHVVHV E 

7KHWDQJHQWLDOVWUDLQFRQVLVWVRIWZRSDUWV7KHILUVWRFFXUVIRUSXUHUDGLDO PRWLRQEHFDXVHWKHHOHPHQWLVHORQJDWHG H- 

 ª’ - U  K  U’ - º¼ U’ - ¬

K  U

7KHVHFRQGFRPHVDERXWGXHWRPRWLRQLQWDQJHQWLDOGLUHFWLRQZKHUHE\ H- 

 ª§ w[  · º ’ - ¸ [ » ¨[  U’ - ¬«© w¹ ¼

 w[  U w-

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

$FFRUGLQJO\WKHWDQJHQWLDOVWUDLQEHFRPHV H-

H-   H- 

K  w[  U U w-

E 

%\DVLPLODUFRQVLGHUDWLRQIRUWKHVKHDUGHIRUPDWLRQLWLVIRXQGLQDQDORJ\ ZLWK D WKDWWKHVWUHVVVWUDLQUHODWLRQVLQSRODUFRRUGLQDWHVDUHJLYHQ E\>@ V*

D   H-  DH U  DH ] D   §¨

VU *

DH-  D   H U  DH ]

V] *

DH-  DH U  D   H ]

W-U * W U] * W- ] *

w § [ ·  wK U ¨ ¸ wU © U ¹ U ww] wK   wU w] w[  w]   w] U w-

K  w[ · wK w]  D  ¸ D w w w] U U U © ¹

J -U J U] J -]

wK w] § K  w[ · D¨  D  ¸  D   wU w] © U U w- ¹ wK w] § K  w[ · D¨   D    ¸ D U U U w w w] © ¹ w[ [  wK    wU U U w-

D 

ZKHUHD P ±P +HQFHIRUWKLWLVDVVXPHGWKDWWKHWKLFNQHVVRIWKH VKHOOK LVVXEVWDQWLDOO\VPDOOHUWKDQWKHUDGLXVRIFXUYDWXUHDQGWKHZDYH OHQJWKVHH)LJ)RUVXFKDFDVHVUFDQEHVHWWR]HURZKHQWKHVKHOO LV IUHH RI H[WHUQDO VWUHVVHV LH D VWDWH RI SODQH VWUHVV ,Q DQDORJ\ ZLWK F LWIROORZVWKDW V-

( H-  PH ]  P

V]

( PH-  H ]  P

w] · ( § K  w[  P ¸  ¨   P © U U ww] ¹ ( § K  w[ w] ·  ¨P  P ¸ U w- w] ¹  P © U

ZKLOVWWKHVKHDUVWUHVVHVUHPDLQXQDOWHUHG

 

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

)LJ'LVSODFHPHQWFRRUGLQDWHVIRUDWKLQZDOOHGVKHOO

7KH SURFHHGLQJ DQDO\VLV IROORZV WKDW RI WKH SUHFHGLQJ VHFWLRQ 7KXV WKH UDGLDOFRPSRQHQWK LVFRQVWDQWRYHUWKHFURVVVHFWLRQZKHUHDVWKHWDQJHQ WLDO DQG D[LDO PRWLRQ ZLOO FRQVLVW RI DQ ³LQSODQH´ SDUW DQG EHQGLQJ SDUW ZKLFKGHSHQGVOLQHDUO\RQWKHUDGLXVFI   [ ]

U [ 0  U  D M-  D ] 0  U  D M ] 

D 

$FFRUGLQJO\WKHUHDUHDJDLQILYHLQGHSHQGHQWYDULDEOHVLHWKHPRWLRQRI WKHPLGSODQHRIWKHVKHOO[0KDQG]0DQGURWDWLRQVM-DQGM]7KHVKHOO UDGLXVLVDDVVXPHGFRQVWDQWLQWKLVDQDO\VLV)LJ:LWK D VXE VWLWXWHGLQWRWKHHTXDWLRQVIRUWKHVWUDLQ  DQGDSSUR[LPDWLQJ

U  D

\

 § \· | ¨  ¸  U D© D¹

WKHVWUDLQVDQGVKHDUDQJOHVEHFRPH

D \ |   U D

UD \ |  U D

E 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV § \·  w[ 0 \ wM H]  ¨   ¸K  D© D¹ D w- D w-

H-

wM w] 0 \ ] w] w]

\· § \ · wK wK §  J U] M ]   ¨   ¸ M-  ¨   ¸ D © D ¹ ww] © D¹ wM \ wM ] \ · w[ 0 § \ · w] §   \ -  ¨  ¸ 0  ¨  ¸ w] D © D ¹ w- D w© D ¹ w]

J UJ ]-

 

7KHUHE\ DUHWHUPVLQYROYLQJ\RPLWWHG)XUWKHUPRUHVLQFH[0DQG]0DUH FRQVWDQWDFURVVWKHFURVVVHFWLRQWKHLUGHULYDWLYHVYDQLVKLQWKHUDGLDOGL UHFWLRQ,WVKRXOGDOVREHQRWHGWKDWw[wU  7KHQH[WVWHSWRXVH+DPLOWRQ¶VSULQFLSOHIRUWKHGHULYDWLRQLVWRH[SUHVV WKHHQHUJLHV7KHNLQHWLFHQHUJ\LVREWDLQHGDV U 

(NLQ

S K   /

UK 

³ ³ ³ [





 K   ]  DG- G\G]

 K   / S /

 ª  §  ·º    ³ ³/ «¬[0  K  ]0  , c ¨© M-  M ]  D [0 M- ¸¹»¼ DG- G]

D 

DQGWKHSRWHQWLDOHQHUJ\ ( SRW

 S K   / V -H-  V ]H ]  W U-J U-  W U]J U]  W ]-J ]- DG- G\G]  ³  K³  ³/  

S K   /

ª (

³ ³ ³ «¬  P H 

 K   /

 -

º  H ]   PH- H ]  * J U-  J U]  J ]- » DG- G\G] ¼

E 

,Q(T D LV , c  KLQWURGXFHG %\VXEVWLWXWLQJ(T  LQ E DQGLQWHJUDWLQJZLWKUHVSHFWWR\ DOHQJWK\H[SUHVVLRQUHVXOWVZKLFKFRQWDLQVWKHGLVSODFHPHQWV[0KHWFDV ZHOODVWKHLUVSDWLDOGHULYDWLYHV7KHVXEVHTXHQWWUHDWPHQWLVDVIRU   WR  LHWKHYDULDWLRQLVXQGHUWDNHQDQGDQLQWHJUDWLRQE\SDUWVVXFK WKDWWKHYDULDWLRQUHIHUVWRWKHILHOGYDULDEOHV8VHLVDOVRPDGHRIWKHIDFW WKDWWKHYDOXHVRIWKHLQWHJUDQGDWWKHOLPLWVDQGSDUHHTXDOGXHWRWKH SHULRGLFLW\ DQG WKDW/DQG±/DUHVRUHPRWHWKDWDOOILHOGYDULDEOHVKDYH GHFD\HG DQG WKH YDULDWLRQV YDQLVK $V LQ (T   D V\VWHP RI OLQHDU GLIIHUHQWLDOHTXDWLRQVUHVXOWVLQWKHLQGHSHQGHQWYDULDEOHV[0K]0M-DQG M]$EEUHYLDWHGWKLVUHDGV

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ ª º « » « » «  S   P   (K » % « » « » « » ¬«  ¼»

ª $ $ $ $ $ º ª[ 0 º « » # » ««] 0 »» « $ $ «# % # » «K » « »« » % # » «M- » «# «$ $ »¼ «¬M ] »¼ ¬ 

 

+HUHLQWKHFRHIILFLHQWVDUH w w   E  w  D E        F/, wW  wV  w] 

$



$

D

$



 w D wV

$ 

$

 E  DD 

w wVw]

$



P w D w]

$ 

$ $ $ $ D w wV



 E D 

w wVw]

$ 

§ w  w · E  D ¨ D     ¸ $  F/, wW  ¹ w] ©  w w w $  $    D    E      F/, wW  w] wV

$

E  DD 

$

w wVw]

$ 

 ª w w  º  w  °½ °­  E  ®   D  «   E     »   ¾ w] ¼ F/, wW  °¿ wV °¯ D ¬ w w ª¬D   E    D  º¼ $  $ D   PE  $  wV w] , c w w w D    E   , c   D  , c     F/, wW  wV w]

w wV 

$ 

$

w wVw]

$ 

$

E  DD 

D  , c

 P  P  D  E    w  V DD w-

D  , c

w , c w w   D , c    wV F/, wW  w]

K  ,c D 

K 

E D 

 6SHFLDO &DVHV 7KH UDWKHU H[WHQVLYH V\VWHPRI (T   SHUPLWV WKH IROORZLQJ FRQFOX VLRQV D  7KHV\VWHPLVV\PPHWULFDVXVXDOIRUZDYHHTXDWLRQVGHULYHGE\PHDQV RI+DPLOWRQ¶VSULQFLSOHIRUVWDWLRQDU\RUXQLIRUPO\PRYLQJPHGLD

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

E  $V WKH ZDOO WKLFNQHVV WHQGV WR ]HUR K Æ  E   DQG , c   DQG M- wK  wV DVZHOODVM] ±wKw]7KHQFH(T  WXUQVLQWRWKH PHPEUDQHHTXDWLRQ>@ ª w   P w  w º   P w ] 0  wK  [      « » 0   w]  F/, wW  ¼  wVw] D wV ¬ wV   P w [ 0 ª w    P w   w º P wK    «     »] 0    wVw] ¬ w]  wV D w] F/, wW  ¼ S%   P 

 w[ 0 P w] 0 ª   w º  «    »K D wV D w] ¬ D F/, wW  ¼

(K

D 



ZKLFKLVIXOO\YDOLGIRUQXPHURXVDSSOLFDWLRQV F  :KHQWKHUDGLXVJURZVODUJHLHD Æ fDOOWHUPVLQYROYLQJDDQGE GLVDSSHDUV 7KH UHPDLQLQJ H[SUHVVLRQV DUH LGHQWLFDO WR WKRVH IRU LQ SODQH ZDYHV E  DQG WKH 0LQGOLQ7LPRVKHQNR EHQGLQJ ZDYH GH VFULSWLRQ   G  8SRQ DSSUR[LPDWLQJ KD |  LQ (T   IRU QRQYDQLVKLQJ K DOO WHUPV LQYROYLQJ E DQG , c D YDQLVKZKLOHWKRVHZLWK , c UHPDLQV,ILQ DGGLWLRQ WKH WHUPV UHSUHVHQWLQJ WKH URWDWLRQDO LQHUWLD LH , c F/, w wWDVLVWKHFDVHIRUWKHVLPSOHEHQGLQJWKHRU\RQHREWDLQV WKH ³'RQHOO0XVKWDUL HTXDWLRQ´ > @ ,W GLIIHUV IURP (T D PHUHO\LQWKDWWKHKWHUPLQWKHODVWUHODWLRQWDNHVWKHIRUP ª § w w c   , «  ¨  w]  © wV ¬D

·  w º K ¸   » ¹ F/, wW ¼

E 

7KHUHE\LVWDNHQLQWRDFFRXQWWKHEHQGLQJRIWKHF\OLQGHUZDOO H  )RU D WUDQVLWLRQ WR D ULQJ ww]   LV HPSOR\HG DQG F/, DV ZHOO DV (    K  DUHUHSODFHGE\ F/,,DQG(UHVSHFWLYHO\RZLQJWRWKHFURVV VHFWLRQDO FRQWUDFWLRQ LQ D[LDO GLUHFWLRQ FI (T   WR   7KHUHE\WZRLQGHSHQGHQWH[SUHVVLRQVUHVXOW6KHDULQD[LDOGLUHFWLRQ w ] 0 w ] 0   wV    P F/, wW 

w ] 0  w ] 0   wV  F7 wW 



F 

DQGPRWLRQLQWDQJHQWLDODQGUDGLDOGLUHFWLRQV ª w   E  w º ª w , c w º [ « K   «   » 0 » F/,, wW ¼ ¬ wV ¬ D wV DF/,, wVwW ¼



 ª ª w §  w , c w º , c w º w · »K [  «   E¨ D  ¸       « » 0  F/,, wV  wW  » wV ¹ F/,, wW «¬ D ©D ¬ D wV DF/,, wVwW ¼ ¼

3%  (K

G 

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

7R DUULYH DW WKHVH H[SUHVVLRQV WKH ODVW WKUHH HTXDWLRQV LQ   DUH DGGHG WR DFFRXQW IRU WKH URWDWLRQDO HQHUJ\ 0RUHRYHU WKH DSSUR[LPD WLRQM- wKw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

[ÖKÖ]Ö SÖ MÖ MÖ H %

]

 MN ] ]

-

H MQV  D H MZ W 

 

+HUHLQLQGH[0LVRPLWWHGIRUEUHYLW\)RUDQLQILQLWHO\ORQJF\OLQGHUWKH DVVRFLDWHGZDYHVSURSDJDWHZLWKWKHVSHHGF] ZN]LQWKHD[LDOGLUHFWLRQ DQGH[KLELWQ QRGHVRYHUWKHSHULPHWHUVHH)LJD7KH\FDQEHLPDJ LQHGDVVFUHZIRUPHGSURSDJDWLQJGLVWXUEDQFHVRI³WKUHDGLQJ´O] SN] ,QWKHOLWHUDWXUHDSSHDUVRIWHQFRVQVDRUVLQQVDLQVWHDGRIHMQVDEXWDOVR FRVQ-RUVLQQ-IRUV\PPHWULFDQGDQWLV\PPHWULFSUREOHPV)RUWKHJHQ HUDOFDVHKRZHYHUWKHQRWDWLRQHMQVDLVPRVWVXLWDEOHDQGIRUWKHGLVSHUVLRQ GLDJUDPVDQGZDYHLPSHGDQFHVWKLVGRHVQRWOHDGWRDQ\GLIIHUHQFHV 8SRQ LQVHUWLQJ   LQ (T   DQG VLPXOWDQHRXVO\ OHWWLQJ M- wK  wV DQGM] ±wKw]DVLVFXVWRPDU\LQFRQMXQFWLRQZLWKVLPSOH EHQGLQJWKHRU\LVREWDLQHG Ö § D  D  M D · §¨ [ ·¸ ¨ ¸ Ö D  M D ¸ ¨ ] ¸ ¨ D ¨ ¸ ¨ MD MD D ¹¸ ¨KÖ ¸ ©  © ¹

§ · ¨ ¸ ¨ ¸ ¨ SÖ Q  Z  U K ¸ © % ¹

D 

7KHPDQLSXODWLRQVUHTXLUHGWRDUULYHDWWKLVV\VWHPRIHTXDWLRQVFRQVLVWVLQ GLIIHUHQWLDWLQJWKHIRXUWKHTXDWLRQZLWKUHVSHFWWRVWKHILIWKZLWKUHVSHFW WR]DQGDGGLQJWKHODVWWKUHHRIWKHUHVXOWLQJH[SUHVVLRQV,Q(T D 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV D

Q   D  N ] D   Q   E  D  N ] D   Q  

D

D  QN ] D  E D  QN ] D

D

Q  E  Q D  N ] D   Q  

D

N ] D   D  Q   Q   E D  Q  

D



E 



 P N ] D   E D  Q N ] D

DQG D   Q   E  ª¬  Q   P N ] D   N ] D   Q   Q  N ] D   Q  º¼ 

,QDGGLWLRQQ ZDF/, ZZULQJLVDQRQGLPHQVLRQDOIUHTXHQF\EDVHGRQ WKHULQJIUHTXHQF\ ZULQJ F/,D S IULQJ 7KHGHWHUPLQDQWRI D LVJLYHQE\ 'HW

 D DD  D  D D D  DD  D D

F 

DQGLWVYDQLVKLQJIXUQLVKHVWKHZDYHQXPEHUVRIWKHIUHHZDYHVZKLFKFDQ H[LVWLQORVVIUHHPHGLDDOVRLQWKHDEVHQFHRIH[FLWDWLRQ:LWK E LQ F DIRXUWKRUGHUSRO\QRPLDOLQ N]D  LVREWDLQHG$FFRUGLQJO\IRXU IUHHZDYHVFDQH[LVWDWPRVWDWHYHU\ IUHTXHQF\$WOHDVWRQHWKHUHRIGRHV QRW SURSDJDWH LPSO\LQJ DQ H[SRQHQWLDOO\ GHFD\LQJ QHDUILHOG )RU WKH UH PDLQLQJ WKUHH SURSDJDWLQJ ZDYHV WKH GLVSHUVLRQ GLDJUDP LQ )LJ  LV REWDLQHGE\HTXDWLQJ F WR]HUR6KRZQLVWKHSKDVHVSHHGIRUWKHD[ LDOGLUHFWLRQLQVWHDGRIWKHDVVRFLDWHGZDYHQXPEHUIRUD3RLVVRQ¶VUDWLRRI P DQGQRUPDOL]HGZLWKUHVSHFWWRF/,7KHIROORZLQJFRQFOXVLRQVFDQ EHGUDZQIURPWKHFXUYHV D 7KH EUDQFKHV GHQRWHG E\ 7 ZKLFK FRPH GRZQ IURP LQILQLW\ DW Q Q P  DSSURDFK DV\PSWRWLFDOO\ F] F/,   P    7KH\ FRU UHVSRQG WR SXUH WUDQVYHUVDO ZDYHV QR FRPSUHVVLRQ RI WKH PDWHULDO VHH6HFW  E 7KH EUDQFKHV GHQRWHG E\ / ZKLFK FRPH GRZQ IURP LQILQLW\ DW Q Q DSSURDFK DV\PSWRWLFDOO\ F]  F/, 7KH\ FRUUHVSRQG WR TXDVLORQJLWXGLQDOZDYHV VHH6HFW  F )RUORZIUHTXHQFLHVLVREWDLQHGWKHUHODWLRQ F]



Z  D   F/,

IRUQ 6LQFHDLVWKHVTXDUHRIWKHUDGLXVRIJ\UDWLRQRIDWKLQ ZDOOHGF\OLQGHUWKLVUHODWLRQFRQVWLWXWHVWKHIOH[XUDOZDYHVSHHGRID F\OLQGULFDOVKHOOYLEUDWLQJDVDEHDP G $ IXQGDPHQWDO UROH SOD\V WKH QRQGLPHQVLRQDO IUHTXHQF\ Q   LH WKHULQJIUHTXHQF\ IULQJ F/,SDZKHUHWKHF\OLQGHUSHULPHWHUHTXDOV RQHZDYHOHQJWKRIWKHORQJLWXGLQDOZDYH

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

H  )RUQ WKHUHH[LVWDWORZIUHTXHQFLHVORQJLWXGLQDOZDYHVRIVSHHG F] F/,   P  (  U  DQG WRUVLRQDO ZDYHV RI VSHHG F] F/,   P   *  U ,QWKHILUVWFDVHDYROXPHHOHPHQWRIWKH F\OLQGHUZDOOPRYHVLQUDGLDOGLUHFWLRQ±EUHDWKLQJPRGH±QHJOHFW LQJ WKH FRPSDUDWLYHO\ PXFK VPDOOHU FURVVFRQWUDFWLRQ ,Q WKH VHF RQGWKHPRWLRQLVWRUVLRQDO

)LJ  6SDWLDO GLVSODFHPHQWV RI PRGHV Q  DQGGLVSHUVLRQGLDJUDP DFFRUGLQJWR F IRUD DQGP 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

I )RUQ t WKHFKDUDFWHULVWLF(T  \LHOGVWKUHHJURXSVRIGLVSHU VLRQFXUYHV7KHILUVWWZR±ORQJDQGVKRUWGDVKHGFXUYHVLQ)LJ ±ZHUHGLVFXVVHGLQD DQGE 7KHWKLUGJURXS±VROLGFXUYHV±LVRQO\ SRVVLEOH DERYH WKH DVVRFLDWHG ULQJUHVRQDQFH IUHTXHQFLHV VHH (T F  $OO WKH VROLG FXUYHV DSSURDFK WKDW RI WKH EHQGLQJ ZDYH VSHHGRIDSODWHRIWKLFNQHVVK ±DOVRGUDZQVROLG±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a

SÖ % YÖU

SÖ % MZKÖ

 

LHWKHUDWLRRIWKHH[FLWLQJSUHVVXUHDPSOLWXGH SÖ % WRWKHUHVXOWLQJUDGLDO YHORFLW\ DPSOLWXGH YÖU  IRU DQ LQILQLWHO\ H[WHQGHG F\OLQGHU )RU GLVFUHWH Q (T D JLYHV =aQ

UK MQ 

 ª D D D  DD  D D º D  «  » DD  D ¬ ¼

 

IURPZKLFKDJRRGDSSUR[LPDWLRQLVIRXQGWREH>@ = aQ |

ª  Q   P    P º ZU K ­°  Q  E  « Q   N ] D   »  ® MQ ¯°    P ¬ ¼

ª ª  P    ª  P   º     º  º½ «   P «  N ] D  Q » N ] D  «  N ] D  Q  Q »Q » °° ¬ ¼ ¬ ¼ »  « ¾ ª  P    º     « »° ª º     Q Q N D Q N D Q ] ] ¼ ¬«  ¼» ¬ ¬« ¼» ¿°

 

7KH ZDYH LPSHGDQFH LQGLFDWHV ZKHWKHU D F\OLQGHU ZLOO EH VWURQJO\ RU ZHDNO\ H[FLWHGE\DSUHVVXUHRIWKHIRUP  VPDOODQGODUJH=aQUH VSHFWLYHO\ )URP =aQ   WKH ZDYHQXPEHUV RI WKH IUHH ZDYHV FDQ EH GH

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

WHUPLQHG DQG LQ WXUQ WKH GLVSHUVLRQ FXUYHV FDQ EH JHQHUDWHG E\ XVLQJ F[  F/, Q  N ] D IRUJLYHQYDOXHVRIQ 5HVRQDQFH)UHTXHQFLHV 7KH UHVRQDQFH IUHTXHQFLHV RI D FLUFXODU ULQJ FDQ EH REWDLQHG IURP (TV FG E\VHWWLQJS% DQGHPSOR\LQJWKHVROXWLRQJLYHQE\   )URP(T F LWLVIRXQGWKDW 

Q Z   D  F7

 

JLYLQJ WKH SRVVLEOH UHVRQDQFH IUHTXHQFLHV IRU WKH SDUWLFOH PRWLRQ LQ D[LDO GLUHFWLRQ Z Q ,,,

Q  F7  D  

D 

)RUWKHRWKHULPSRUWDQWPRWLRQFRPSRQHQWVWKHGLVSHUVLRQHTXDWLRQLVRE WDLQHGLQDVLPLODUPDQQHUDV  ,WLVVXIILFLHQWWRSXWN] VLQFHQR] GHSHQGHQFH H[LVWV 6RPH IXUWKHU PDQLSXODWLRQV \LHOG WKH FKDUDFWHULVWLF HTXDWLRQ   Q  ª¬  E    Q  º¼  Q  ª Q      E Q    º  Q  E    Q  »¼ ¬«

 

%\IRFXVVLQJRQWKLQZDOOHGF\OLQGHUV E  WKHDSSUR[LPDWLRQV Q Q, Q

 Q,,

Z Q,  D | Q    F/,  Q  Q    Z Q,,   D |E F/, Q  



EF 

UHVXOWIRUWKHTXDVLORQJLWXGLQDODQGIOH[XUDOPRWLRQUHVSHFWLYHO\7KXVWKH UHVRQDQFHVDUHGHWHUPLQHG ,WVKRXOGEHSRLQWHGRXWKRZHYHUWKDWWKHULQJUHVRQDQFHVDUHQRWDFFX UDWHO\SUHGLFWHGE\PHDQVRIWKHDSSUR[LPDWLRQIRUWKHZDYHLPSHGDQFHLQ  IRUN] 7KHHUURUDPRXQWVWR IRUQ IRUQ  Q DQGLVVPDOOHUWKDQZKHQQ t $V IRU VOHQGHU EHDPV WKH FDOFXODWLRQRIWKH HLJHQIUHTXHQFLHV RIDF\ OLQGULFDOWXEHRIOHQJWKO]FDQEHUHDGLO\PDGHRQO\IRUWZRLGHDOERXQGDU\ FRQGLWLRQV7KLVPHDQVWKDWWKHHLJHQIXQFWLRQV

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV [ -  ] [ P  Q VLQ

PS ] VLQ Q-  O]

K -  ] KP  Q VLQ

PS ] FRV Q-  O]

] -  ] ] P  Q FRV

 

PS ] FRV Q-  O]

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

Z D O] œ PS F/, PS D

Z O] F]

N ] O] œ PO]

O]

DWWKHVHSRLQWV)RUWKHIOH[XUDOPRWLRQRISULPDU\LQWHUHVWWKHUHVRQDQFH IUHTXHQFLHVDUHFORVHO\DSSUR[LPDWHGE\>@ Q Q P

Z Q P D  F/,



|

 PS D  O]   E  ª PS D  O]  Q  º  ¬ ¼ ª PS D  O]   Q  º ¬ ¼

^

Q    P    P °½  ¾    P °¿

D 

ZKHUHDVIRUWKHRWKHUW\SHVRIPRWLRQ Q Q P |

 P ª  PS D  O]  Q º¼   ¬ 

Q Q P |   ª PS D  O]  Q  º  ¬ ¼

E 

7KHIRUPXODHLQ  HQFRPSDVVIRUQ WKHYHU\LPSRUWDQWULQJIUH TXHQF\ Q DVPSDO] WHQGVWR]HUR2WKHUH[SUHVVLRQVIRUUHVRQDQFHIUH

+DPLOWRQ¶V3ULQFLSOHIRUWKH'HULYDWLRQRIWKH(TXDWLRQVRI0RWLRQ

TXHQFLHVDVZHOODVFRPSDULVRQRIGLIIHUHQWVKHOOHTXDWLRQVFDQEHIRXQGLQ >@ )URPWKHUHVXOWVRIDIHZFDOFXODWHGH[DPSOHVLWLVFOHDUWKDWWKHUHVR QDQFHVIRUORQJWXEHV O]!D WHQGWRRFFXUMXVWDERYHWKHFRUUHVSRQGLQJ ULQJ IUHTXHQFLHV RI (T E  $GGLWLRQDOO\ LW LV UHDOL]HG WKDW ZQ P!ZQ P HYHQ LI Q  Q 7KXV LW LV SRVVLEOH WKDW WZR PRGHV RI WKH VDPH QXPEHU P RI QRGHV LQ D[LDO GLUHFWLRQ EXW RI GLIIHULQJ QRGHV LQ FLUFXPIHUHQWLDOH[KLELWWKHKLJKHVWUHVRQDQFHIRUWKHVPDOOHUQ

)LJ1RWDWLRQVIRUDF\OLQGHUVHJPHQW

(TXDWLRQV DE DOVRDSSO\IRUVHJPHQWVRIF\OLQGULFDOVKHOOVRZ LQJWRWKHIDFWWKDWWKHEDVLFGLIIHUHQWLDOHTXDWLRQVDUHWKHVDPH)RUDF\ OLQGULFDOVHJPHQWRIDUFDQGHGJHOHQJWKVEDQGO]DVVNHWFKHGLQ)LJ WKHHLJHQIXQFWLRQV [ P P  VLQ

PS ] P SV VLQ   O] E

K KP P  VLQ

PS ] P SV FRV   O] E

[

]

] P P  FRV

 

PS ] P SV VLQ   O] E

DSSO\IRUP «SURYLGHGWKDWWKHERXQGDU\FRQGLWLRQVIXOILO

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

[

K



[

K



w [ w]  w [ wV 

w K w]  w K wV 

 

w] w] w] wV

 ]

 O] 

 V

 E

7KLVPHDQVWKDWWKHVXSSRUWVSUHVHQWQRPRPHQWVZLWKUHVSHFWWRUDGLDODQG D[LDOPRWLRQVDQGQRIRUFHVWRWKHWDQJHQWLDO,WLVUHDGLO\ YHULILHGWKDWWKH HLJHQIXQFWLRQVLQ  VDWLVI\WKHVSDWLDOGHSHQGHQFLHVRIWKHF\OLQGUL FDOVKHOOHTXDWLRQVDQGOHDGWRDFKDUDFWHULVWLFHTXDWLRQRIWKHVDPHIRUP DV WKDWIRUDFRPSOHWHF\OLQGHUWKHRQO\ DOWHUDWLRQEHLQJWKDWQLVUHSODFHGE\ PSDEDQGPE\ P ,WLVHDV\ WRGHPRQVWUDWHWKDWDVWKHUDGLXVRIFXUYDWXUHWHQGVWRLQILQLW\ D Æ f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

Z P  P

  º ­ª ½ § · P P S S   § ·   °« ¨ ° »    º ° ° « D © O] ¸¹ D ¨© E ¸¹ »  ª § · S S P P K § · ° ° ¼  «   » ¾ | F/, ® ¬ ¨ ¸ ¨ ¸    «© O] ¹ © E ¹ » ° ° ª§ PS · § PS · º ¬ ¼ » ° «¨ ° ¸ ¨ ¸ °¯ ¬«© O] ¹ © E ¹ ¼» °¿

 

6WUXFWXUH%RUQH6RXQG,QWHQVLW\ )XQGDPHQWDO(TXDWLRQV 7KHZHOOWKXPEHGGHILQLWLRQRIWKHLQWHQVLW\YHFWRUIRUDLUERUQHVRXQGFDQ UHDGLO\EHH[WHQGHGWRWKHVWUXFWXUHERUQHVRXQGFDVH>@ - DLU

S˜Y

SY  SY  SY  [

\

]

 

,Q WKH VWUXFWXUHERUQH FDVH KRZHYHU LW LV QHFHVVDU\ WR WDNH LQWR DFFRXQW WKDW WKH G\QDPLF TXDQWLWLHV DUH QRW UHSUHVHQWHG E\ WKH SUHVVXUH EXW WKH

6WUXFWXUH%RUQH6RXQG,QWHQVLW\

VWUHVVWHQVRU,Q&DUWHVLDQFRRUGLQDWHVWKHUHIRUHWKHJHQHUDOGHILQLWLRQRI WKHWKUHHFRPSRQHQWVRIWKHLQWHQVLW\YHFWRUUHDGV -[

 ª¬V [ Y[  W [\ Y \  W [] Y] º¼

-\

 ª¬W [\ Y[  V \ Y \  W \] Y] º¼

-]

 ª¬W [] Y[  W \] Y \  V ] Y] º¼ 

 

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

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

8SRQVXEVWLWXWLQJWKHVHH[SUHVVLRQVLQWR(T  LVREWDLQHG -[

wM w] wM · ( § w[ 0  \ [  P 0  P \ ] ¸ [0  \M [  ¨ w[ w] w] ¹   P © w[ ª§ º wM w] wM · wK · § w[ 0  * «¨ M [   \ [  0  \ ] ¸ ]0  \M ] »  ¸K  ¨ w[ ¹ w] w[ w[ ¹ © w] »¼ ¬«©







 



+HUHLQ WKH GLIIHUHQWLDWLRQ ZLWK UHVSHFW WR WLPH LV GHQRWHG E\ D GRW LH Y[ [0  \M [  Y \ K DQG Y] ]0  \M ]  (TXDWLRQ   IXUQLVKHV WKH LQ WHQVLW\ LQ D SODQH \  FRQVW 2I JUHDWHU SUDFWLFDO LQWHUHVW KRZHYHU LV WKH FURVVVHFWLRQDOO\LQWHJUDWHGSRZHUSHUXQLWZLGWK K

:[c

 ³ - [ G\

 

K

:LWK(T  LQVHUWHGLQ  WKHSRZHUSHUXQLWZLGWKEHFRPHV :[c K

w] · w] · ( § w[ 0 § w[  P 0 ¸ [0  *¨ 0  0 ¸ ]0  ¨ w] ¹ w[ ¹   P © w[ © w] wM · (, c § wM [ wK · § wM [ wM ] · §   P ] ¸ M [  *¨ M [   M ] ¸ K  *, c¨  ¨ w] ¹ w[ ¹ w[ ¸¹   P © w[ © © w]

 

,Q WKLV H[SUHVVLRQ , c   K LV WKH UDGLXV RI J\UDWLRQ VTXDUHG DQG VWHPV IURPWKHLQWHJUDWLRQRI\WHUPVZKHUHDVDOO\WHUPVYDQLVK 7KHWHUPVRIWKHILUVWOLQHLQ(T  ERWKGHSHQGRQO\RQWKHPRWLRQRI WKH QHXWUDO OD\HU RI WKH VWUXFWXUH DQG WKXV UHSUHVHQW WKH SRZHU FDUULHG E\ WKH LQSODQH ZDYHV FI 6HFW,QFRQWUDVWWKRVHRIWKHVHFRQGOLQH DUHDOODVVRFLDWHGZLWKWKHIOH[XUDOPRWLRQDQGFDQEHUHZULWWHQVXFKWKDW RQO\WKHWUDQVODWRU\PRWLRQSHUSHQGLFXODUWRWKHVXUIDFHKDSSHDUV$OVRLW LVVXLWDEOHWRDOWHUWKHIDFWRU* M[wKw[ WRLPSURYHRQLWVSUHVHQWDS SHDUDQFHDVf ˜ 8SRQEDFNWUDFNLQJWKLVH[SUHVVLRQLWLVVHHQWKDW wK · § *¨ M [  ¸K w[ ¹ ©   K4[ K

 W [\K G\ K³



K W [\ G\ K³

% c w § w K w K ·   ¨ ¸K K w[ © w[  w]  ¹

 

,Q WKLV WKH IDFW LV XVHG WKDW WKH FURVVVHFWLRQDOO\ LQWHJUDWHG VKHDU VWUHVV HTXDOVWKHWRWDOIRUFHDWWKHFURVVVHFWLRQ4[JLYHQE\(T H VHH)LJ $QRWKHUSRVVLELOLW\WRDUULYHDWWKLVIRUP ZRXOGEHWRUHZULWHWKHILUVW OLQHLQ(T  DV

6WUXFWXUH%RUQH6RXQG,QWHQVLW\ wK · § *K ¨ M [  ¸ w[ ¹ ©

§   P w M ] w w · %c  U , M] ¨ *,   %c  ¸ M [   w[w] w[ ¹ © w]

 

:LWKWKHDSSUR[LPDWLRQVM[ wKw[DQGM] wKw]DSSOLFDEOHIRUWKLQ SODWHVWKHULJKWKDQGVLGHRI  WXUQVLQWR  ZLWKWKHQHJOLJLEOH GLIIHUHQFHRIWKHVPDOOWHUP U , M[ DVVRFLDWHGZLWKWKHDSSUR[LPDWLRQDQG (T  ILQDOO\EHFRPHV>@ :[c K

(  P

w] 0 § w[ 0 ¨ w[  P w] ©

· § w[ 0 w] 0 ¸ [ 0  *¨ w]  w[ ¹ ©

· ¸] 0 ¹

%c ª§ w K w K · wK w K wK w § w K w K · º  «¨   P  ¸    P  ¨  ¸K » K «© w[ w] ¹ w[ w[w] w] w[ © w[  w]  ¹ » ¬ ¼

D 

)RUWKHRQHGLPHQVLRQDOFDVHDQGSXUHEHQGLQJLWLVUHDGLO\IRXQGWKDW :[%

§ w K wK w K · %¨    K ¸ © w[ w[ w[ ¹

LVLQDJUHHPHQWZLWKWKHSUHYLRXVUHVXOWVLQ  RU D  (TXDWLRQ D  GHPRQVWUDWHV WKDW PHDVXUHPHQW RI VWUXFWXUHERUQH VRXQGSRZHULVQRWVRVLPSOHHYHQIRUWKLQSODWHV7KLVLVGXHWRWKHDYHU DJLQJ RI ODWHUDO PRWLRQV [ ] DQG WKHLU GHULYDWLYHV PHDVXUHG DW WKH SODWH VXUIDFHV\ r KZKLFK\LHOGWKHTXDQWLWLHV[0DQG]00RUHRYHULVUH TXLUHGWKHWUDQVODWLRQK WRLWVWKLUGRUGHUVSDWLDOGHULYDWLYHZLWKDKLJKGH JUHHRIDFFXUDF\FI)LJVDQG ,IEHQGLQJZDYHQHDUILHOGVDUHGLVUHJDUGHGWKHQ w K w[ 

 N %K

Z

Pc K %

D 

DSSOLHV LQ WKH RQHGLPHQVLRQDO FDVH IRU UHJLRQV PRUH WKDQ KDOI D ZDYH OHQJWK DZD\ IURP VWUXFWXUDO GLVFRQWLQXLWLHV DQG H[FLWDWLRQV 7KLV PHDQV WKDW E EHFRPHV :[%

§ wK wK · Z Pc% ¨K  K ¸ © w[ w[ ¹

E 

ZKLFKFDQEHIXUWKHUVLPSOLILHGE\PDNLQJXVHRIWKHFKDLQUXOH w § wK · ¨K ¸ wW © w[ ¹

wK w w K w[ wW w[wW

K

wK wK K w[ w[

F 

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

DQGREVHUYLQJWKDWHYHU\DUELWUDU\IXQFWLRQRIWLPHJ W RVFLOODWLQJDURXQG ]HURZLOOYDQLVKZKHQDYHUDJHG w J W wW

W

  w J W GW 7 ³W wW

 ª J W  J W ¼º |  7¬

G 

,PSOLFLW LV WKDW WKH DYHUDJH LV WDNHQ RYHU D ORQJ WLPH :LWK WKLV UHODWLRQ WKHQ w § wK · ¨K ¸ wW © w[ ¹

 K

wK wK K w[ w[

K

œ

wK w[

K

wK w[

H 

)URP(T E >@WKHUHIRUH :[%

Z P %

wK K w[

Z P %

wK K w[

I 

DSSOLHVIRUWKHIDUILHOGLQWKHRQHGLPHQVLRQDOFDVH 8SRQHPSOR\LQJWZRDFFHOHURPHWHUVZLWKLGHQWLFDODPSOLWXGHDQGSKDVH FKDUDFWHULVWLFVSODFHGDVPDOOGLVWDQFH'[DSDUWDJRRGDSSUR[LPDWLRQIRU WKHSRZHUWUDQVPLWWHGLVJLYHQE\ :[%

Z P %

K  K  W ³ D  D GW   '[ f

 

SURYLGHGWKDW'[O%+HUHLQKKDUHWKHPHDVXUHGGLVSODFHPHQWV DQGDDWKHDFFHOHUDWLRQV8VHLVDOVRPDGHRIWKHDSSUR[LPDWLRQ wK K  K |  K Q w[ '[

W

³ DGW  



)RUDQDUURZIUHTXHQF\EDQGK |DZ VXFKWKDW W

:[% |

Pc%  D  D ³ D  D GW Z '[ f

D 

:LWKWKHH[FHSWLRQRIWKHIDFWRULQIURQWWKLVLVWKHVDPHIRUPXODDVLV XVHGLQWKHDLUERUQHVRXQGFDVHZLWKWKHDFFHOHUDWLRQVXEVWLWXWHGLQVWHDGRI SUHVVXUH2XWVLGHUHJLRQVZLWKVLJQLILFDQWQHDUILHOGVLWLVWKXVSRVVLEOHWR DVVHVVWKHSRZHUFDUULHGE\EHQGLQJZDYHVDSSO\LQJWKHVDPHVLJQDOSURF HVVLQJ HTXLSPHQW DV LV XVHG IRU DLUERUQH VRXQG LQWHQVLW\ $OVR DV LQ WKH DLUERUQHVRXQGLQWHQVLW\FRQWH[W>@LWLVILQDOO\SRVVLEOHWRSXW W

³ D GW 

f

J W Ÿ D

wJ  wW

6WUXFWXUH%RUQH6RXQG,QWHQVLW\

ZKLFKPHDQVWKDW :[%

º wJ wJ wJ Pc% ª wJ J    J    J   J » « Z'[ ¬« wW wW wW wW »¼ w J J   wJ   wJ º Pc% ª wJ J    « »  wW  wW »¼ Z'[ «¬ wW wW W Pc%  D ³ D GW  Z'[ f

E 

$VLQ(T G WKHDYHUDJHGWLPHGHULYDWLYHVFDQFHO )URP DWUDQVLWLRQWRSRZHUVSHFWUDOGHQVLWLHVLWLVSRVVLEOHWRVKRZWKDW WKH SRZHU VSHFWUXP :[% Z  LV SURSRUWLRQDO WR WKH LPDJLQDU\ SDUW RI WKH FURVVSRZHUVSHFWUXPEHWZHHQWKHWZRDFFHOHUDWLRQV7KLVLVPRVWUHDGLO\ GHPRQVWUDWHGLQWKDWDVLQJOHIUHTXHQF\LVFRQVLGHUHGLH D D

 ª DÖH MZ W  D H  MZ W º¼  ¬  ª DÖ H MZ W  D H MZ W º¼  5H ª¬ DÖ H MZ W º¼ ¬

5H ª¬ DÖH MZ W º¼

6XEVWLWXWLRQLQWR(T D \LHOGV :[% Z

Pc% DÖ DÖ VLQ M   M  Z'[

F 

LQ ZKLFK M  M  LV WKH XVXDOO\ VPDOO SKDVH GLIIHUHQFH EHWZHHQ WKH WZR DFFHOHUDWLRQV 3RZHU7UDQVPLVVLRQLQ7KLQ:DOOHG&\OLQGULFDO6KHOOV ,QF\OLQGULFDOFRRUGLQDWHVWKHJHQHUDOH[SUHVVLRQVIRUWKHFRPSRQHQWVRI WKHLQWHQVLW\YHFWRUUHDG --U -]

 ªV -[  W U-K  W - ]] º  ¬ ¼   ª º  W U-[  V UK  W U]]  ¬ ¼  ªW - ][  W U]K  V ]] º  ¬ ¼

 

DQGDUHDQDORJRXVWRWKRVHLQ(T  7KHIROORZLQJDQDO\VLVLVFRQ ILQHG WR WKLQZDOOHG F\OLQGHUV IUHH IURP H[FLWDWLRQV RQ WKH F\OLQGHU ZDOO )RU VXFK FDVHV WKHUH LV QR SRZHU WUDQVPLWWHG LQ WKH UDGLDO GLUHFWLRQ LH

6XUYH\RI:DYH7\SHVDQG&KDUDFWHULVWLFV

-U 7KHVWUHVVDQGVWUDLQUHODWLRQVLQ(TV  WR  PRUHRYHU FDQDOVREHHPSOR\HGLQWKLVFRQWH[WRZLQJWRWKHYDQLVKLQJUDGLDOVWUHVV VU :LWKWKRVHH[SUHVVLRQVLQWURGXFHGLQ  VRPH PDQLSXODWLRQV \LHOGWKHSULQFLSDOSRZHUWUDQVPLVVLRQLQD[LDOGLUHFWLRQDV :]c K

w[ ( § w] 0 P 0 wV   P ¨© w]

· §  w[ 0  w] 0 ¸ ] 0  *¨   E w]    E wV ¹ ©

· ¸[0 ¹



wM · wK · (, c § wM ] § wM ] wM- · §  P - ¸ M ]  * ¨ M ]   M¸K  *, c¨ ] wV ¹ w w] ¸¹   P  ¨© w] © ¹ © wV



(, c P §  ,c · ¨K[ 0  KM ] ¸ D  P D © ¹



*, c ª§ wM ] wM- ·  § w] 0 w[ 0 · º [0  ¨ M- »    « D «¬©¨ wV w] ¸¹ w] ¹¸ »¼ © wV

 

7KHILUVWOLQHRIWKLVH[SUHVVLRQGHVFULEHVWKHLQSODQHLQWHQVLW\LQWKHDE VHQFHRIUDGLDOPRWLRQRIWKHF\OLQGHU,QWKHVHFRQGLVFRQWDLQHGWKDWDV VRFLDWHG ZLWK EHQGLQJ PRWLRQ 7KLV LV FOHDUO\ VHHQ IURP D FRPSDULVRQ RI (TV  DQG  ZKHUHE\LWLVWDNHQLQWRDFFRXQWWKDWWKHSRZHULQ ]GLUHFWLRQLVFRQVLGHUHGLQWKHODWWHU7KHODVWWZROLQHVILQDOO\UHSUHVHQW PL[HGWHUPVZKLFKYDQLVKDVWKHUDGLXVDWHQGVWRLQILQLW\D Æ f 7RSURFHHGZLWKWKHDQDO\VLVWKHFURVVVHFWLRQDOIRUFHLVLQWKLVFDVHRE WDLQHG IURP WKH HTXDWLRQ RI PRWLRQ IRU WKH V\VWHP LH IURP   %\ FDOFXODWLQJ * M]  wKw]  IURP WKH ODVW RI WKH H[SUHVVLRQV DQG VXEVH TXHQWO\ LQWURGXFLQJ WKH DSSUR[LPDWLRQV M]  wKw] DQG M-  wKwV WKH SRZHUWUDQVPLWWHGLVREWDLQHGDV º ª w[ º ( ª w] 0  w[ w] ] 0  P 0 ]0 »  * « 0 [0  0 [0 » « wV w w ] V   P  ¬« w] ¼» ¬« ¼»

:]c K

w K · wK § w’ K  w K · w K wK º (, c ª§ w K « » K    P P  ¸ ¨    ¨   ¸   P «© w] F/, w]wW ¹ wV ¹ w] © w] wVw] wV » ¬ ¼ ( Pª  , c wK º *, c ª w] 0 wK w ] 0 º K »    « «K] 0  K »    P  D ¬« D wV  »¼ w] ¼» D «¬ wV wV 

 

3UDFWLFDOPHDVXUHPHQWVDFFRUGLQJWRWKLVH[SUHVVLRQDUHKLJKO\ FRPSOL FDWHG DQG WLPHFRQVXPLQJ 0RVW RIWHQ WKHUHIRUH WKH VLPSOLILHG RQH GLPHQVLRQDOIRUPXODH  WR F DUHHPSOR\HGWRDVVHVVWKHRUGHU RIPDJQLWXGHDQGDERYHDOOWKHGLUHFWLRQRIWKHLQWHQVLW\ $VLQWKHFDVHRIDLUERUQHVRXQGLQWHQVLW\VWUXFWXUHERUQHLQWHQVLW\GXH WR PXOWLSOH WRQDO FRKHUHQW VRXUFHV PXVW EH WUHDWHG FDXWLRXVO\ $Q\ SKDVH

5HIHUHQFHV

VKLIW EHWZHHQ WKH VRXUFHV PD\ UHVXOWLQKLJKO\GLIIHULQJGLUHFWLYLW\RIWKH LQWHQVLW\

5HIHUHQFHV >@0RUVH309LEUDWLRQDQGVRXQG&K0F*UDZ+LOO1HZ@6FKRFK$3HULRGLVFKH:HOOHQLPVFKXEVSDQQXQJVIUHLHQSODQSDUDOOH OHQ6FKLFKWHQ(UJHEQLVVHGHUH[DNWHQ1DWXUZLVVHQVFKDIWHQS >@ $FKHQEDFK -'  :DYH SURSDJDWLRQ LQ HODVWLF VROLGV 1RUWK+ROODQG $PVWHUGDP >@ 1DDNH +- DQG 7DPP .  6RXQG SURSDJDWLRQ LQ SODWHV DQG URGV RI UXEEHUHODVWLFPDWHULDOV$FXVWLFDS >@ *|W] -hEHUGHQ6FKDOOGXUFKJDQJGXUFK0HWDOOSODWWHQLQ)OVVLJNHLWHQ EHLVFKUlJHP(LQIDOOHLQHU:HOOH$NXVWLVFKH=HLWVFKULIWS >@ )LUHVWRQH )$  7ULFNV ZLWK WKH VXSHUVRQLF UHIOHFWRVFRSH 1RQ GHVWUXFWLYHWHVWLQJS >@7DPP.DQG:HLVV28QWHUVXFKXQJHQEHUSHULRGLVFKH:HOOHQH[ SRQHQWLHOOH XQGNRPSOH[H1DKIHOGHULPEHJUHQ]WHQ)HVWN|USHU$FXVWLFD S >@ (SVWHLQ 36  2Q WKH WKHRU\ RI HODVWLF YLEUDWLRQVLQ SODWHVDQGVKHOOV -RXUQDORI0DWKHPDWLFVDQG3K\VLFVS >@ +XEHU 07  'LH *UXQGODJHQ HLQHU UDWLRQHOOHQ %HUHFKQXQJ GHU NUHX]ZHLVH EHZHKUWHQ (LVHQEHWRQSODWWH =HLWVFKULIW GHV gVWHUHLFKLVFKHQ ,Q JHQLHXUXQG$UFKLWHNWHQ9HUHLQVS >@ +HDUPRQ 5)6 DQG $GDPV (+  7KH EHQGLQJ DQG WZLVWLQJ RI DQ LVRWURSLFSODWHV%ULWLVK-RXUQDORI$SSOLHG3K\VLFVS >@ 7LPRVKHQNR 63 DQG :RLQRZVN\.ULHJHU 6  7KHRU\ RI SODWHV DQG VKHOOV QGHG &K0F*UDZ+LOO$XFNODQG >@+HFNO08QWHUVXFKXQJHQDQRUWKRWURSHQ3ODWWHQ$FXVWLFD  >@ +HFNO 0  (LQIDFKH $QZHQGXQJ GHV +DPLOWRQVFKHQ 3ULQ]LSV EHL .|USHUVFKDOOSUREOHPHQ$FXVWLFD >@0LQGOLQ5',QIOXHQFHRIURWDWRU\LQHUWLD DQGVKHDURQIOH[XUDO PR WLRQVRILVRWURSLFHODVWLFSODWHV-RXUQDORI$SSOLHG0HFKDQLFVS >@ -XQJHU0/DQG)HLW'6RXQGVWUXFWXUHVDQGWKHLULQWHUDFWLRQ&K 0,73UHVV&DPEULGJH0$ >@/RYH$(+$WUHDWLVHRQWKHPDWKHPDWLFDOWKHRU\RIHODVWLFLW\&K ;;,,,'RYHU3XEOLFDWLRQV1HZ@ 3DYLF *  0HDVXUHPHQW RI VWUXFWXUH ERUQH ZDYH LQWHQVLW\ SDUW , )RUPXODWLRQRIWKHPHWKRGV-RXUQDORI6RXQGDQG9LEUDWLRQS >@ 3DYLF * 7KH LQIOXHQFH RI FXUYDWXUH RQ VWUXFWXUHERUQH DFRXVWLFDO SRZHU SURSDJDWLRQ LQ D F\OLQGULFDO FLUFXODU VKHOO 3URF WK ,QWHUQDWLRQDO &RQJUHVVRQ$FRXVWLFV3DSHU'7RURQWR >@)DK\)-6RXQGLQWHQVLW\ QGHG ( )16SRQ/RQGRQ >@0D\VHQK|OGHU:.|USHUVFKDOOHQHUJLH*UXQGODJHQ]XU%HUHFKQXQJ YRQ(QHUJLHGLFKWHQXQG,QWHQVLWlWHQ6+LU]HO9HUODJ6WXWWJDUW

'DPSLQJ

'DPSLQJ0HFKDQLVPVDQGWKHLU0DWKHPDWLFDO 'HVFULSWLRQ 7KHSUHYLRXVFKDSWHUZKLFKGHDOWZLWKWKHYDULRXVW\SHVRIZDYHVWKDWFDQ RFFXULQDVROLGERG\LQDOOFDVHVPDGHXVHRIVRPHIRUPRI+RRNH¶VODZ WKDW LV LW DOZD\V LQYROYHG SURSRUWLRQDOLW\ EHWZHHQ VWUHVV DQG VWUDLQ +RRNH¶VODZOLNHPRVWODZVRISK\VLFVLVH[DFWRQO\ IRULGHDOVLWXDWLRQV ZKLFKUHSUHVHQWOLPLWLQJFRQGLWLRQVIRUSUDFWLFDOVLWXDWLRQV)RUWKHWRSLFV WUHDWHGLQWKHSUHYLRXVFKDSWHUWKHGHYLDWLRQVIURP+RRNH¶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³'DPSLQJ´DQG³GLVVLSDWLRQ´XVXDOO\UHIHURQO\WRFRQYHUVLRQRI PHFKDQLFDOHQHUJ\LQWRKHDW7KHVHWHUPVDUHJHQHUDOO\QRWDSSOLHGWRHQ HUJ\ORVVHVWKDWRFFXUDVDUHVXOWRIUHIOHFWLRQDWGLVFRQWLQXLWLHV VHH&KDS WHU ZKLFKSURFHVVLVXVXDOO\GHVFULEHGDV³DWWHQXDWLRQ´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

'DPSLQJ

SKHQRPHQDKDVFRQFHUQHGSK\VLFLVWVIRUDORQJWLPH$VHDUO\ DV2 ( 0H\HU >@ VXJJHVWHG WKDW D YLVFRXV IULFWLRQ IRUFH VKRXOG EH WDNHQ WR DFWLQDGGLWLRQWRWKHHODVWLFIRUFHV6LQFHVXFKDIULFWLRQIRUFHLVSURSRU WLRQDOWRWKHWLPHGHULYDWLYHRIWKHVWUDLQ(T  ZRXOGWKHQEHUHZULWWHQ DVWKHODWHUGHQRWHG.HOYLQ9RLJWPRGHO>@ GH · § ' ¨H ¸  GW ¹ ©

V

 

)RUWKHVLQXVRLGDOWLPHYDULDWLRQVZKLFKDUHRIJUHDWHVWLQWHUHVW HÖ FRV Z W 

H

 

LVREWDLQHG V

'HÖ FRV Z W  Z- VLQ Z W 'HÖ   Z -  FRV Z W  DUFWDQ Z- 

 

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³VWUDLQ KLVWRU\´ )ROORZLQJ %ROW]PDQQ DQG DVVXPLQJ WKDW WKH HI IHFWVRIWKHSULRUVWUDLQVPD\EHVXSHUSRVHGOLQHDUO\WKHGHSHQGHQFHRIWKH VWUHVVV W DWWLPHWRQWKHVWUDLQH W DWWLPHWDQGRQWKHVWUDLQH W'W DW WKHHDUOLHUWLPH W'W DVFDQEHZULWWHQDV f

V W

'H W  ³ H W  'W M 'W G 'W 

 



7KHIXQFWLRQM 'W GHVFULEHVWKH³DIWHUHIIHFW´RIDVWUDLQDQGWKHIRUP RIWKLVIXQFWLRQGHWHUPLQHVWKHVWUHVVVWUDLQUHODWLRQ2QHPD\ QRWHWKDWWKH DERYHUHGXFHVWR+RRNH¶VODZIRUM 'W {DVRQHZRXOGH[SHFW 2IWKHPDQ\³DIWHUHIIHFWIXQFWLRQV´M 'W WKDWDUHSRVVLEOHLQSULQFLSOH RQO\RQHWKHVRFDOOHG³UHOD[DWLRQIXQFWLRQ´LVSK\VLFDOO\PHDQLQJIXO,ID VWUDLQ LV LPSRVHG RQ D PDWHULDO WKHQ WKHUH RFFXU FHUWDLQ PROHFXODU SURF HVVHV GLVSODFHPHQWV GLVWRUWLRQV RI FU\VWDO ODWWLFHV FKDQJHV LQ PROHFXODU VWUXFWXUH H[FLWDWLRQ RI FHUWDLQ PROHFXODU RVFLOODWLRQV HWF  ZKLFK DUHH[

'DPSLQJ0HFKDQLVPVDQGWKHLU0DWKHPDWLFDO'HVFULSWLRQ

FLWHGJUDGXDOO\DQGZKLFKWKHQDOVRGHFD\ JUDGXDOO\,IWKHDSSOLHGVWUDLQ IRU H[DPSOH FDXVHV D PROHFXOH FKDLQ WR RVFLOODWH RQH PD\ DVVXPH WKDW WKHVHRVFLOODWLRQVZLOOGHFD\H[SRQHQWLDOO\7KHFRUUHVSRQGLQJDIWHUHIIHFW IXQFWLRQWKHQPD\EHZULWWHQDV M 'W

' 'W  W  H W

 

ZKHUH'LVDFRQVWDQWDQGW LVWKHVRFDOOHGUHOD[DWLRQWLPHZKLFKLQWKH SUHVHQWH[DPSOHLVDVRUWRIGHFD\ WLPHRIPROHFXODURVFLOODWLRQV7KHIRUP RIWKHVWUHVVKLVWRU\IRUDJLYHQVWUDLQLVVKRZQLQ)LJIRUWZRH[DP SOHV

)LJ  9DULDWLRQV RI VWUDLQ DQG VWUHVV ZLWK WLPH IRU D SUHVFULEHG VWHSIXQFWLRQ GLVSODFHPHQW OHIW  DQG VLQXVRLGDO GLVSODFHPHQW ULJKW  3DUDPHWHUV '  ' ZW 

7KH OHIWKDQG SDUW RI WKH ILJXUH LQGLFDWHV WKH IRUFH WKDW LV UHTXLUHG WR SURGXFHDVWHSIXQFWLRQMXPSLQWKHVWUDLQIURPWRH2QHILQGVWKDWWKH LQLWLDO IRUFH LV UHODWLYHO\ ODUJH EXW WKDW WKH IRUFH GHFUHDVHV DV WKH ERG\ ³JHWVXVHGWR´LWVQHZVWDWH 7KHULJKWKDQGSDUWRIWKHILJXUHLOOXVWUDWHVWKHVWUHVVKLVWRU\IRUDSHUL RGLF VWUDLQ %HFDXVH WKLV FDVH LV SDUWLFXODUO\ LPSRUWDQW LW LV RI LQWHUHVW WR DQDO\VHLWLQVRPHGHWDLO8SRQLQWURGXFLQJ(TV  DQG  LQWR   WKHVWUHVVEHFRPHV

'DPSLQJ

V W

'HÖ FRV Z W 

' f HÖ FRV Z W  'W H 'W  W G 'W W ³

' · ZW § ¨ '  Z W    ¸ HÖ FRV Z W  ' Z W    HÖ VLQ Z W © ¹

 

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

 GV   V ' GW Q GV GH · § V  W ' ¨H W ¸ GW GW ¹ ©

 

UHVSHFWLYHO\ +HUHLQ Q GHQRWHV WKH G\QDPLF YLVFRVLW\ ZKLOH W DQG W DUH UHOD[DWLRQWLPHV7KH0D[ZHOOPRGHOLVSULPDULO\XVHG LQ WKHGHVFULSWLRQ RIFUHHSDQGWKH=HQHUPRGHOLVDFRPELQDWLRQRIWKDWRI(T  DQGWKH 0D[ZHOOPRGHO

&RPSOH[0RGXOXVDQG:DYHQXPEHUV

&RPSOH[0RGXOXVDQG:DYHQXPEHUV )RUWXQDWHO\ WKH GLIILFXOWLHV DVVRFLDWHG ZLWK WKH VWUHVVVWUDLQ UHODWLRQV GLV FXVVHG DERYH FDQ EH DYRLGHG LI RQH OLPLWV RQHVHOI WR SHULRGLF SURFHVVHV DQGLQWURGXFHVDFRPSOH[PRGXOXV$VKDVEHHQSRLQWHGRXWHDUOLHUKRZ HYHURWKHUWLPHGHSHQGHQFLHVFDQEHKDQGOHGE\PHDQVRIVXSHUSRVLWLRQ (TXDWLRQ   DV ZHOO DV (T   LQGLFDWHV WKDW WKH SULPDU\ HIIHFW RI GDPSLQJLVWKHSURGXFWLRQRIDSKDVHGLIIHUHQFHEHWZHHQVWUHVVDQGVWUDLQ 2QHPD\H[SUHVVWKLVIDFWYHU\FRQYHQLHQWO\LQFRPSOH[QRWDWLRQE\ ZULW LQJ V W

5H ^ 'HÖH MZ W `

 

'cHÖ FRV Z W  'ccHÖ VLQ Z W 

ZKHUH '

' c  M' cc

' c   MK

D 

UHSUHVHQWVWKHFRPSOH[PRGXOXVRIHODVWLFLW\7KHTXDQWLW\ K  'cc  'c LV WKHVRFDOOHGORVVIDFWRUZKLFKLVXVHGZLGHO\WKURXJKRXWWKHUHVWRIWKLV ERRN LQ WH[WERRNV RQ SRO\PHUV RIWHQ WKH ORVV IDFWRU LV GHQRWHG G  ,W VKRXOGEHQRWHGKRZHYHUWKDWWKHFRPSOH[PRGXOXVPD\OHDGWRFDXVDOLW\ SUREOHPV E\ D WUDQVIRUPDWLRQ IURP IUHTXHQF\ GRPDLQ WR WLPH GRPDLQ ZKHQWKHORVVIDFWRULVQRWNQRZQSUHFLVHO\ 7DEOH)UHTXHQF\GHSHQGHQFHRIFRPSOH[PRGXOLDQGORVVIDFWRU 7\SH 9RLJW.HOYLQ 0D[ZHOO %ROW]PDQQ FRQVWDQWORVVIDFWRU K

'c ' Z Q   '  '   Z Q   '  ' 

'   Z W  '

'cc 'ZZQ  ' '   Z Q  '  '

ZW   Z W  K '

K Z'ZQ 'ZW '  '  'Z W  K

,Q 7DEOH  DUH VKRZQ WKH UHDO DQG LPDJLQDU\ SDUWV RI WKH FRPSOH[ G%P@ /HYHOGHFD\IRUORQJLWXGLQDOZDYHV '/c  KO>G%P@ (  ( c  MK 

/S / I S /I / DUFWDQ /SM 

'DPSLQJ

4XDVL/RQJLWXGLQDO:DYHVDQG7RUVLRQDO:DYHV 2IWKHPDQ\PHWKRGVDYDLODEOHIRUWKHDQDO\VLVRIEHDPYLEUDWLRQVWKHRQH XVHGLQWKLVVHFWLRQ±ZKLFKGHGXFHVWKHYLEUDWLRQVRIDILQLWHEHDPIURP WKHEHKDYLRXURIDQLQILQLWHRQH±PD\EHVRPHZKDWFXPEHUVRPHEXWSUR YLGHVFRQVLGHUDEOHSK\VLFDOLQVLJKW&RQVLGHUDURGVXFKDVWKDWVKRZQLQ )LJH[FLWHGDWRQHHQGE\WKHIRUFH )

^

`

Ö MZ W  5H )H

,QDQLQILQLWHO\ORQJURGVXFKDQH[FLWDWLRQZRXOGJLYHULVHWRDZDYHPR WLRQ YÖ  [

)Ö  M N[ H  =

 

ZKHUH=UHSUHVHQWVWKHFRPSOH[GULYLQJSRLQWLPSHGDQFH ZKLFKWDNHVRQD FRPSOH[ YDOXH EHFDXVH WKH SURSDJDWLRQ VSHHG LV FRPSOH[  DQG N LV WKH FRPSOH[ZDYHQXPEHU)RUTXDVLORQJLWXGLQDOZDYHVWKHGULYLQJSRLQWLP SHGDQFHDWDIUHHHQGLV6UF/,,

)LJ:DYHSURSDJDWLRQDORQJDEHDP

:KHQWKHZDYHJLYHQE\(T  DUULYHVDWWKHHQGRIWKHURGWKHYH ORFLW\DPSOLWXGHLV )Ö  M NO H  =

7KHZDYHQRZLVUHIOHFWHGUHVXOWLQJLQDEDFNZDUGWUDYHOOLQJZDYHJLYHQ E\ )Ö M N O [ U O H   =

5HVRQDQW9LEUDWLRQVRI'DPSHG%HDPV

ZKHUH UO UHSUHVHQWV WKH FRPSOH[  UHIOHFWLRQ FRHIILFLHQW DW WKH ORFDWLRQ [ O7KLVZDYHLVUHIOHFWHGRQFHDJDLQZKHQLWUHDFKHV[ DQGWKHGRX EO\UHIOHFWHGZDYHREH\V )Ö M N O [ U O U  H   =

ZKHUHUGHQRWHVWKHUHIOHFWLRQFRHIILFLHQWDW[ 7KLVGRXEO\UHIOHFWHG ZDYH LV DJDLQ UHIOHFWHG DW [  O DQG WKLV SURFHVV LV UHSHDWHG DJDLQ DQG DJDLQLQGHILQLWHO\,QWKHVWHDG\VWDWHZKHUHDOOWKHVHFRPSRQHQWVDUHSUH VHQWVLPXOWDQHRXVO\RQHWKXVREWDLQV Y [

) ª  M N[   M N O  [  M N O  [  M N O  [ H  U O H  U  U O H  U  U O H !º¼  ¬ =

 

2QHPD\QRWHWKDWWKHILUVWWKLUGILIWK«WHUPVRIWKLVH[SUHVVLRQIRUP D JHRPHWULF VHULHV ZLWK WKH UDWLR UUOHMNO DQG WKH VDPH KROGV DOVR IRU WKH VHFRQGIRXUWKVL[WK«WHUPV7KXVE\DSSO\LQJWKHVXPPDWLRQIRUPXOD IRULQILQLWHJHRPHWULFVHULHVRQHREWDLQV Y [

 M N O  [ ) H  M N[  U O H  =   U  U O H  M NO

 

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|

)   = MNO

 

)RU TXDVLORQJLWXGLQDO ZDYHV = NO  ZU6O DQG WKH DERYH UHVXOW LPSOLHV WKDWDVKRUWURGZKLFKLVIUHHDWERWKHQGVEHKDYHVHVVHQWLDOO\DVDPDVVDV RQHZRXOGH[SHFW )RUDURGWKDWLVIUHHDWRQHHQGDQGFODPSHGDWWKHRWKHU U UO   RQHILQGVE\PXOWLSO\LQJ(T  E\ HMNOWKDW Y [

M

) VLQ N O  [  = FRV N O  [

 

'DPSLQJ

,QWKHVSHFLDOFDVHRIVKRUWURGV NO WKHYHORFLW\DW[ REH\V Y  | MNO

)  =

DQGIRUTXDVLORQJLWXGLQDOZDYHVIRUZKLFK = Y  | M

  6 (U DQG N

Z O)  6(

Z U  ( 

 

WKXVVXFKVKRUWURGVEHKDYHOLNHSXUHVSULQJV 7KHEHKDYLRXURI(T  LQWKHYLFLQLW\RIPLQLPDRIWKHGHQRPLQD WRULVRISDUWLFXODULPSRUWDQFHLQUHODWLRQWRPHDVXUHPHQWV,QRUGHUWRILQG WKHVHPLQLPDRQHPD\DVVXPHWKDWDWWKHWZRHQGVRIWKHURGWKHUHRFFXU QRHQHUJ\ORVVHVLH_U_ EXWRQO\WKHSKDVHVKLIWVJDQGJOUHVSHFWLYHO\ VHH 6HFW,IRQHLQWURGXFHVWKHFRPSOH[ZDYHQXPEHUN N¶±MN¶¶DV GHILQHGLQ(T  WKHQRQHPD\UHZULWHWKHGHQRPLQDWRURI(T   DV   H  N ccO ª¬FRV N cO  J   J O  M VLQ N cO  J   J O º¼ 

 

:LWK VPDOO GDPSLQJ WKDW LV IRU N ccO   WKLV IXQFWLRQ WDNHV RQ LWV PLQLPXPYDOXHLHWKHYHORFLW\UHDFKHVLWVPD[LPXPIRU N cO  J   J O

QS 

%\ FRPSDULQJ WKLV UHVXOW ZLWK (T   RQH ILQGV WKDW LW FRUUHVSRQGV SUHFLVHO\WRWKHUHVRQDQFHIUHTXHQFLHVRIXQGDPSHGURGV,IWKHGDPSLQJLV QRWVPDOORQHPXVWILQGWKHPLQLPDRI(T  E\VHWWLQJWKHGHULYDWLYH RIWKHHTXDWLRQZLWKUHVSHFWWRN¶HTXDOWR]HUR7KHVHUHVRQDQFHIUHTXHQ FLHV WXUQ RXW WR EH VRPHZKDW ORZHU WKDQ WKRVH RI VLPLODU OLJKWO\ GDPSHG URGV :KHWKHU WKH PLQLPD RI WKH GHQRPLQDWRU RI (T   FRUUHVSRQG WR PD[LPD RI WKH YHORFLW\ DOVR GHSHQGV RQ WKH REVHUYDWLRQ SRLQW 7KH QX PHUDWRURI(T  LVSHULRGLFLQ[WKDWLVWKHUHRFFXUQRGHVDQGDQWL QRGHV RI YLEUDWLRQ 7KLV LV LOOXVWUDWHG LQ )LJ D ZKLFK VKRZV WKH IUH TXHQF\ GHSHQGHQFHRIWKHYHORFLW\IRUYDULRXVDPRXQWVRIGDPSLQJDQGLQ )LJEZKLFKVKRZVWKHVSDWLDOGHSHQGHQFHRIWKHYHORFLW\%RWKWKHVH ILJXUHVSHUWDLQWRDURGWKDWLVIUHHDWERWKHQGVDQGH[FLWHGE\DIRUFHRI FRQVWDQW IUHTXHQF\ LQGHSHQGHQW  DPSOLWXGH 7KH FRUUHVSRQGLQJ YHORFLW\ REH\V 

Y [



) FRVK N cc O  [  FRV N c O  [  = FRVK N ccO  FRV N ccO

 

5HVRQDQW9LEUDWLRQVRI'DPSHG%HDPV

7KH DEVFLVVD RI )LJ D LV N cO  WKH ZDYHQXPEHU PXOWLSOLHG E\ WKH EHDPOHQJWK DQGWKXVLVSURSRUWLRQDOWRWKHIUHTXHQF\DQGWKHDEVFLVVDRI )LJELV N c[ 7KHRUGLQDWHRIWKHVHILJXUHVLVWKHORJDULWKPRIWKHUDWLR RIWKHYHORFLW\ WRDQDUELWUDU\UHIHUHQFHYHORFLW\ Y2QHPD\QRWHWKDWWKH FXUYHV EHFRPH IODWWHU ZLWK LQFUHDVLQJ IUHTXHQF\ DQG HYHQWXDOO\ EHKDYH OLNHH[SRQHQWLDOO\GHFD\LQJIXQFWLRQV)RU N cc  N c WKDWLVIRUK | WKHFXUYHVZRXOGH[KLELWQRUHFRJQL]DEOHSHULRGLFLW\DWDOO

)LJ  4XDVLORQJLWXGLQDO RU WRUVLRQDO YLEUDWLRQV RI IUHHIUHH EHDPV DV GHWHU PLQHGIURP(T  D 9DULDWLRQZLWKIUHTXHQF\DW [ OE 9DULDWLRQZLWKSRVL WLRQIRUNAO S

)RU VPDOO GDPSLQJ RQH PD\ REWDLQ VLPSOLILHG H[SUHVVLRQV IRU WKH EH KDYLRXULQWKHYLFLQLW\RIUHVRQDQFHVE\GHYHORSLQJ(T  LQDVHULHV DERXWDUHVRQDQFHFRQGLWLRQ,QWKLVGHYHORSPHQWKRZHYHURQHPXVWWDNH LQWRDFFRXQWWKDWERWKWKHQXPHUDWRUDQGWKHGHQRPLQDWRUGHSHQGRQIUH TXHQF\ $VVXPH WKDWWKHYHORFLW\LVPHDVXUHGDWDQDQWLQRGHLHDPD[L PXPLQ)LJEWKXVDWDSRVLWLRQZKHUHWKHFXUYHFRUUHVSRQGLQJWRWKH VSDWLDOYDULDWLRQLVYHU\IODWDQGWKXVXQDIIHFWHGE\VPDOOFKDQJHVLQ N c[  %HFDXVHRIWKLVDVVXPSWLRQWKHDQDO\VLVWKDWIROORZVLVQRWYDOLGIRUQRGDO SRVLWLRQVVLQFHWKHUHVPDOOFKDQJHVLQIUHTXHQF\ OHDGWRODUJHFKDQJHVLQ WKHQXPHUDWRUDQGGHQRPLQDWRURI(T  )RUWKHDQWLQRGHVLWLVIRXQG WKDW

'DPSLQJ Y

Y%

  U  U O H  N ccO H

 M  N cO J  J O



,I WKH GDPSLQJ LV VPDOO DQG RQO\ WKH EHKDYLRXU LQ WKH YLFLQLW\ RI UHVR QDQFHVLVRILQWHUHVWWKHQRQHPD\XVHWKHDSSUR[LPDWLRQV H  N ccO |   N ccO  M  N cO J J   K N cO DQG H |   'N cO  DQGZULWH 

Y% |

O

Y    U  U O   N ccO   M 'N cO

6LQFHWKLVH[SUHVVLRQKDVLWVPD[LPXPYDOXHYPD[DWUHVRQDQFH ' N cO      U  U O   N ccO Y% | Y PD[   U  U O   N ccO   M 'N cO

  'N cO  M N ccO    U  U O   U  U O

IURPZKLFKLWLVIRXQG Y% Y PD[



|

 ª 'N c º   « » c ¬« N N cc  N c    U  U O   U  U O N cO »¼





 

,QWKHYLFLQLW\RIWKHUHVRQDQFHIUHTXHQF\DSORWRIY%YHUVXVIUHTXHQF\ WKXVKDVWKHW\SLFDOUHVRQDQFHUHVSRQVHFKDUDFWHUVHH)LJ7KHZLGWK RIWKHUHVRQDQFHFXUYHKHUHGHSHQGVRQWKHHQHUJ\ORVVHVGXHWRUHIOHFWLRQ DQGLQWHUQDOGDPSLQJ,IQRHQHUJ\ORVVHVDUHGXHWRUHIOHFWLRQV _UUO_   WKHQ(T  UHGXFHVWR Y% Y PD[



|

 § 'N c ·  ¨ ¸ © N cc ¹





 

,I WKH ZDYHQXPEHUV DUH H[SUHVVHG LQ WHUPV RI WKH IUHTXHQF\ WR ZKLFK WKH\DUHSURSRUWLRQDOWKHQ' N c  N c EHFRPHV'IIQZKHUH'IUHSUHVHQWVWKH IUHTXHQF\ GHYLDWLRQPHDVXUHGIURPWKHSHDNIUHTXHQF\ IQ,IRQHDOVRVHWV N cc  K N c  VHH(T  RQHREWDLQV Y% Y PD[



|

 § 'I ·  ¨ ¸ © K IQ ¹





 

7KHKDOIYDOXHEDQGZLGWKRIWKHUHVRQDQFHFXUYHWKXVLVE KIQVHH)LJ  &OHDUO\PHDVXUHPHQWRIWKLVUHVRQDQWEHKDYLRXUFRQVWLWXWHVDVLPSOH

5HVRQDQW9LEUDWLRQVRI'DPSHG%HDPV

PHDQVIRUGHWHUPLQLQJWKHGDPSLQJSURYLGHGWKDWWKHORVVIDFWRULVQRWWRR JUHDWDQGWKHPHDVXUHPHQWVDUHFDUULHGRXWDWDQDQWLQRGH

)LJ9LEUDWLRQDOUHVSRQVHRIOLJKWO\GDPSHGV\VWHPVLQWKHYLFLQLW\ RIDUHVR QDQFH

)RUWKHYLEUDWLRQVLQWKHYLFLQLW\ RIQRGHVRQHILQGVWKDWDWFRQVWDQWIUH TXHQF\ Y.



ª cc  U O H  N O  [  | Y PLQ «  « cc   U O H  N O  [ «¬





N '[ 



º » » »¼

 

IRUYDQLVKLQJO\VPDOOGDPSLQJ N cc   WKLVUHGXFHV IRUUO | WR Y.



| Y PLQ



ª § N '[ «  ¨ « ¨©   U O ¬

· ¸¸ ¹



º » » ¼

Y PLQ



ª § S «  ¨ « ¨©   U O ¬

)RU FRPSOHWH UHIOHFWLRQ DW WKH HQG _UO_   N cc O  [ PD\EHXVHGWRREWDLQ

H  N cc O  [

Y.



| Y PLQ



ª § '[ ·  N c º «  ¨ » ¸  «¬ © O  [Q ¹ N cc »¼



· § '[ · º ¸¸ ¨ ¸ »  ¹ © O ¹ »¼

 

   WKH DSSUR[LPDWLRQ

 

$V FDQ EH REVHUYHG WKHUH H[LVWV D FORVH UHODWLRQ EHWZHHQ WKH YLEUDWLRQV QHDUDUHVRQDQFHDQGQHDUDQRGH7KHUHVRQDQFHIUHTXHQF\ IQLVDQDORJRXV WRWKHORFDWLRQ[QRIDQRGHDQGWKHIUHTXHQF\LQWHUYDO'I LVDQDORJRXVWR WKHGLVWDQFH'[7KHIDFWWKDWWKHH[SUHVVLRQ«RFFXUVLQWKHQXPHUDWRU

'DPSLQJ

LQRQHFDVHDQGLQWKHGHQRPLQDWRULQWKHRWKHUPDNHVOLWWOHGLIIHUHQFHSDU WLFXODUO\LIWKHYHORFLW\LVUHSUHVHQWHGORJDULWKPLFDOO\ 7KHUHDUHDYDLODEOHPDQ\PHWKRGVIRUDQDO\VLQJWKHYLEUDWLRQVRIDURG LQDGGLWLRQWRWKDWXVHGDERYH$PRQJWKHPRVW LPSRUWDQWRIWKHVHLVWKH VRFDOOHG³IRXUSROH´UHSUHVHQWDWLRQ7KLVUHSUHVHQWDWLRQFDQEHGHYHORSHG E\ FRQVLGHULQJ WKH ZDYH ILHOG LQ D URG WR EH FRPSRVHG RI IRUZDUG DQG EDFNZDUGSURSDJDWLQJZDYHVIRUZKLFKWKHYHORFLW\PD\EHH[SUHVVHGDV Y [

Y  H  M N[  Y  H M N[ 

 

ZKHUHYDQGY DUHXQNQRZQIRUWKHSUHVHQW7KHFRUUHVSRQGLQJIRUFHWKDW DFWVLQWKHEHDPLVJLYHQE\ ) [

= Y  H  M N[  = Y  H M N[

D 

ZKHUHWKHFKDQJHLQWKHDOJHEUDLFVLJQUHVXOWVIURPWKHGLIIHUHQWGLUHFWLRQV RIZDYHSURSDJDWLRQ7KHDERYHPXVWKROGIRUDOOORFDWLRQVDQGWKXVDOVR IRU[ 2QHWKHUHIRUHREWDLQVWKHIROORZLQJFRQGLWLRQVIRUYDQGY Y

Y  Y  ) 

= Y  Y 

,I RQH LQWURGXFHV WKH UHVXOWLQJ YDOXHV RI Y DQG Y LQWR (TV   DQG D WKHQRQHREWDLQVWKDWDWWKHORFDWLRQ[ OIRUH[DPSOH YO )O

) VLQ NO =  M= Y  VLQ NO  )  FRV NO Y  FRV NO  M

 

(TXDWLRQV  DUHDIRXUSROHUHSUHVHQWDWLRQRIWKHURG7KH\UHODWHWKH YHORFLW\ DQGWKHIRUFHDWLWVEHJLQQLQJWRWKHYHORFLW\DQGIRUFHDWLWVHQG ,QYHUVLRQRI(TV  JLYHV Y )

)O VLQ NO = M= YO VLQ NO  ) O FRV NO YO FRV NO  M

D 

2IFRXUVH(TV D DQG  DOZD\VOHDGWRWKHVDPHUHVXOW ,WGHSHQGVRQWKHSDUWLFXODUSUREOHPWREHVROYHGZKHWKHU(T  RU WKH IRXUSROH UHSUHVHQWDWLRQ RI (T D  LV PRUH DGYDQWDJHRXV IRU WKH DQDO\VLV (TXDWLRQ   LV SUHIHUDEOH LI WKH UHIOHFWLRQ FRHIILFLHQWV DUH NQRZQ,IRQHLVFRQFHUQHGZLWKWKHWUDQVPLVVLRQIURPRQHDUELWUDU\V\V WHPYLDWKHURGWRDQRWKHUV\VWHPKRZHYHUWKHQWKHIRXUSROHUHSUHVHQWD WLRQJHQHUDOO\LVPRUHFRQYHQLHQW

5HVRQDQW9LEUDWLRQVRI'DPSHG%HDPV

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

Y

 

2QHPD\XVHWKLVHTXDWLRQRUWKHHTXLYDOHQWH[SUHVVLRQ Y

D 

Y FRVK N[  Y  VLQK N[  Y FRV N[  Y  VLQ N[

GHSHQGLQJ RQ ZKLFK RI WKH WZR OHDGV WR VLPSOHU PDQLSXODWLRQV 7KH IRXU XQNQRZQVYYYMYMRUYYYYUHVSHFWLYHO\PD\EHHYDOXDWHGIURP WKHIRXUERXQGDU\FRQGLWLRQV)RUH[DPSOHIRUDIUHHIUHHEHDPWKDWLVH[ FLWHGE\DIRUFH)DW[ VHH6HFWRQHREWDLQV 

0 O { 

%N ª Y  H  M NO  Y  H M NO  Y  M H  NO  Y  M H NO º¼ MZ ¬





) O {

%N ª MY  H M NO  MY  H M NO  Y  M H NO  Y  M H NO º¼ MZ ¬ 

%N ª Y   Y   Y  M  Y  M º¼ 0  {  MZ ¬



 





)  {

%N ª MY   MY   Y  M  Y  M º¼ MZ ¬

) 

%\ VROYLQJ(T  IRUWKHIRXUXQNQRZQVRQHILQGVDIWHUVRPHOHQJWK\ PDQLSXODWLRQWKDWEHQGLQJYLEUDWLRQVRIIUHHIUHHEHDPVREH\

'DPSLQJ

Y

VLQK NO FRV N O  [  VLQ N[  FRVK NO VLQ N O  [  MZ )  VLQ NO FRVK N O  [  VLQK N[  FRV NO VLQK N O  [      FRV NO FRVK NO %N

 

)RUWKHORFDWLRQ[ OWKLVUHVXOWUHGXFHVWR Y O

MZ )  VLQK NO  VLQ NO   %N   FRV NO FRVK NO

 

:KHQNO ÆLHIRUYHU\VKRUWEHDPVRQHPD\VKRZE\XVLQJDVHULHVH[ SDQVLRQLQFOXGLQJWHUPVXSWRWKHIRXUWKSRZHURI NO WKDW Y O |

)   MZ P O

D 

7KLVUHVXOWFRUUHVSRQGVWRULJLGERG\PRWLRQLQFOXGLQJURWDWLRQDQGWUDQV ODWLRQRIDULJLGEHDPH[FLWHGODWHUDOO\DWRQHHQG )LJXUH D VKRZV KRZ WKH YHORFLW\ DW WKH HQG [  O YDULHV ZLWK IUH TXHQF\DVFDOFXODWHGIURP(T  )LJXUHEVKRZVKRZWKHYHORFLW\ YDULHVDORQJWKHEHDPDWDFRQVWDQWIUHTXHQF\FRUUHVSRQGLQJWR N cO  S DV FDOFXODWHG IURP (T   ,Q WKHVH ILJXUHV WKH RUGLQDWH DJDLQ LV WKH ORJDULWKPRIWKHDEVROXWHYDOXHRIWKHYHORFLW\DQGWKHDEVFLVVDDUH N cO DQG Z  Pc  % WKHDEVFLVVDRI)LJDLVSUR N c[ UHVSHFWLYHO\%HFDXVH N c SRUWLRQDOWRWKHVTXDUHURRWRIWKHIUHTXHQF\

)LJ  )OH[XUDO YLEUDWLRQV RI D IUHHIUHH EHDP D  9DULDWLRQ ZLWK IUHTXHQF\ DW [ ODVFDOFXODWHGIURP(T  E 9DULDWLRQZLWKSRVLWLRQIRUN¶O SDV FDOFXODWHGIURP(T  

5HVRQDQW9LEUDWLRQVRI'DPSHG%HDPV

2QHPD\GHULYH(T  ±EXWQRW(T  ±FRQVLGHUDEO\PRUHUDS LGO\ZLWKWKHDLGRIWKHVRFDOOHGHLJKWSROHHTXDWLRQV>@ ,QRUGHUWRGH YHORSWKHVHHTXDWLRQVFRQYHQLHQWO\RQHPD\UHZULWH(T D LQWHUPV RIWKHIROORZLQJIXQFWLRQV  FRVK N[  FRV N[ & N[    VLQK N[  VLQ N[ 6 N[  

 FRVK N[  FRV N[ F N[    VLQK N[  VLQ N[ V N[  

7KHVHIXQFWLRQVKDYHWKHSURSHUW\WKDWGLIIHUHQWLDWLRQRIDQ\ RIWKHP JLYHV DQRWKHU IXQFWLRQ RI WKH VHW LQ WKH VHTXHQFH & N[  V N[  F N[  6 N[  & N[ ,QDGGLWLRQ&   6   V   F   ,IWKHWUDQVYHUVHYH ORFLW\DQGWKHDQJXODUYHORFLW\DUHZULWWHQDV Y[ Z[

D & N[  E 6 N[  J F N[  G V N[ N ª¬D V N[  E & N[  J 6 N[  G F N[ º¼ 

RQHPD\REVHUYHWKDWWKHYHORFLW\ Y DWWKHEHJLQQLQJRIWKHEHDPLVHTXDO WRDZKHUHDVWKHDQJXODUYHORFLW\DWWKHEHJLQQLQJRIWKHEHDPLVZ NE 7KHPRPHQWDQGWKHVKHDUIRUFHVLPLODUO\PD\EHIRXQGWREHVLPSO\UH ODWHG WR WKH FRHIILFLHQWV LQ WKH IRUHJRLQJ H[SUHVVLRQV $IWHU VRPH PDWKH PDWLFDOPDQLSXODWLRQRQHPD\GHWHUPLQHWKDW Y[ Z[ 0[ )[

   6 N[  0  F N[  )  V N[ N M: c M: N N  Y  NV N[  Z& N[  0  6 N[  )  F N[ M:

M:

M:

 Y  M: F N[  Z V N[  0  & N[  )  6 N[ N N Y  M: N6 N[  Z M: F N[  0  NV N[  )  & N[ 

Y  & N[  Z

 

ZKHUH : %N   Z %P  ,QYHUVLRQLHH[SUHVVLQJWKHSDUDPHWHUVDWWKHEHJLQQLQJRIWKHEHDPLQ WHUPVRIWKRVHDWWKHHQGPD\HDVLO\ EHDFFRPSOLVKHGKHUH DVIRUWKHIRXU SROH HTXDWLRQV  EHFDXVH WKH GHWHUPLQDQW RI WKH FRHIILFLHQWV LV HTXDO WR XQLW\LQYLHZRIUHFLSURFLW\+HUHWKHLQYHUVLRQPHUHO\OHDGVWRDFKDQJH LQWKHDOJHEUDLFVLJQRIWKH6 N[ DQGV N[ WHUPV 6HYHUDODSSUR[LPDWLRQVDUHRILQWHUHVWLQUHODWLRQWRH[SHULPHQWDOPHDV XUHPHQWV)RU N cO !SWKHYDULDWLRQZLWKIUHTXHQF\RIWKHYHORFLW\DWWKH [ OHQGRIWKHEHDPPD\EHZULWWHQDV

'DPSLQJ



Y O |

Z) %N





  FRVK N ccO  FRV N cO

 

DQGWKHVSDWLDOYDULDWLRQLQWKHUHJLRQS  N c[  N cO ±SPD\EHDSSUR[L PDWHGE\ 

Y [ |

Z) %N





FRVK N cc O  [  VLQ N c O  [  FRVK N ccO  FRV N cO

 

,W LV HDVLO\ YHULILHG WKDW WKHVH HTXDWLRQV DUH DOVR REWDLQHG E\ VHWWLQJ U UO  M LQ (T   $V KDV DOUHDG\ EHHQ PHQWLRQHG RQH PD\ DSSO\ WKHVLPSOHUUHODWLRQVRI6HFWWREHQGLQJYLEUDWLRQVRIORQJEHDPVLI RQHLQWURGXFHVWKHDSSURSULDWHUHIOHFWLRQFRHIILFLHQWV7KHGLIIHUHQFHVEH WZHHQWKHGLIIHUHQWZDYHW\SHVRQO\EHFRPH HYLGHQWLIRQHGHWHUPLQHVWKH IUHTXHQFLHV DQG WKH ORVV IDFWRU IURP WKH ZDYHQXPEHUV N c DQG N cc  )RU EHQGLQJZDYHVRQHILQG N cc  N c  K VHH(T E XQOLNHIRUORQJLWX GLQDODQGWRUVLRQDOZDYHV 7KH GDPSLQJ RI EHQGLQJ ZDYHV FDQ EH GHWHUPLQHG IURP KDOIYDOXH EDQGZLGWKPHDVXUHPHQWVH[DFWO\DVIRUORQJLWXGLQDODQGWRUVLRQDOZDYHV SURYLGHG WKH EHDP LV QRW WRR KLJKO\ GDPSHG 7KH VDPH DQDO\VLV DV ZDV FDUULHGRXWLQ6HFWLQWKLVFDVHOHDGVWR Y%   | Y PD[   M 'N cO N ccO

%HFDXVH N

 

Z  Pc  % RQHILQGVWKDWKHUH 'N Y% Y PD[



|

 § 'I ·  ¨ ¸ © K IQ ¹





 N 'I 7KXV  IQ

D 

2QHPD\REVHUYHWKDWWKHUHODWLRQ EHWZHHQEDQGZLGWKDQGORVVIDFWRUIRU EHQGLQJ ZDYHV LV WKH VDPH DV WKDW IRU ORQJLWXGLQDO DQG WRUVLRQDO ZDYHV QDPHO\ E K IQ 

 

7KHYLEUDWLRQVLQWKHYLFLQLW\RIDYLEUDWLRQQRGHREH\ WKHVDPHUHODWLRQV DVGHULYHGLQ6HFW7KHUHIRUH(TV  WR  DSSO\GLUHFWO\WR EHQGLQJZDYHVIRUORFDWLRQVRXWVLGHRIWKHQHDUILHOGUHJLRQV 7KH EHKDYLRXU RI EHDP YLEUDWLRQV LQ WKH YLFLQLW\ RI WKH YDULRXV UHVR QDQFHVLVRILQWHUHVWQRWRQO\LQUHODWLRQWRH[SHULPHQWVDQGIRUGHWHUPLQD WLRQRIWKHORVVIDFWRUEXWDOVREHFDXVHDQ\YLEUDWLRQRIDEHDPFDQEHUHS

0HDVXUHPHQWRI&RPSOH[0RGXOL

UHVHQWHG E\ D VXP RI LWV UHVRQDQW RU QDWXUDO YLEUDWLRQV 7KLV SUREOHP LV WUHDWHGLQGHWDLOLQ6HFWZKHUHLWLVVKRZQWKDWWKHYHORFLW\RIDEHDP PD\EHH[SUHVVHGDV Y [

Y QM Q [

f

¦ Z   MK  Z Q 

 Q





 

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|

Y QM Q [  Q



Y QM Q [  Q

Z  Z  MZ K

Z Q  Z Z Q  Z  MZ QK



 

,QWKHYLFLQLW\RIDUHVRQDQFHZ ZQZQZ |ZQDQGZLWKZQZ 'Z LWLVIRXQGWKDW Y [ |

Y QM Q [

   MZ K   M 'Z KZ Q  Q

ZKLFKFRUUHVSRQGVGLUHFWO\WR(T D 

0HDVXUHPHQWRI&RPSOH[0RGXOL %HFDXVH WKH FRPSOH[PRGXOXVRIHODVWLFLW\RIDPDWHULDOLVDYHU\LPSRU WDQW PHFKDQLFDO SURSHUW\ PDQ\ WHFKQLTXHV DUH DYDLODEOH IRU LWV PHDVXUH

'DPSLQJ

PHQW 2QO\ WKH PRVW LPSRUWDQW RQHV ZLOO EH GHVFULEHG KHUH ,Q HVVHQFH WKHUH H[LVW WKUHH EDVLFDOO\ GLIIHUHQW W\SHV RI PHDVXUHPHQW PHWKRGV )RU ORZIUHTXHQF\ PHDVXUHPHQWV RQ D VDPSOH RI VPDOO GLPHQVLRQV RQH PD\ FRQVLGHUWKHWHVWVDPSOHDVDVSULQJRQHWKHQQHHGVRQO\WRGHWHUPLQHWKH VDPSOH¶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

(

6 O

6 (c  K  O

 

0HDVXUHPHQWRI&RPSOH[0RGXOL

6UHSUHVHQWVWKHFURVVVHFWLRQDODUHDDQGOWKHOHQJWKRIWKHWHVWVDPSOHDQG IURP WKH SKDVH DQJOH M EHWZHHQ WKH IRUFH DQG WKH GLVSODFHPHQW RQH PD\ ILQGWKHORVVIDFWRUDV K WDQM

 

)LJ0HDVXUHPHQWRIFRPSOH[PRGXOXVYLDWKHVWUHVVVWUDLQFXUYH

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

'DPSLQJ

SODFHPHQWIRUH[DPSOHE\PHDQVRIDQHFFHQWHUVHHWKHORZHUSDUWRI)LJ 7KLVDUUDQJHPHQWDOVRSHUPLWVWKHDSSOLFDWLRQRIDVWDWLFSUHORDGZLWK RXWJUHDWGLIILFXOW\ 7KHPRVWVLJQLILFDQWSUREOHPVZLWKWKLVW\SHRIPHDVXUHPHQWDUHDVVR FLDWHGZLWKWKHDFFXUDF\WRZKLFKWKH SKDVH DQJOHPXVWEHPHDVXUHGDQG ZLWKVXSSRUWLQJWKHVDPSOH³ULJLGO\´$³ULJLGVXSSRUW´LVMXVWDVLPSRVVL EOHWRUHDOL]HDVHJDQDEVROXWHYDFXXP,QRUGHUWRPHDVXUHORVVIDFWRUV RIWKHRUGHUK |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¶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

0HDVXUHPHQWRI&RPSOH[0RGXOL

)UHTXHQF\5HVSRQVH0HDVXUHPHQWVLQD0DVV6SULQJ 0DVV5LJ 'XHWRWKHODUJHGLIILFXOWLHVLQUHDOL]LQJDULJLGVXSSRUWDVHPSOR\HGLQ)LJ LVFRPPRQO\XVHGDPHDVXUHPHQWULJZLWKG\QDPLFDOO\IUHHWHUPLQDOV 7KHVLPSOHVWIRUPRIVXFKDULJLVWKHPDVVVSULQJPDVVV\VWHP7ZRH[ DPSOHVRIVXFKDULJW\SHDUHVKRZQLQ)LJ:KHQWKHPDVVHVFDQEH FRQVLGHUHG ULJLG WKH VSULQJ HOHPHQW FRPSDUDWLYHO\ ³PDVVOHVV´ DQG WKH VXVSHQVLRQRIWKHULJLWVHOIDVZHOODVWKDWRIDQ\ VWDWLFSUHORDGG\QDPLFDOO\ IUHHWKHHTXDWLRQVRIPRWLRQUHDG V Y  Y  )   MZ V MZ P Y  Y   Y  MZ MZ P Y 

 

7KLVPHDQVWKDWDOOWKHVXVSHQVLRQVDUHPDGHH[WUHPHO\VRIWVXFKWKDWWKHUH LVQRG\QDPLFLQWHUIHUHQFHLQWKHPHDVXUHPHQWUDQJHIURPWKHVXSSRUWRU SUHORDGERG\

)LJ  0HDVXUHPHQW RI IUHTXHQF\ UHVSRQVH IXQFWLRQV D) D) RU WUDQVIHU IXQFWLRQDDLQDPDVVVSULQJPDVVV\VWHPcVXVSHQVLRQdSUHORDGDQGeSXO OH\V

)URP(T  LVREWDLQHG Y

)

MZ P  V  MZ  V P  P  Z  PP

Y

)

V  MZ  V P  P  Z  PP

 

'DPSLQJ

$V FDQ EH VHHQ WKHUH LVDQ DQWLUHVRQDQFH DW Z $ V  P ZKLFKPHDQVWKDW WKH YHORFLW\ Y YDQLVKHV $OVR D UHVRQDQFH RFFXUV DW Z 5 V P  P  P  P  $W WKRVH WZR IUHTXHQFLHV WKH G\QDPLF VWLIIQHVV FDQEHGHWHUPLQHGIURP(T  $ PRUHGLUHFWZD\WRGHWHUPLQHGWKH G\QDPLF VWLIIQHVV KRZHYHU LV REWDLQHG YLD WKH IUHTXHQF\ UHVSRQVH IXQF WLRQVD)DQGD)RUYLDWKHWUDQVIHUIXQFWLRQVDD,QWURGXFLQJD MZY LQVWHDGRIWKHYHORFLWLHVLQ(T  \LHOGV V Z  P V

  P D  )     P  P D  ) 

Z  P

V Z  P

P D   )     P  P D   ) 

 

    D  D 

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| D DOZD\V SUREOHPDWLF ZLWK UHVSHFW WR WKH ODVW IRUP VLQFH WKH GLIIHUHQFH LQ WKH GHQRPLQDWRU EH FRPHVVPDOO 7KH PHDVXUHPHQW RI PRGXOL YLD WUDQVIHU IXQFWLRQV LV GXH WR )LW]JHUDOG >@ ZKRXVHGDVSHFLILFULJ+HUHE\WKHPHFKDQLFDOTXDQWLWLHVDUHSUR SRUWLRQDO WR HOHFWULFDO UHVLVWDQFHV DFFXUDWHO\ GHWHUPLQHG IURP D EULGJH FLUFXLW

0HDVXUHPHQWRI&RPSOH[0RGXOL

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

)LJ6FKHPDWLFDUUDQJHPHQWRIWRUVLRQDOSHQGXOXP

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

'DPSLQJ /

OQ $Q  $Q  

D

)RUVPDOOGDPSLQJRQHILQGVIURP(T  WKDW  

/  S 

K

)RUODUJHGDPSLQJLQFOXVLRQRIDFRUUHFWLRQOHDGVWR K

/ S   /  S





 

%HFDXVH / FDQ EH GHWHUPLQHG UDWKHU DFFXUDWHO\ IURP PHDVXUHPHQW RYHU VHYHUDO SHULRGV WKH YLEUDWLRQ GHFD\ WHFKQLTXH LVDVLPSOHDQG UHDVRQDEO\ H[DFWPHWKRGIRUWKHORVVIDFWRUK 7KHVHFRQGTXDQWLW\RILQWHUHVWWKHWRUVLRQDOVWLIIQHVV7 RIWKHVDPSOHLV JLYHQE\ 7

S I Q



§ / 4 ¨   © S

· ¸ ¹

 

ZKHUH 4 UHSUHVHQWV WKH PDVV PRPHQW RI LQHUWLD RI WKH GLVF 7KH DEVROXWH YDOXH RI WKHVKHDUPRGXOXVLVGHWHUPLQHGIURPWKHWRUVLRQDOVWLIIQHVVDQG WKHGLPHQVLRQVRIWKHVDPSOH)RUULEERQVKDSHGWHVWVDPSOHVDVDUHRIWHQ XVHG *

*   K

7

O  E K   E  K 

 

ZKHUHOUHSUHVHQWVWKHOHQJWKKWKHZLGWKDQGEWKHWKLFNQHVVRIWKHWHVW VDPSOHZLWK K!E )RUKLJKSRO\PHUVWKH*HUPDQVWDQGDUG',1 UHFRPPHQGVULEERQGLPHQVLRQVO |PP K |PP E |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

0HDVXUHPHQWRI&RPSOH[0RGXOL

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

)LJ  0HDVXUHPHQW RI G\QDPLF VWLIIQHVV DQG ORVV IDFWRU YLD UHVRQDQFH IUH TXHQF\DQGEDQGZLGWK

7KHUHDOSDUWRIWKHPRGXOXVRIHODVWLFLW\PD\EHFDOFXODWHGIURP (c

S  I 5 P

O  6

 

ZKHUHO UHSUHVHQWVWKHWKLFNQHVVRIWKHVDPSOHDQG6LWVVXUIDFHDUHD)RUD VDPSOHUHVWLQJRQDULJLGVXSSRUWPUHSUHVHQWVWKHPDVVDWRSWKHVDPSOH )RU D WHVW VDPSOH UHVWLQJ RQ D VRIWO\ VXSSRUWHG PDVV PX DQG FDUU\LQJ D PDVVPPLVJLYHQE\ P PPX PPX FI)LJ:KHQWKHPDVV

'DPSLQJ

RIWKHWHVWVDPSOHPV FDQQRWEHQHJOHFWHGDILUVWFRUUHFWLRQFDQEHPDGHE\ DGGLQJPVWRWKHPDVVPIRUDVHWXSDVLQ)LJZKHUHDVWKHVDPSOH PDVV LV SURSRUWLRQDOO\ DGGHG WR WKH WZR PDVVHV RI WKH PDVVVSULQJPDVV VHWXS7KHORVVIDFWRULVJLYHQE\ K

E  I5

 

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± IRU H[DPSOHLQRUGHUWRLQYHVWLJDWHWKHORDGGHSHQGHQFHRIWKHEHKDYLRXURID IRDP PDW ±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³SXPSHG´ LQDQGRXW$WKLJKIUHTXHQFLHVWKHDLUFDQQRORQJHUEHPRYHGEDFNDQG IRUWK UDSLGO\ HQRXJK LW WKHQ VXIIHUV VRPH FRPSUHVVLRQ DQG PD\ PDNH D

0HDVXUHPHQWRI&RPSOH[0RGXOL

FRQVLGHUDEOH FRQWULEXWLRQ WR WKH WRWDO VWLIIQHVV )RU VDPSOHV ZLWK FORVHG FHOOVRQHFDQQRWGLVWLQJXLVKEHWZHHQWKHVWLIIQHVVRIWKHPDWHULDOLWVHOIDQG WKDWRIWKHHQFORVHGDLURQHFDQRQO\PHDVXUHWKHWRWDOVWLIIQHVVZKLFK± OLNHWKDWRIUXEEHU±GHSHQGVYHU\VWURQJO\ RQWKHJHRPHWU\ RIWKHVDPSOH 6XFKVDPSOHVPD\EHKDYHPXFKOLNHOLTXLGVLHWKH\ PD\ H[KLELWD3RLV VRQ¶VUDWLRQHDU 0HDVXUHPHQWVRQ%HDPV $VZDVGLVFXVVHGLQGHWDLOLQ&KDSWHURQEHDPVDUHHQFRXQWHUHGSULPDU LO\TXDVLORQJLWXGLQDOZDYHVWRUVLRQDOZDYHVDQGEHQGLQJZDYHV6XUIDFH ZDYHV RFFXU RQO\ DW UHODWLYHO\ KLJK IUHTXHQFLHV 7KXV RQH PD\ XVH WKH WKUHHDIRUHPHQWLRQHGW\SHVRIZDYHVIRUPHDVXUHPHQWVRQEHDPV,QSUDF WLFH RQH XVXDOO\FRQFHUQVRQHVHOIRQO\ZLWKEHQGLQJZDYHVQRWRQO\EH FDXVH WKHVH DUH PRVW LPSRUWDQW ZLWK UHVSHFW WR VRXQG UDGLDWLRQ EXW DOVR EHFDXVHWKH\FDQEHH[FLWHGPRVWHDVLO\DQGPRVW³FOHDQO\´VLQFHWKHLULP SHGDQFH LV WKH VPDOOHVW RI WKRVH RI WKH WKUHH ZDYH W\SHV 7KH DYDLODEOH EHQGLQJZDYH PHDVXUHPHQW WHFKQLTXHV KDYH EHHQ GHYHORS SULPDULO\ IRU VWXG\LQJGDPSLQJOD\HUVDQGVLPLODUQRLVHFRQWUROWUHDWPHQWV>@ 7KHVH WHFKQLTXHV ZLOO EH GLVFXVVHG LQ VRPH GHWDLO LQ WKH IROORZLQJ SDUD JUDSKV 7HFKQLTXHV SHUWDLQLQJ WR WRUVLRQDO DQG TXDVLORQJLWXGLQDO ZDYHV ZLOOEHWUHDWHGRQO\FXUVRULO\ +DOI9DOXH%DQGZLGWK %HQGLQJZDYH UHVRQDQFHV FDQ EH PHDVXUHG EHVW RQ IUHHO\ VXVSHQGHG EHDPVZLWKH[FLWDWLRQDSSOLHGWRRQH HQGDQGWKHUHVXOWLQJYLEUDWLRQVRE VHUYHGDWWKHRWKHUDVVKRZQVFKHPDWLFDOO\LQ)LJ

)LJ0HDVXUHPHQWRIIOH[XUDOUHVRQDQFHIUHTXHQF\DQGEDQGZLGWK

'DPSLQJ

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± RU RI FRXUVH E\ FDUU\LQJ RXW WKH PHDVXUHPHQWV LQ D YDFXXP $ SUDFWLFDOO\ VLPSOH DUUDQJHPHQW LV WR LQVHUW WKH WHVW VDPSOH LQ DQ XSVLGH GRZQKHOGSODVWLFEDJILOOHGZLWKKHOLXPFI>@ 2QH PD\ GHWHUPLQH WKH FRPSOH[ PRGXOXV VLPSO\ E\ PHDVXULQJ WKH UHVRQDQFHIUHTXHQFLHVIQ DQGWKHFRUUHVSRQGLQJKDOIYDOXHEDQGZLGWKVE$ W\SLFDODPSOLWXGHIUHTXHQF\FXUYHLVVKRZQLQ)LJ

0HDVXUHPHQWRI&RPSOH[0RGXOL

)LJ([SHULPHQWDOO\GHWHUPLQHGYLEUDWLRQUHVSRQVHRIDIUHHIUHHEHDP ZLWK DWKLQGDPSLQJOD\HUK |[IUHTXHQF\GHSHQGHQW 

,Q YLHZ RI (T H  WKH UHDO SDUW RI WKH EHQGLQJ VWLIIQHVV LV IRXQG IURP %c

PcO  I Q

 S



Q  





 

DQGWKHORVVIDFWRUIURP K

E  IQ

 

ZKHUH Pc UHSUHVHQWVWKHPDVVSHUXQLWOHQJWKRIWKHWHVWEHDPOUHSUHVHQWV LWVOHQJWKVXFKWKDW Pc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cc  N c  LH K |  WKH ULVH LQ OHYHO KDUGO\ EHFRPHV PRUH WKDQ  G% DOUHDG\ DW WKH WKLUGUHVRQDQFH$FRQVLGHUDWLRQRIKLJKRUGHUWHUPVVKRZVWKDWWKHUHSUH VHQWDWLRQRIEHDP YLEUDWLRQVWKURXJKDUHVRQDQFHFXUYHLHWKHXVHRI(T  LQVWHDGRI  LVIXOO\LQDSSURSULDWH IRU N cc O!7KHIUDFWLRQDO HUURULQKLVDSSUR[LPDWHO\ QKLQRWKHUZRUGVLIWKHIUHTXHQF\LQWHUYDO EHWZHHQUHVRQDQFHVLVRIWKHVDPHRUGHUDVWKHKDOIYDOXHEDQGZLGWKWKHQ RQHFDQQRORQJHUXVHWKLVPHWKRG

'DPSLQJ

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

OQ  |  I7 Z7

 

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

0HDVXUHPHQWRI&RPSOH[0RGXOL

)LJ([SHULPHQWDOO\REVHUYHGGHFD\SURFHVVQHDUDUHVRQDQFHIUHTXHQF\RI +]

,QSULQFLSOHUHYHUEHUDWLRQWLPHPHDVXUHPHQWVDUHDSSOLFDEOHXSWRUHOD WLYHO\KLJKYDOXHVRIWKHORVVIDFWRUK+RZHYHUEHFDXVHRIWKHLQHUWLDRI WKHXVXDOLQVWUXPHQWDWLRQWKLVWHFKQLTXHJHQHUDOO\LVXVHIXORQO\IRUVPDOO GDPSLQJ $WWHQXDWLRQRI9LEUDWLRQVZLWK'LVWDQFH ,QWKHOLPLWLQJFDVHRIYHU\ORQJKLJKO\ GDPSHGEHDPVZDYHSURSDJDWLRQ RFFXUVDVRQLQILQLWHO\ORQJEHDPV7KHYHORFLWLHV RUDFFHOHUDWLRQ DWWKH WZRSRLQWV[DQG[DUHUHODWHGE\ Y

YH

 M N [  [

Y H

 N cc [  [  MN c [  [

H



 

7KXVWKHSKDVHGLIIHUHQFHEHWZHHQWKHVHWZRSRLQWVLVJLYHQE\ M 

N c [  [

S [  [  O

 

ZKHUHOUHSUHVHQWVWKHZDYHOHQJWKDQGWKHUHGXFWLRQLQDPSOLWXGHREH\V OQ

Y Y 1S  ORJ  G%  N cc [  [ G% Y Y

 

%\ PHDVXULQJWKHSKDVHDQGWKHDPSOLWXGHUHGXFWLRQRQHPD\WKHUHIRUHGH WHUPLQHWKHUHDODQGLPDJLQDU\SDUWVRIWKHZDYHQXPEHU

'DPSLQJ

)RUGDPSLQJWKDWLVQRWH[WUHPHO\KLJKRQHPD\REWDLQWKHUHDOSDUWRI WKHEHQGLQJVWLIIQHVVIURP % c Z  Pc

§[ [ · Z  Pc ¨   ¸ © M  ¹

 N c



 

DQGWKHORVVIDFWRUIURP K



N cc Nc

'%c O  

 

ZKHUH '%c  UHSUHVHQWV WKH UHGXFWLRQ LQ YLEUDWLRQ OHYHO LQ GHFLEHOV SHU XQLW   OHQJWKWKDWLV [  [ '%c  ORJ «ª Y  Y »º G%  ¬ ¼ ,Q RUGHU WR FDUU\ RXW WKLV PHDVXUHPHQW RQH W\SLFDOO\ H[FLWHV WKH WHVW EHDP DW RQH HQG DQG VFDQV WKH YLEUDWLRQV DORQJ WKH URG ZLWK D PRYDEOH VHQVRUDVGHSLFWHGLQ)LJ

)LJ 6FDQQLQJRISURSDJDWLQJIOH[XUDOZDYHVIRUPHDVXUHPHQWRIZDYHOHQJWK DQGGDPSLQJ

,IRQHPRYHVWKHVFDQQLQJVHQVRUDORQJWKHEHDP DWDFRQVWDQWYHORFLW\ WKHQRQHPD\GHWHUPLQH '%c DQGWKXV N cc GLUHFWO\IURPWKHVORSHRIWKHUH FRUGHG YLEUDWLRQ OHYHO 0HDVXUHPHQW RI WKH SKDVH GLIIHUHQFH PD\ EH DF FRPSOLVKHGVLPSO\E\IHHGLQJWKHH[FLWLQJYROWDJHDQGWKHVLJQDOIURP WKH VHQVRU WR WKH WZR SHUSHQGLFXODU SDLUV RI SODWHV RI DQ RVFLOORVFRSH DQG E\ PDUNLQJRQWKHWHVWEHDPWKHSRLQWVDWZKLFKWKHSKDVHGLIIHUHQFHZLWKUH VSHFWWRWKHH[FLWDWLRQDPRXQWVWRƒRUƒ1DWXUDOO\WKHSKDVHFDQEH UHJLVWHUHGGLUHFWO\DOVRE\PHDQVRIDWZRFKDQQHOVLJQDODQDO\VHU

0HDVXUHPHQWRI&RPSOH[0RGXOL

6XVSHQVLRQRIWKHWHVWVDPSOHDQGDWWDFKPHQWRIWKHH[FLWHULVUHODWLYHO\ XQFULWLFDO IRU WKLV W\SH RI PHDVXUHPHQW EXW WKH VFDQQLQJ VHQVRU PXVW EH UHODWLYHO\ OLJKW,WLVXVXDOO\XVHIXOWRHPEHGRQHHQGRIWKHEHDPLQVDQG RU VRPH VLPLODU PDWHULDO LQ RUGHU WR UHGXFH WKH UHIOHFWLRQ RI ZDYHV IURP WKHHQGRIWKHEHDP ,IUHIOHFWLRQVDWWKHHQGDQGWKHDVVRFLDWHGVWDQGLQJZDYHVDUHQRWHOLPL QDWHGWKHQWKHDPSOLWXGHDWWHQXDWLRQLVQRWJLYHQE\ H  N cc[ EXWE\WKHH[ SUHVVLRQLQWKHQXPHUDWRURI(T  ,QWKDWFDVHWKHDPSOLWXGHPD[LPD  FRVK N cc O  [  ZKLOH WKH OLH RQ D FXUYH JLYHQ E\ ª¬FRVK N cc O  [  º¼ PLQLPDDUHJLYHQE\  VLQK N cc O  [ )URPWKHVHWZRFRQGLWLRQVRQHPD\ GHWHUPLQH N cc DQGWKXVK 7KHWHFKQLTXHRIPHDVXULQJWKHOHYHOUHGXFWLRQZLWKGLVWDQFHDORQJWKH EHDPJHQHUDOO\\LHOGVDFFHSWDEO\DFFXUDWHUHVXOWVRQO\IRUK N cO !  :LWK DJRRGVDQGWHUPLQDWLRQWKHOLPLWPD\EHVRPHZKDWORZHU2QWKHRWKHU KDQGWKHKDOIYDOXHEDQGZLGWKPHWKRGLVRQO\JRRGXSWRDERXWK N cO   VHH DOVR )LJV D DQG E ,Q WKH UDQJH EHWZHHQ WKHVH WZR OLPLWV RQH WKXVQHHGVWRYDU\WKHOHQJWKRIWKHWHVWVDPSOHVRPHZKDWLQRUGHUWREH DEOHWRXVHRQHRUWKHRWKHURIWKHVHWZRPHWKRGV 



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c OLQVWHDG RI(T  7KHVDPHHUURUHVWLPDWHVDSSO\KHUHDVEHIRUH&DUHPXVWEH WDNHQLQDOOFDVHVWKDWWKHFRUUHFWZDYHIRUPLVH[FLWHGDQGPHDVXUHG3DU WLFXODUO\ IRU LQKRPRJHQHRXV EHDPV LW LV YHU\ GLIILFXOW WR DYRLG H[FLWLQJ VHYHUDO ZDYH W\SHV VLPXOWDQHRXVO\ HYHQ LI V\PPHWU\ LV FDUHIXOO\ PDLQ WDLQHG0HDVXUHPHQWVRQTXDVLORQJLWXGLQDODQGWRUVLRQDOZDYHVDUHJHQHU DOO\XVHGIRUYHU\OLJKWO\GDPSHGPHGLDEHFDXVHRIWKHYHU\VPDOODVVRFL DWHG UDGLDWLRQ GDPSLQJ DQG DW KLJK IUHTXHQFLHV LQ YLHZ RI WKH ODUJH DVVRFLDWHGZDYHOHQJWKV

'DPSLQJ

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ÌiÀˆ>Ê

i˜ÃˆÌÞ œ`ՏÕà Ž}ɓÎÊ œv

>Ã̈VˆÌÞ É“ÓÊ

-…i>À *œˆÃܘ½ œ`ՏÕà ÃÊ,>̈œ ɓÓ

V “ÉÃ

V/ “ÉÃ

œÃÃÊ>V̜À iÝÕÀ>Ê

,i“>ÀŽÃ

œ˜}ˆÌÕ`ˆ ˜>

Õ“ˆ˜ˆÕ“Ê ÓÇää i>`Ê ££Îää

ÇÓ Ê£ä™Ê £Ç Ê£ä™Ê

ÓÇ Ê£ä™Ê ÈÊÊ£ä™Ê

ä°Î{Ê ä°{ÎÊ

xÓää £Óxä

Σää ÇÎä

ä°Îq£ä Ê£ä ‡Ó xqÎä Ê£ä £q{ Ê£ä‡Î

Ê£ä‡{ ÊÓÊÊ£ä‡Ó

Àœ˜Ê -ÌiiÊ œ`Ê

œ««iÀÊ

Çnää Çnää £™Îää n™ää

ÓääÊ Ê£ä™Ê Ó£äÊ Ê£ä™Ê nä Ê£ä™Ê £ÓxÊ Ê£ä™Ê

ÇÇ Ê£ä™Ê ÇÇ Ê£ä™Ê Ón Ê£ä™Ê {È Ê£ä™Ê

ä°ÎäÊ ä°Î£Ê ä°{ÓÎÊ ä°ÎxÊ

xäxä x£ää Óäää ÎÇää

Σää Σää £Óää ÓÎää

‡{ £q{ Ê£ä ‡{ ä°ÓqΠʣä ÊÎÊʣ䇇{Ê ‡Î ÓÊÊ£ä ÓqÇ Ê£ä‡{

Ó‡È Ê£ä‡{

Q{°£™]ÊÓ£]ÊÓ{RÊ

Ê ÊÓÊÊ£ä‡Î

>}˜ÃˆÕ“Ê À>ÃÃÊ ˆVŽiÊ -ˆÛiÀÊ ˆÃ“ÕÌ…Ê  Ș}iÊVÀÞÃÌ>Ê Q{°Ó{R Q{°£™R Q{°Ó{R Q{°ÓÓ]ÊÓÎR Q{°Ó{R Q{°Ó{R Q{°Ó{R

‡x

ä°Óq£ Ê£ä‡Î Ê{ÊÊ£ä‡{

Ê£ä‡{ Ê£ä‡Î Ê£ä‡Î Î Ê£ä‡Î ÊnÊÊ£ä‡{ ÊÎÊÊ£ä‡{ ÊÓä Ê£ä‡{

Q{°£™]ÊÓä]ÊÓ{R Q{°£™RÊV…i“° «ÕÀi Q{°£™Rʘ̈“œ˜

0HWDOV $OWKRXJK WKH GHQVLW\ PRGXOXV RI HODVWLFLW\ DQG 3RLVVRQ¶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

'DPSLQJ

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±@ DUHVWLOORQJRLQJEXWLQWHUSUHWDWLRQRIWKHKHDWFRQ GXFWLRQSKHQRPHQDLVUHODWLYHO\ZHOOLQKDQG,WPHUHO\ LQYROYHVWHPSHUD WXUHFKDQJHVSURGXFHGE\ VWUDLQVDQGWKHUHVXOWLQJKHDWFRQGXFWLRQDQGWKH FODVVLFDOHTXDWLRQRIKHDWFRQGXFWLRQDQGOLQHDUWKHUPRG\QDPLFUHODWLRQV DSSO\ 8SRQ FDUU\LQJ RXW WKH FRUUHVSRQGLQJ DQDO\VLV >@ LW LV IRXQG WKDW WKH GDPSLQJGXHWRKHDWFRQGXFWLRQDSSHDUVDVDUHOD[DWLRQSURFHVV7KHUHE\ WKHUHOD[DWLRQWLPHVDUH W W



O  S &Y U  /  K  S &Y U  /

 

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±LVRWKHUPDOFDVH±EXWWRRORQJWRUHDO L]HLGHDODGLDEDWLFFRQGLWLRQVWKHQWKHUHLVDVPDOOHQHUJ\UHPDLQGHUZKLFK VHUYHV WR KHDW WKH PDWHULDO 7KH IDFW WKDW WKH ZDYHOHQJWK DSSHDUV LQ (T  RQWKHRQHKDQGDQGWKHSODWHWKLFNQHVVRQWKHRWKHUVWHPVIURPWKH

([SHULPHQWDO'DWD

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³QRUPDO´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

'DPSLQJ

)LJ 'HSHQGHQFHRIPRGXOXVRIHODVWLFLW\DQGORVVIDFWRURISRO\YLQ\OFKOR ULGHRQIUHTXHQF\DQGWHPSHUDWXUH>@

)LJXUHLOOXVWUDWHVWKHIDFWWKDWWKHORVVIDFWRULVJUHDWHVWZKHUHWKH PRGXOXV RI HODVWLFLW\ FKDQJHV PRVW UDSLGO\ ZLWK IUHTXHQF\ 7KLV VRUW RI EHKDYLRXULVH[DFWO\ZKDWRQHH[SHFWVIRUDUHOD[DWLRQSURFHVVLQYLHZRI (TV   DQG   FI )LJ D DQG 7DEOH  7KH IDFW WKDW LQ UHDOLW\ WKHUHDUHSUHVHQWDPXOWLWXGHRIUHOD[DWLRQSURFHVVHVDQGQRWMXVWDVLQJOH RQHGRHVQRWDIIHFWWKHIRUHJRLQJFRQVLGHUDWLRQV 6HYHUDORWKHU³ODZV´DSSO\WRDPRUSKRXVKLJKSRO\PHUVLQDGGLWLRQWR WKH UHODWLRQ EHWZHHQ WKH ORVV IDFWRU DQG WKH IUHTXHQF\ GHSHQGHQFH RI WKH PRGXOXV RI HODVWLFLW\ 7KH PRVW LPSRUWDQW RI WKHVH ODZV DUH EULHIO\ GH VFULEHGEHORZ $OOSODVWLFVKDYHDVRFDOOHG³IUHH]LQJ´SRLQWEHORZZKLFKWKHLUSURSHU WLHV DUH PXFK OLNH WKRVH RI JODVV 7KLV IUHH]LQJ SRLQW RU JODVVWUDQVLWLRQ WHPSHUDWXUHLVQRWXQLTXHO\GHILQHG,WLQFUHDVHVZLWKLQFUHDVLQJIUHTXHQF\ DVHYLGHQWLQ)LJ%HORZWKHJODVVWUDQVLWLRQWHPSHUDWXUHPRVWKLJK SRO\PHUV KDYH D PRGXOXV RI HODVWLFLW\ RI DERXW  ˜  1P DQG D UHOD WLYHO\ VPDOO ORVV IDFWRU XVXDOO\ OHVV WKDQ  3ODVWLFV WKHUHIRUH DUH QRW VXLWHGIRUGDPSLQJDSSOLFDWLRQVEHORZWKHLUWUDQVLWLRQWHPSHUDWXUHV $ERYH WKH JODVVWUDQVLWLRQ WHPSHUDWXUH WKHUH RFFXUV D PRUH RU OHVV EURDGWHPSHUDWXUHUHJLRQ±WKHVRFDOOHG³UXEEHUHODVWLF´UHJLRQ±LQZKLFK SODVWLFVEHKDYHQHLWKHUHQWLUHO\OLNHVROLGVQRUOLNHOLTXLGV%HFDXVHSODV WLFVH[KLELWUDWKHUKLJKORVVIDFWRUVLQWKLVUHJLRQVHH7DEOHLWLVWKLV UHJLRQWKDWLVEHVWIRUGDPSLQJDSSOLFDWLRQV7KHPRGXOLRIHODVWLFLW\LQWKLV WUDQVLWLRQUHJLRQDUHRIWKHRUGHURIWR1P IRU³OLQHDU´PDWHULDOV WKDW LV IRU PDWHULDOV ZLWK ORQJ QRQFURVVOLQNHG PROHFXOHV )RU FURVV OLQNHGPDWHULDOVWKHPRGXOLDUHRIWKHRUGHURI1P7KHPRGXOXVRI HODVWLFLW\RISODVWLFV ZKLFKLVRILPSRUWDQFHIRUGDPSLQJOD\HUWUHDWPHQWV PD\EHLQFUHDVHGVRPHZKDWE\WKHDGGLWLRQRIILOOHUVHJYHUPLFXOLWH%\

([SHULPHQWDO'DWD 7DEOH/RVVIDFWRUDQGPRGXOXVRIHODVWLFLW\>1P@ DWPD[LPXPGDPSLQJIRU VRPHKLJKSRO\PHUV 3RO\YLQ\OFKORULG SXUH  3RO\VW\UHQH 3RO\LVREXW\OHQH 1LWULOH5XEEHU +DUG5XEEHU 3RO\YLQ\OFKORULG ZLWK  SODVWL FL]HU

K  K K K K

   

K 

( ˜ ( ( ( (



˜ ˜ ˜ ˜

( ˜

DWƒ&DQG

+]

DWƒ&DQG DWƒ&DQG DWƒ&DQG DWƒ&DQG

+] +] +] +]

DWƒ&DQG

+]

WKLVPDWHULDOVZLWKORVVPRGXOLXSWRDERXW ( cc ( cK  1PFDQEHRE WDLQHG ,Q DGGLWLRQ RQH PD\ REWDLQ VLJQLILFDQW FKDQJHV LQ WKH WHPSHUDWXUH YDULDWLRQV RI WKH SURSHUWLHV RI SODVWLFV E\ DGGLQJ ³SODVWLFL]HUV´ 2QH PD\ WKXV H[SHFW LQ SULQFLSOH WR EH DEOH WR REWDLQ PDWHULDOV ZLWK RSWLPLVHG GDPSLQJEHKDYLRXURYHUSUHVFULEHGWHPSHUDWXUHDQGIUHTXHQF\UHJLRQVE\ XVLQJDSSURSULDWHFRPELQDWLRQVRIFURVVOLQNHGSRO\PHUVILOOHUVDQGSODV WLFL]HUV +RZHYHU ERWK H[SHULHQFH DQG WKHRUHWLFDO FRQVLGHUDWLRQV > @ VKRZWKDWWKHWHPSHUDWXUHUHJLRQLQZKLFKWKHORVVIDFWRULVKLJKWKDW LVWKH³WHPSHUDWXUH EDQGZLGWK´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

'DPSLQJ

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

'HQVLW\ NJP $VEHVWRV FRQ  FUHWH

0RGXOXV RI HODV F/,,PV WLFLW\1P ˜ ˜

/RVVIDFWRU

5HPDUNV

˜

$VSKDOW



2DN



˜ ˜ ˜ ˜

   |˜

ƒ&  6RIW %LWXPLQRXV FRQ WHQW ƒ&  6RIW %LWXPLQRXV FRQ WHQW ƒ&  +DUG %LWXPLQRXV FRQ WHQW

)LEHU PDWV LQ FOXGLQJ PDWUL[ DQG DLU VWLII QHVV )LU )HOW *\SVXPERDUG *ODVV 3UHVVHGZRRG SDQHOV 3ODVWHU &RUN /LJKWFRQFUHWH 3OH[LJODV 3RURXV FRQ FUHWH 6DQGGU\ 'HQVHFRQFUHWH 3O\ZRRG %ULFN



˜



|˜

  

˜ ˜ ˜ ˜ ˜

˜ ˜ ˜

|˜ |˜ ˜ ˜ ˜

    

˜ |˜ ˜ ˜ ˜

˜ ˜ ˜ ˜ ˜

˜  ˜ ˜ ˜

>@

|   

|˜ ˜ ˜ |˜

˜ ˜ ˜ ˜

 ˜ |˜ ˜

>@ >@ >@ >@

˜ ˜ ˜ ˜

|

>@ >@ >@

>@

3ODWHVZLWK$WWDFKHG/D\HUV

3ODWHVZLWK$WWDFKHG/D\HUV :KHUHDVSUHYLRXVGLVFXVVLRQVGHDOWZLWKPRUHRUOHVVKRPRJHQHRXVPDWH ULDOV WKH GLVFXVVLRQ WR IROORZ GHDOV ZLWK YHU\ LQKRPRJHQHRXV DUUDQJH PHQWV3ODWHVZLWKDWWDFKHGOD\HUVLQSDUWLFXODUDUHRIJUHDWLPSRUWDQFHIRU SUDFWLFDOQRLVHFRQWURODSSOLFDWLRQV7\SLFDOFRQILJXUDWLRQVXVXDOO\ FRQVLVW RIDEDVHSODWH XVXDOO\RIPHWDO WRZKLFKDUHDWWDFKHGRQHRUPRUHOD\HUV RIDYLVFRHODVWLF HJDKLJKSRO\PHU GDPSLQJPDWHULDODQGSRVVLEO\ DG GLWLRQDO PHWDO SODWHV 6XFK V\VWHPV HVVHQWLDOO\ UHVXOW LQ D ³GLYLVLRQ RI OD ERXU´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

 (cG  (c G  H 





 G[  (cG  (c G   G[

 

ZKHUH[UHSUHVHQWVWKHPD[LPXPGLVSODFHPHQWGXULQJDSHULRG (c DQGG UHSUHVHQWWKHUHDOSDUWRIWKHPRGXOXVRIHODVWLFLW\DQGWKHWKLFNQHVVRIWKH EDVLFSODWHDQG (c DQGGUHSUHVHQWWKHVHSURSHUWLHVRIWKHDWWDFKHGOD\HU +HUHLQDQGLQWKHIROORZLQJLVFRQVLGHUHGWKHHQHUJ\SHUXQLWDUHDDQGWR DYRLGFRQIXVLRQ:LVLQWURGXFHG ,IWKHORVVHVRIWKHEDVHSODWHDUHQHJOLJLEOHWKHQLQYLHZRI(T  WKH HQHUJ\WUDQVIRUPHGLQWRKHDWSHUF\FOHRIYLEUDWLRQPD\EHIRXQGWRREH\ 

:GLVV

SK (c G 

G[  G[

7KXVWKHORVVIDFWRUK/IRUORQJLWXGLQDOORDGLQJLVJLYHQE\

D 

'DPSLQJ

K/

:GLVV S :5

K

(c G   (cG  (c G 

 

$VRQHPD\ QRWHWKLVORVVIDFWRULVSURSRUWLRQDOWRWKHORVVPRGXOXV K (c  )RUWKHEHVWPDWHULDONQRZQVRIDUWKLVTXDQWLW\KDVDYDOXHRIDERXW 1P)RUVKHHWVWHHOZLWK( ˜1PRQHILQGVWKDWIRUG GWKH ORVV IDFWRU DPRXQWV RQO\ WR K/ |  ˜  7KXVTXDVLORQJLWXGLQDOZDYHV DUHYHU\GLIILFXOWWRGDPSXQOHVVWKH\FDQEHFRQYHUWHGLQWRRWKHUW\SHVRI ZDYHV ,QWKHFDVHRIEHQGLQJZDYHVZKLFKLVFRQVLGHUDEO\ PRUHLPSRUWDQWIRU VRXQGUDGLDWLRQWKHDWWDFKHGOD\HUDOVRGHIRUPVSULPDULO\LQH[WHQVLRQDV VKRZQLQ)LJ,I[0 UHSUHVHQWVDGLVSODFHPHQWDWWKHFHQWUHOLQHRIWKH DWWDFKHGOD\HUWKHQWKHHQHUJ\SHUXQLWDUHDWUDQVIRUPHGLQWRKHDWLQRQH F\FOHLVJLYHQE\ 

:GLVV

SK (c G 

G[ 0  G[

 

)LJ)OH[XUHRIEHDPZLWKYLVFRHODVWLFOD\HU

7KHUHYHUVLEOHPHFKDQLFDOHQHUJ\LQEHQGLQJDFFRUGLQJWR(T E  PD\EHH[SUHVVHGDV :5

 G K %  G[ 





 wE %   w[

D 

3ODWHVZLWK$WWDFKHG/D\HUV

ZKHUH% UHSUHVHQWVWKHEHQGLQJVWLIIQHVVRIWKHFRPELQHGV\VWHPFRQVLVW LQJRIWKHSODWHDQGWKHDWWDFKHGOD\HUDQGEUHSUHVHQWVWKHDQJOHRIIOH[ XUH,ID GHQRWHVWKHGLVWDQFHIURPWKHQHXWUDOILEHUWRWKHFHQWUHOLQHRIWKH DWWDFKHG OD\HU WKHQ [0  DE 7KXV WKH ORVV IDFWRU K% IRU EHQGLQJ ZDYHV REH\V K%

:GLVV S :5

K

(c G  D   %

 

7KLV H[SUHVVLRQ FRQWDLQV WKH WZR TXDQWLWLHV D DQG % WKDW PXVW VWLOO EH HYDOXDWHG$VDQDSSUR[LPDWLRQ>@RQHPD\WDNHD  GG WKDWLV RQHPD\DVVXPH D WREHDSSUR[LPDWHO\HTXDOWRWKHGLVWDQFHIURPWKHPLG GOHRIWKHEDVLFSODWHWRWKHPLGGOHRIWKHDWWDFKHGOD\HUFI6HFW 2QHPD\DSSUR[LPDWH%DV %|

(G  ( G  D   

D 

ZKHUHWKHVHFRQGWHUPLVRILPSRUWDQFH RQO\IRUUHODWLYHO\WKLFNDWWDFKHG OD\HU G!G  )LJXUHVKRZVWKHGHSHQGHQFHRIWKHORVVIDFWRUUDWLR K%KRQWKH WKLFNQHVVUDWLREDVHGRQWKHH[DFWDQDO\VLVRI2EHUVW>@$OVRVKRZQLQ WKLVILJXUHDUHSRLQWVIRU (c  (  ˜ &OHDUO\(T  LVDYHU\ JRRG DSSUR[LPDWLRQ

)LJ  /RVV IDFWRU RI EHDPV RU SODWHV ZLWK VLQJOH YLVFRHODVWLF OD\HU DIWHU 2EHUVW 3RLQWV VKRZ YDOXHV FDOFXODWHG IURP WKH DSSUR[LPDWH UHODWLRQV (TV   DQG D (c  (  ˜  

'DPSLQJ

,W LV LQWHUHVWLQJ WR QRWH WKDW WKH ORVV IDFWRU IRU EHQGLQJ ZDYHV LV LQGH SHQGHQWRIIUHTXHQF\DVLWLVIRUORQJLWXGLQDOZDYHVEXWWKDWLWLVFRQVLG HUDEO\KLJKHUIRUEHQGLQJZDYHV:KHUHDVIRU (cK  1PG GDQG ( ˜1PLWZDVSUHYLRXVO\IRXQGWKDWWKHORQJLWXGLQDOZDYHORVV IDFWRULVK/ |˜IRUWKHVDPHFRQILJXUDWLRQDQGPDWHULDOVWKHORVVIDF WRUIRUEHQGLQJZDYHVWXUQVRXWWREHK% |˜±WKXVPRUHWKDQ WLPHVJUHDWHU ([SHULPHQWDO PHDVXUHPHQWV PDGH RQ SODWHV ZLWK DWWDFKHG OD\HUV KDYH EHHQIRXQGWREHLQJRRGDJUHHPHQWZLWK)LJ7KHUHIRUHWKHVHUHOD WLRQVPD\DOVREHXVHGIRUWKHGHVLJQRIGDPSLQJWUHDWPHQWV$VRQHPD\ QRWH WKH WRWDO GDPSLQJ LQFUHDVHV ZLWK LQFUHDVLQJ YDOXHV RI WKH SURGXFW (c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³VSDFHU´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

)LJ6SDFLQJOD\HUIRULQFUHDVLQJWKHHIIHFWRIDYLVFRHODVWLFOD\HU

3ODWHVZLWK$WWDFKHG/D\HUV

3ODWHVZLWK0XOWL/D\HU7UHDWPHQWV ,WKDVEHFRPHFRPPRQSUDFWLFHWRGDPS SODWHOLNHVWUXFWXUHVE\ DWWDFKLQJ D OD\HU RI YLVFRHODVWLF PDWHULDO DQG SODFLQJ DWRS WKDW OD\HU D VHFRQGDU\ ³FRQVWUDLQLQJ´SODWHRIVWUXFWXUDOPDWHULDO6XFKDUUDQJHPHQWVDUHRIWHQRE WDLQHGLQSUDFWLFHE\XVHRIVRFDOOHG³GDPSLQJWDSH´ DVHOIDGKHVLYHUHOD WLYHO\WKLFNPHWDOIRLOZKHUHWKHDGKHVLYHLVDYLVFRHODVWLFPDWHULDO RULQ VDQGZLFKFRQILJXUDWLRQVZKLFKLQFRUSRUDWHDQLQWHUOD\HURIDKLJKO\GLVVL SDWLYH SODVWLF PDWHULDO $OVR FRQFHLYDEOH DUH PDQ\ RWKHU ³FRQVWUDLQHG OD\HU´FRQILJXUDWLRQVZLWKRUZLWKRXWVSDFHUVVHH)LJ,QDOORIWKHVH FRQVWUDLQHGOD\HUFRQILJXUDWLRQVWKHYLVFRHODVWLFPDWHULDOLVVXEMHFWHGSUL PDULO\ WR VKHDU ORDGLQJ VHH )LJ   FDXVHG E\ WKH FRYHULQJ SODWH H[WHQGLQJOHVVWKDQWKHXSSHUVXUIDFHRIWKHEDVLFSODWH7KHDVVRFLDWHGGLV SODFHPHQWGLIIHUHQFH[ JLYHVULVHWRWKHVKHDUDQJOHJ [G%\DQDORJ\ WR(T  RQHILQGVWKDWWKHHQHUJ\ DJDLQSHUXQLWDUHD WKDWLVFKDQJHG LQWRKHDWZLWKLQWKHYLVFRHODVWLFOD\HULQRQHF\FOHLVJLYHQE\ 

:GLVV

SK *c G  J 



SK*c G 

[  G

 

ZKHUH*UHSUHVHQWVWKHVKHDUPRGXOXVRIWKHPLGGOHOD\HU 7KHUHYHUVLEOHVWUDLQHQHUJ\KHUHDVSUHYLRXVO\LVJLYHQE\ :5

 GE %  G[



   %N E  

 

ZKHUH% UHSUHVHQWVWKHIOH[XUDOVWLIIQHVVRIWKHHQWLUHFRPSRVLWHSODWHDQG LVPDGHXSRIVHYHUDOSDUWVDVZLOOEHGLVFXVVHGODWHU

)LJ9DULRXVFRQILJXUDWLRQVLQZKLFKYLVFRHODVWLFPDWHULDOVDFWLQVKHDU

'DPSLQJ

7KH [GLIIHUHQWLDWLRQ LQGLFDWHG LQ (T   PD\ EH FDUULHG RXW HDVLO\ IRUWKHFDVHZKHUHWKHHQWLUHFRPSRVLWHSODWHYLEUDWHVRQO\LQIOH[XUHDQG ZKHUH WKH ZDYHOHQJWKV RI WKH PRWLRQV RI WKH WZR SODWHV DQG RI WKH LQWHU OD\HUDOODUHWKHVDPH)RUWKHFDVHRISODQHZDYHVIURPZKLFKRQHPD\ REWDLQ DQ\ GHVLUHG ZDYHIRUP E\ VXSHUSRVLWLRQ WKH [GHSHQGHQFH RI DOO PRWLRQVLVJLYHQE\ HMN[ +HUH N  Z  Pc  % LVWKHEHQGLQJZDYHQXPEHU ZKLFKGHSHQGVRQWKHWRWDOPDVVDQGWRWDOEHQGLQJVWLIIQHVVRIWKHFRPSRV LWHSODWH 7KHORVVIDFWRURIDFRPSRVLWHSODWHLQZKLFKWKHHQHUJ\ORVVHVDUHGXH SULPDULO\ WR VKHDU RI D YLVFRHODVWLF LQWHUOD\HU PD\ EH IRXQG IURP (TV    DQG  WREHJLYHQE\ K K



*c G  J  %N  E





 

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

[

D E  [ 

 

2QHPD\GHWHUPLQH[IURPWKHIRUFH*JWKDWDFWVRQWKHFRYHUSODWHDQG WKHUHVWRULQJIRUFHLQWKLVSODWHZKLFKH[WHQGVORQJLWXGLQDOO\%\ UHSODFLQJ WKHLQHUWLDIRUFHLQ(T  E\WKHVKHDUIRUFHDQGXVLQJ(T  RQHRE WDLQV ( G

G [ G[ 

* J  

ZKHUH(UHSUHVHQWVWKHPRGXOXVRIHODVWLFLW\RIWKHFRYHULQJSODWH

 

3ODWHVZLWK$WWDFKHG/D\HUV

)LJ'HIRUPDWLRQRIYLVFRHODVWLFOD\HUEHWZHHQWZRSODWHV

6LQFHGLIIHUHQWLDWLRQZLWKUHVSHFWWR[ DJDLQLVHTXLYDOHQWWRPXOWLSOLFD WLRQE\ NRQHILQGVIURP(TV  DQG  WKDW J E

D G

  *  ( G G  N 

 

1RWH WKDW WKH FRPSOH[ PRGXOXV *  *c   MK  DSSHDUV DERYH DQG QRW WKHUHDOSDUW *c RIWKHVKHDUPRGXOXVDVLQ(TV  DQG  %\VXE VWLWXWLQJ(T  LQWR  WKHORVVIDFWRURIWKHFRPSRVLWHSODWHFDQEH ZULWWHQDV K K

( G D  J G %     MK J G





 

LQWHUPVRIDVRFDOOHG³VKHDUSDUDPHWHU´JGGHILQHGDV JG

*c  ( G G  N 

 

ZKHUHLQGH[GGHQRWHVWKDWWKHFRYHULQJSODWHLVWKLQ7KHVKHDUSDUDPHWHU DOVRPD\EHH[SUHVVHGDV JG

*c ( G G Z

% Pc

*cO   S  ( G G 

D 

'DPSLQJ

ZKHUH O UHSUHVHQWV WKH EHQGLQJ ZDYHOHQJWK RI WKH FRPSRVLWH SODWH 2QH PD\QRWHWKDWJGDQGWKHUHIRUHDOVRWKHORVVIDFWRURIWKHFRPSRVLWHSODWH GHSHQGVRQWKHIUHTXHQF\,QFRQWUDVWWKHORVVIDFWRURIDSODWHZLWKDVLQ JOH GDPSLQJ OD\HU ZDV IRXQG WR EH IUHTXHQF\ LQGHSHQGHQW 7KH YDULDWLRQ RIWKHORVVIDFWRUZLWKIUHTXHQF\DVREWDLQHGIURP (T  LVVKRZQLQ )LJ$VRQHPD\REVHUYHWKHUHRFFXUVDPD[LPXP LQWKHGDPSLQJ 7KHIUHTXHQF\DWZKLFKWKLVPD[LPXPLVREWDLQHGLVIRXQGWRREH\ I PD[

  *c   K S ( G G 

%  F/G *c |   K  Pc  G  G (

 

ZKHUH F/, UHSUHVHQWVWKHORQJLWXGLQDOZDYHVSHHGLQWKHEDVHSODWH

)LJ)UHTXHQF\GHSHQGHQFHRIORVVIDFWRURIFRQVWUDLQHGOD\HUV\VWHP

6LQFH F/, |  ˜ PVHFIRUPHWDOSODWHVWKHVKHDUPRGXOXVRIWKHLQWHU OD\HUPXVWEHPXFKVPDOOHUWKDQWKHPRGXOXVRIHODVWLFLW\RIWKHFRYHULQJ SODWHLIWKHIUHTXHQF\FRUUHVSRQGLQJWRWKHGDPSLQJPD[LPXP LVWROLHLQ WKHWR+]IUHTXHQF\UHJLRQRUSULPDU\ LQWHUHVW7KHGHFUHDVHLQ WKHGDPSLQJRQERWKVLGHVRIWKHPD[LPXPRFFXUVUHODWLYHO\VORZO\VXFK WKDWWKH³KDOIYDOXHEDQGZLGWK´DPRXQWVWRDERXWRFWDYHV

3ODWHVZLWK$WWDFKHG/D\HUV

7KHUH LV DQRWKHU LPSRUWDQW GLIIHUHQFH EHWZHHQ VLQJOHOD\HU DQG PXOWL OD\HU GDPSLQJ WUHDWPHQWV LQ DGGLWLRQ WR WKH IUHTXHQF\ GHSHQGHQFH GLV FXVVHGDERYH:KHUHDVIRUDVLQJOHOD\HUWUHDWPHQWWKHGDPSLQJLVSURSRU WLRQDO WR WKH SURGXFW K (c  VHH (T   WKH GDPSLQJ RI PXOWLOD\HU WUHDWPHQWVGRHVQRWYDU\DVWKHSURGXFW K *c ,QIDFW(T  LQGLFDWHV WKDWWKHVKHDUPRGXOXV *c GRHVQRWDIIHFWWKHPDJQLWXGHRIWKHORVVIDFWRU PD[LPXP *c RQO\ GHWHUPLQHVWKHIUHTXHQF\DWZKLFKWKLVPD[LPXPRF FXUV ,Q VHOHFWLQJ YLVFRHODVWLF PDWHULDOV KHUH RQH WKXV PD\ SULPDULO\ LJ QRUHWKHPDJQLWXGHRI *c DQGRQO\FKRRVHDPDWHULDOZLWKWKHODUJHVWSRV VLEOHORVVIDFWRUK(YHQUHODWLYHO\VRIWSODVWLFVZLWKK!PD\EHXVHG HIIHFWLYHO\VHH7DEOH 7KHPD[LPXPGDPSLQJREWDLQHGDWWKHRSWLPXPIUHTXHQF\LVJLYHQE\ KRSW

( G D  K  %   K 





 

)RU WKLFN EDVH SODWHV IRU ZKLFK % | ( G DQG WKLQ FRYHULQJOD\HUV IRUZKLFKG!!GG!!GDQGD | G(T  UHGXFHVWR KRSW

K  ( G   (G     K  





D 

7KHJUHDWHVWORVVIDFWRUREWDLQDEOHZLWKK  ODUJHUYDOXHVDUHH[WUHPHO\ UDUH  LV DERXW  ( G( G %\ FKRRVLQJ DSSURSULDWH GLPHQVLRQV DS SUR[LPDWHO\RIWKLVLVREWDLQHGRYHUDUDWKHUZLGHEDQGRIIUHTXHQ FLHV7KH³DSSURSULDWHGLPHQVLRQV´KRZHYHUYDU\IRUGLIIHUHQWEDVHSODWHV DQGPXVWEHFDOFXODWHGVHSDUDWHO\IRUHDFKVSHFLDOFDVH $QLQFUHDVHLQWKHWHPSHUDWXUHUHVXOWVLQDGHFUHDVHLQ *c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

'DPSLQJ  K

%  % §   ·  ¨ ¸  D © (G ( G ¹

J

*c §   ·  ¸  ¨ G  N © (G ( G ¹

 

DQG D 

UHVSHFWLYHO\WKHODWWHURIZKLFKLVDQDORJRXVWRWKDWLQ(T  +HUHLQ % DQG % DUH WKH XQDVVHPEOHG EHQGLQJ VWLIIQHVVHV RI WKH EDVH DQG WRS SODWHVUHVSHFWLYHO\DQGD | G GG WKHGLVWDQFHWRWKHQHXWUDOOD\HU :LWKWKHVHSDUDPHWHUVWKHORVVIDFWRURIWKHV\VWHPUHDGV K K

JK 

    MK J  JK ª¬ J   K º¼



 

6LQFH *c D  G  N  %  %

JK

D 

WKH DQDORJ\ WR   LV HYLGHQW 6LPLODUO\ WKH IUHTXHQF\ IRU GDPSLQJ PD[LPXPLVREWDLQHGE\LQWURGXFLQJ 

J RSW



 

  K   K

DQGWKHPD[LPXPE\ KRSW

K

KJRSW    JRSW  KJRSW



D 

,WLVUHDGLO\FRQILUPHGWKDWWKHH[SUHVVLRQVIRUDWKLQWRSSODWHDUHUHJDLQHG IRUK )RUWKHDSSOLFDWLRQRIWKHH[SUHVVLRQVLQ  DQG  WKHUHLVRQH GLIILFXOW\ZKLFK UHODWHV WREHQGLQJVWLIIQHVVRIWKHDVVHPEOHGV\VWHPUH TXLUHGIRUWKHEHQGLQJZDYHQXPEHULQ(T D $OWKRXJKWKHFRPSOH[ EHQGLQJVWLIIQHVVFDQEHH[SOLFLWO\ZULWWHQDV %

§

%  % ¨¨  ©

JK · ¸ J   J ¸¹

J   MK 

 

WKHGLIILFXOW\LVQRWUHPRYHGVLQFHJLVUHTXLUHGLQYROYLQJNZKLFKLQWXUQ GHSHQGVRQ%,WZRXOGEHSRVVLEOHWRFDOFXODWHNGLUHFWO\ VHH6HFW 

3ODWHVZLWK$WWDFKHG/D\HUV

EXW WKLV LPSOLHV KLJK RUGHU SRO\QRPLDOV DQG KHQFH DQ LWHUDWLYH SURFHGXUH DSSHDUV HDVLHU 7KLV PHDQV WKDW N LV DSSURDFKHG E\ XVLQJ % %% RU WKHEHQGLQJVWLIIQHVVIURP J JRSWDVWKHLQLWLDOYDOXH)RUWKHVSHFLDOFDVH RIWZRHTXDOSODWHVLH( (DQGG G!!GWKHJHRPHWU\SDUDPHWHU LVDSSUR[LPDWHO\ K |DQGWKHQFH KRSW

K

      K

  

 

)LJ([SHULPHQWDOO\REVHUYHGORVVIDFWRUVRIDVDQGZLFKSODWHDQGRIDSODWH ZLWKDVLPSOHOD\HU GG | 

)LJ /RVVIDFWRUVIRUDGDPSLQJOD\HUFRQVWUDLQHGE\DVHJPHQWHGWRSSODWH * §   ·  3DUDPHWHU D  ¨ ¸ G  © (G (G  ¹

'DPSLQJ

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

3ODWHVZLWK$WWDFKHG/D\HUV

7RDFKLHYHWKLVLWLVVXLWDEOHWRHPSOR\+DPLOWRQ¶VSULQFLSOHDVZDVGRQHLQ 6HFWVDQG 8SRQDVVXPLQJWKDWWKHPRWLRQVRIWKHWZROD\HUV\VWHPGHSLFWHGLQ)LJ FDQEHGHVFULEHGE\PHDQVRIWKHORQJLWXGLQDOPRWLRQ[/RIWKHSODQH \   WKH WUDQVYHUVDO PRWLRQ [ DQG D URWDWLRQ E WKH PRWLRQ LQ WKH [ GLUHFWLRQ[FDQEHZULWWHQDV [ [ /  \ E 



ZKLFKLVYDOLGIRUDQ\SODQH\7KHDFWXDOSRVLWLRQRIWKHSODQH\ ZLWK UHVSHFWWRWKHLQWHUIDFHEHWZHHQWKHWZROD\HUVLVQRWVSHFLILHG,WLVDUEL WUDU\ DQG QHHG QRW FRLQFLGH ZLWK WKH QHXWUDO OD\HU 7KLV PHDQV KRZHYHU WKDW WKH GLVSODFHPHQW [/ DOVR GHSHQGV RQ WKH FKRLFH RI WKH SODQH \   ZKLFKZLOOEHIXUWKHUGLVFXVVHGVXEVHTXHQWO\

)LJ  1RWDWLRQV IRU D WZROD\HU V\VWHP ZLWK GHQVLWLHV U U DQG @DQGDOPRVWDOLNH WKDW LQ (T D  7KH H[SUHVVLRQ   FDQ EH XVHG WR FRPSXWH WKH EHQGLQJZDYHQXPEHUDQGWKHORVVIDFWRUIRUFRPSOH[@ FDQ EHDFKLHYHGE\OHWWLQJ(YDQLVK7KLVPHDQVWKDWWKHLQWHUPHGLDWHOD\HULV DVVXPHGWREHYHU\VRIWLQFRPSUHVVLRQ,WLVDVVXPHGPRUHRYHUWKDWDOO LQHUWLD WHUPV H[FHSW ZP( FDQ EH QHJOHFWHG LH 0  0   DQG U U +HUHE\WKHWZRHTXDWLRQVLQ  FDQEHGHFRXSOHGE\ VH OHFWLQJKVXFKWKDW

'DPSLQJE\PHDQVRI5HVRQDQW6\VWHPV

D  D 

DJ  D 

 

ZKLFKFDQEHVKRZQWREHHTXLYDOHQWWR K

(G   J  ( G J G  G   ª¬ (G   J  ( G J º¼



 

ZKHUH J  * (GGN  (TXDWLRQ   SUHVHQWV WKH SRVLWLRQ RI WKH QHXWUDOOD\HURIDWKUHHOD\HUEHDPVWUXFWXUH:LWKWKHDSSUR[LPDWLRQVLQ WURGXFHGDERYH(T  \LHOGV ª  (G N  º Ö  « N ( (  Z P(  » [ /1)    J ¼ ¬  ­°  ª (G( G § G G · º ½° Ö  ®Z P(  N « %  %  J ¨   G  ¸ » ¾[  (G   J  ( G J ©   ¹ »¼ °¿ «¬ °¯

  SÖ $

7KHILUVWRIWKHWZRHTXDWLRQVGHVFULEHVWKHTXDVLORQJLWXGLQDOZDYHDWWKH QHXWUDOOD\HURIWKHWKUHHOD\HUHGEHDP7KHVHFRQGGHVFULEHVWKHEHQGLQJ ZDYH ZKHUHE\ WKH H[SUHVVLRQZLWKLQEUDFNHWVFRUUHVSRQGVWRWKHEHQGLQJ VWLIIQHVV LQ   +HQFH WKH DSSUR[LPDWLRQ LQWURGXFHG UHFRYHU WKH VLP SOLILHGIRUPXODHRI6HFW ,WVKRXOGEHSRLQWHGRXWWKDW  LPSOLHVWKHVROXWLRQVRIRQHVHFRQG DQGRQHWKLUGRUGHUHTXDWLRQLQNVLQFHDOVRJGHSHQGVRQN&RPPRQO\ DQLQLWLDOYDOXHLVLQVHUWHGLQJDQGDQLWHUDWLYHSURFHGXUHLVXVHG

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c

V c [ 6  [ 0 

 

'DPSLQJ

ZKHUH ) c GHQRWHVWKHH[FLWLQJIRUFH[6DQG[0UHSUHVHQWWKHGLVSODFHPHQWV RIWKHWZRHQGVRIWKHVSULQJDQG V c GHQRWHVWKHFRPSOH[VSULQJVWLIIQHVV 6LQFHDOOVSULQJVKHUHDFWLQVKHDURQHPD\ZULWH Vc * )

E O

E *)c   MK)  O

 

ZKHUH*) UHSUHVHQWVWKHVKHDUPRGXOXVK)WKHORVVIDFWRURIWKHVSULQJPD WHULDOOWKHGLVWDQFHEHWZHHQWKHEHDP DQGHDFKDWWDFKHGPDVVDQGEWKH ZLGWKRIWKHVSULQJPDWHULDO

)LJ5RGZLWKDWWDFKHGGDPSHGVSULQJPDVVV\VWHPV

)URP1HZWRQ¶VODZRQHILQGVWKDWWKHPRWLRQRIWKHPDVVREH\V )c

Z  Pc[ 0 

 

ZKHUHWKHIRUPRIHMZWLVXQGHUVWRRGIRUWKHPRWLRQ,IRQHQRZLQWURGXFHV WKHYHORFLW\ Y6 MZ[6RQHREWDLQV )c

MZ PcY6  Z   Z

D 

ZKHUH 

Z

Vc Pc

*)c E   MK) Z   MK) OPc

E 

UHSUHVHQWV WKH VTXDUH RI WKH FRPSOH[ QDWXUDO IUHTXHQF\ RI HDFK VSULQJ PDVVV\VWHP ,QRUGHUWRGHWHUPLQHWKHYLEUDWRU\PRWLRQRQHRQO\QHHGVWRDSSO\(TV  DQG  DIWHUH[WHQGLQJWKHVHWRLQFOXGHWKHIRUFHDVJLYHQE\(T D  7KH ORQJLWXGLQDO PRWLRQ RI D EHDP ZLWK DWWDFKHG VSULQJV DQG PDVVHVWKXVREH\V

'DPSLQJE\PHDQVRI5HVRQDQW6\VWHPV



G) G[

MZU 6Y6 

GY  (6 6 G[

MZ Pc Y6  Z   Z

 

MZ ) 

'LIIHUHQWLDWLRQZLWKUHVSHFWWR[WKHQOHDGVWR  § G  Y6 Pc Z    MK)  U   Z  ¨ ( ¨© U 6 Z   Z   MK) Z  G[ 

· ¸¸ Y6 ¹



 

$V RQH ZRXOG H[SHFW IRU Pc  WKLV HTXDWLRQ UHGXFHV WR WKH XVXDO ZDYH HTXDWLRQ 7KH DWWDFKHG VSULQJV DQG PDVVHV UHVXOW LQ FKDQJLQJ WKH ZDYHQXPEHU ZKLFKIRUWKHRULJLQDOXQWUHDWHGEHDPLVJLYHQE\ N / Z   F Z  U  ( WR WKHQHZYDOXHNZKLFKREH\V N



  ª Q Pc   Q  K) Pc «  MK) N   « U 6   Q   K  U 6   Q    K  ) ) ¬  /

º » » ¼

 

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

'DPSLQJ

)LJ /RVVIDFWRUDQGSURSDJDWLRQYHORFLW\RIEHDPZLWKDGGHGVSULQJPDVV V\VWHPV Pc U6 

2QH PD\ UHDGLO\ H[WHQG (TV   WR   WR RWKHU V\VWHPV DW WDFKHGWRWKHEHDPLIWKHVHV\VWHPVDFWLQGHSHQGHQWO\ RIHDFKRWKHU2QH PHUHO\QHHGVWRXVHWKHLQSXWLPSHGDQFH =c

)c Y6

RIDQDWWDFKHGV\VWHP SHUXQLWOHQJWKRIWKHEHDP LQ(T  WRREWDLQ WKHGLIIHUHQWLDOHTXDWLRQ G  Y6 U§ =c ·  Z  ¨  M ¸ Y6  G[ (© ZU 6 ¹



 

ZKLFKUHSUHVHQWVDJHQHUDOL]DWLRQRI(T  7KHUHDOSDUWRIWKHLQSXW LPSHGDQFHGHWHUPLQHVWKHPDJQLWXGHRIWKHGDPSLQJ,IWKLVLPSHGDQFHLV SXUHO\UHDOWKHGDPSLQJFDQEHYHU\ ODUJH7KHGULYLQJSRLQWLPSHGDQFHRI DSODWHLVDQH[DPSOHRIVXFKDUHDOLPSHGDQFHVHH(T  7KXVRQH PD\ REWDLQ YHU\ FRQVLGHUDEOH DPRXQWV RI GDPSLQJ RI TXDVLORQJLWXGLQDO

'DPSLQJE\PHDQVRI5HVRQDQW6\VWHPV

ZDYHVE\ PHDQVRIWKLQGDPSHGSODWHVDWWDFKHGQRUPDOWRWKHD[LVRIWKH EHDP/RVVIDFWRUVRIDUHDFKLHYDEOHZLWKVXFKFRQILJXUDWLRQV 7KH GDPSLQJ RI WKH SUHYLRXVO\ GLVFXVVHG V\VWHPVPD\ DOVREHGHULYHG E\ XVHRIWKHHQHUJ\ PHWKRGVRI6HFW,IWKHORDGLQJE\WKHDGGHGDW WDFKHGVSULQJVDQGPDVVHVLVQRWWRRJUHDWWKHQWKHUHYHUVLEOHVWUDLQHQHUJ\ VWRUHGLQDEHDPRIOHQJWKOPD\EHZULWWHQDV  U 6O Y6  

:5cc

7KHHQHUJ\ORVWE\WKHEHDPSHUXQLWWLPHLVJLYHQE\  Q 5H ^Y6 ) ` 

Q  Y6 5H ^= ( `  

ZKHUH Q LV WKH QXPEHU RI DWWDFKHG V\VWHPV DQG =( UHSUHVHQWV WKH GULYLQJ SRLQWLPSHGDQFHRIVXFKV\VWHPV7KXVWKHHQHUJ\ORVWSHUSHULRG7RIWKH YLEUDWLRQLV cc :GLVV

Q  Y6 7 5H ^= ( ` 

 QS Y6 5H ^= ( `  Z

,QYLHZRIWKHGHILQLWLRQ(T  WKHORVVIDFWRURIWKHWRWDODUUDQJH PHQWWKXVLVJLYHQE\ K

Q 5H ^= ( `  O ZU 6

 

6LQFHQ=(OFRUUHVSRQGVWRWKHGULYLQJSRLQWLPSHGDQFHSHUXQLWOHQJWK RQH ILQGV WKDW (TV   DQG   JLYH YHU\ QHDUO\ WKH VDPH UHVXOW 7KH VPDOO GLVFUHSDQF\ PD\ EH DVFULEHG WR WKH IDFW WKDW WKH DWWDFKHG V\V WHPVDOVRVWRUHUHYHUVLEOHHQHUJ\ZKLFKKHUHKDVEHHQQHJOHFWHG 7KH VDPH NLQG RI DQDO\VLV DV ZDV FDUULHG RXW DERYH IRU TXDVL ORQJLWXGLQDOZDYHFDQDOVREHSHUIRUPHG IRUWRUVLRQDODQGIOH[XUDOZDYHV )RUWKLVSXUSRVHWKHIRUFHVGXHWRWKHH[WHUQDOORDGVDUHDGGHGWRWKHLQHU WLDIRUFHV)RUEHQGLQJZDYHVLVREWDLQHG § G Y =c ·  N %  ¨  M ¸Y 6¹ G[ ZU ©



 

LQVWHDGRI(T  +HUH = c UHSUHVHQWVWKHGULYLQJSRLQWLPSHGDQFHSHU XQLWOHQJWKRIWKHDGGHGV\VWHPU6GHQRWHVWKHEHDPPDVVSHUXQLWOHQJWK DQGN%LVWKHEHQGLQJZDYHQXPEHURIWKHRULJLQDOXQWUHDWHGEHDP&OHDUO\ WKHGDPSLQJLVDJDLQGHWHUPLQHGE\WKHUHDOSDUWRIWKHLPSHGDQFH)RUDW WDFKHGVLPSOHVSULQJPDVVV\VWHPVWKHGDPSLQJDJDLQH[KLELWVDGHILQLWH SHDNDQGWKHRYHUDOOEHKDYLRXULVPXFKOLNHWKDWLQGLFDWHGLQ)LJ

'DPSLQJ

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

)LJ/RVVIDFWRURIEHDPZLWKVKHHWPHWDOVWULSUHVRQDWRUV

$QRWKHUV\VWHPLQZKLFKWKHGDPSLQJLVRIDUHVRQDQWQDWXUHFRQVLVWVRI WZRSODWHVWKDWDUHLQWHUFRQQHFWHGE\UHVLOLHQWHOHPHQWV6XFKSODWHVLQHI IHFW DUH WZR FRXSOHG V\VWHPV ZKLFK PD\ EH GHVFULEHG E\ PHDQV RI WZR FRXSOHG EHQGLQJZDYH HTXDWLRQV 7KH H[DFW H[SUHVVLRQV REWDLQHG IRU WKH WZR EHQGLQJ ZDYHQXPEHUV DUH UDWKHU FRPSOLFDWHG WKH JHQHUDO FDVH VHH 6HFW+RZHYHULWLVIRXQGWKDWWKHGDPSLQJLVJUHDWHVWDW Z

Z

V cc Pcc  Pcc  Pcc Pcc

'DPSLQJE\PHDQVRI5HVRQDQW6\VWHPV

ZKHUH V cc UHSUHVHQWVWKHVSULQJFRQVWDQWRIWKHLQWHUOD\HUSHUXQLWDUHDDQG Pcc  DQG Pcc GHQRWH WKH PDVVHV SHU XQLW DUHD RI WKH WZR SODWHV Z WKHQ UHSUHVHQWV WKH UHVRQDQFH IUHTXHQF\ RI WKH WZR PDVVHV LQWHUFRQQHFWHG E\ WKHVSULQJ$ERYHDQGEHORZZ WKHGDPSLQJLVUHODWLYHO\VPDOO7KLVIDFW LV VRPHWLPHV RI LPSRUWDQFH IRU PXOWLOD\HU SODWHV ZLWK YHU\ VRIW RU WKLFN LQWHUOD\HUV IRU ZKLFK WKH UHODWLYHO\ KLJK ORVV IDFWRUV LQGLFDWHG LQ 6HFW  LV QRW REWDLQHG DERYH Z 2QH WKHUHIRUH VKRXOG EH FDUHIXO QRW WR PDNHWKHLQWHUOD\HUVRIPXOWLOD\HUFRPSRVLWHVWRRVRIW  'DPSLQJ E\ 7KLFN /D\HUV %DOODVW )RUWKLFNOD\HUVGDPSLQJLVDFKLHYHGE\UDGLDWLRQRIVWUXFWXUHERUQHVRXQG HQHUJ\WRDPDWHULDOZLWKKLJKORVVHV7KHUHLVWKXVDFHUWDLQUHVHPEODQFH WRDLUERUQHVRXQGUDGLDWLRQWUHDWHGLQ&KDSWHU)RUWKHIROORZLQJLWLVDV VXPHGWKDWWKHZDYHOHQJWKOS!RIWKHYLEUDWLQJVWUXFWXUHLVODUJHUWKDQWKDW RIWKHWKLFNORVV\OD\HUF0I:LWKWKHTXDQWLWLHVRI&KDSWHUWKLVPHDQV WKDWWKHUDGLDWLRQHIILFLHQF\LVDSSUR[LPDWHO\XQLW\V|7KLVDSSUR[LPD WLRQLPSOLHVWKDWWKHOD\HUFDQEHWUHDWHGDVDRQHGLPHQVLRQDOZDYHJXLGH RIGHQVLW\U0!DQGWKHFRPSOH[VSHHGRIVRXQGF0:KHQWKHZDYHJXLGHLV IUHHDWWKHUHPRWHHQGDWWKHGLVWDQFHGIURPWKHLQWHUIDFHZLWKWKHVWUXFWXUH DVGHSLFWHGLQ)LJWKHQWKHLQSXWLPSHGDQFHSHUXQLWDUHDLV =c

M U 0 F 0 WDQ Z G  F 0  F 0

F0c   MK0

 

+HUHLQK0LVWKHORVVIDFWRURIWKHOD\HUPDWHULDO,IWKHEDVHVWUXFWXUHSHU IRUPV EHQGLQJ YLEUDWLRQV F0 LV JLYHQ E\ WKH ORQJLWXGLQDO ZDYH VSHHG RI WKHOD\HUZKHUHDVIRUORQJLWXGLQDOYLEUDWLRQVWKHVKHDUZDYHVSHHGLVWREH FKRVHQ 8SRQVXEVWLWXWLQJ(T  LQWR  RU  WKHORVVIDFWRURI WKHDVVHPEO\FDQEHHVWLPDWHGDSSUR[LPDWHO\$QH[DPSOHLVVKRZQLQ)LJ  ZKHUH H[WHQVLRQDO ZDYHV LQ DQ DOXPLQLXP URG DUH GDPSHG )RU WKH GDPSLQJ LV HPSOR\HG D VRIW UXEEHU OD\HU IRU ZKLFK F0 |  PV DQG K0 |  7KH ILUVW WKLFNQHVV UHVRQDQFH RI WKH OD\HU LV DW DERXW  +] ZGF0 S$VFDQEHREVHUYHGWKHORVVIDFWRUGLPLQLVKHVUDSLGO\EHORZ WKLVIUHTXHQF\VLQFHRQO\OLWWOHHQHUJ\LVWUDQVPLWWHGWRWKHOD\HU$ERYHWKH ILUVWWKLFNQHVVUHVRQDQFHWKHVXEVHTXHQWUHVRQDQFHVDUHEDUHO\ YLVLEOHRZ LQJ WR WKH KLJK ORVV IDFWRU ,Q WKLV IUHTXHQF\ UDQJH LW LV SRVVLEOH WR VHW 5H > = 0c @ | U 0 F0c 

'DPSLQJ

)LJ'DPSLQJE\VWUXFWXUHERUQHVRXQGUDGLDWLRQWRDWKLFNOD\HU OLQHU RI VRIWPDWHULDO OS!F0I 

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c @ | U 0 F0c +HUHU0 LVWKHGHQVLW\RIVDQGDQGF0LVWKHDSSURSUL DWH ZDYH VSHHG ZKLFK FDQ RQO\ EH TXDQWLILHG ZLWK GLIILFXOW\ ,W LV VXJ

'DPSLQJDW-RLQWV

JHVWHGKRZHYHUWKDWIRUWKLQGU\XQFRPSDFWHGVDQGOD\HUVF/ |PV F7 | PVDQGK |>@7KHVHYDOXHVUHSUHVHQWWKHPRVWIDYRXU DEOHFDVHVLQFHVDQGLVPRVWRIWHQKXPLG%HVLGHVWKLVWKHVDQGLVLQHYLWD EO\ FRPSDFWHGWKURXJKJUDYLW\DQGE\EHLQJYLEUDWHG%RWKHIIHFWVWKHUH IRUH OHDG WR KLJKHU ZDYH VSHHG DQG WKHQFH OHVV GDPSLQJ IRU ORZ DQG LQWHUPHGLDWHIUHTXHQFLHV1HYHUWKHOHVVFRPPHUFLDOO\DYDLODEOHGULHGVDQG UHPDLQV DQ DSSUHFLDEOH GDPSLQJ PDWHULDO IRU LQWHUPHGLDWH DQG KLJK IUH TXHQFLHV

'DPSLQJDW-RLQWV ,WLVZHOONQRZQWKDWPDFKLQHVDQGWHFKQLFDODSSOLDQFHVFRQVLVWLQJRIVHY HUDO VWUXFWXUDO SDUWV H[KLELW VXEVWDQWLDOO\ ODUJHU ORVVHV WKDQ WKRVH RI WKH FRQVWLWXWLYH PDWHULDOV $V DQ H[DPSOH WKH ORVV IDFWRU RI D EXLOGLQJ LV DO PRVW DOZD\V DERXW  ˜  ZKHUHDV V\VWHPV PDGH RI VWHHO RU DOXPLQLXP KDYLQJDPDWHULDOORVVIDFWRUOHVVWKDQVKRZ K |  ˜  I  +] K |  ˜  I ! +]

LQV\VWHPVFRQVLVWLQJRIIHZWKLFNVXEVWUXFWXUHVVXFKDVDVKLS¶VKXOO K | IRUV\VWHPVFRQVLVWLQJRIVHYHUDOWKLFNRUWKLQSDUWVVXFKDVDFDUERG\DQG K |  ˜  I  +] K | 

 I ! +]

IRUV\VWHPVRIVHYHUDOWKLQVXEVWUXFWXUHV 6XFK ³EXLOWLQ´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

'DPSLQJ

'DPSLQJE\5HODWLYH0RWLRQ1RUPDOWRWKH,QWHUIDFH 5DWKHUFRPSUHKHQVLYHO\LQYHVWLJDWHGLVWKHGDPSLQJGXHWRUHODWLYHPRWLRQ QRUPDO WR DQ LQWHUIDFH ZLWK D IOXLG ILOP JDV SXPSLQJ RU VTXHH]H ILOP GDPSLQJ  >@ 7KH PRVW HVVHQWLDO PHFKDQLVP EHKLQG WKH HQHUJ\ ORVVFDQEHXQGHUVWRRGIURPFDOFXODWLQJWKHGDPSLQJRIDSODWHYLEUDWLQJLQ DYLVFRXVPHGLXP ZLWKSUHVFULEHGYHORFLW\VHH)LJ6WDUWLQJSRLQWLV WKHOLQHDUL]HG1DYLHU6WRNH¶VHTXDWLRQVIRUDFRPSUHVVLEOHPHGLXPRIGHQ VLW\ UVSHHG RIVRXQGFDQGNLQHPDWLFYLVFRVLW\ Q QDLU |˜PV QZDWHU |PV  § wY wY · wY[ wS   U Q ¨ [  [ ¸  wW w[ w\ ¹ © w[  § w Y\ w Y\ · wY \ wS U   U Q ¨    ¸  ¨ w[ wW w\ w\ ¸¹ © wY \  wS wY U [  U   w[ w\ F wW U

 

5HVWULFWLQJWKHDQDO\VLVWRKDUPRQLFYLEUDWLRQVVXFKWKDWDOOILHOGYDULDEOHV DUHLQWKHIRUP H  MN [ H  MN \ H MZ W DVHWRIKRPRJHQHRXVOLQHDUHTXDWLRQVUHVXOWV IURP  E\HPSOR\LQJWKHSKDVRUQRWDWLRQ7KLVVHWRIHTXDWLRQVKDV QRQWULYLDO VROXWLRQV DV WKH GHWHUPLQDQW WR WKHFRHIILFLHQW PDWUL[YDQLVKHV LHZKHQ [

\



N \

§Z · Z   N [ | ¨ ¸  N [    F   MZQ  F © F ¹ 

N

D 

 MZ  MZ §   M ·  N [ | |¨ ¸  Q Q © G ¹

 \

7KHDSSUR[LPDWLRQVLQWURGXFHGFRPHVDERXWDVWKHNLQHPDWLFYLVFRVLW\ QLV VPDOO DQG WKH WKLFNQHVV RI WKH UHVXOWLQJ DFRXVWLF ERXQGDU\ OD\HU G Q  Z XVXDOO\ LV RI WKH RUGHU RI P ZKHUHDV WKH ZDYHOHQJWKV DUH VRPHRUGHUVRIPDJQLWXGHORQJHU7KHJHQHUDOVROXWLRQWR(T  IRU KDUPRQLFPRWLRQFDQEHZULWWHQDV YÖ[ YÖ \ SÖ

$H $ H $ H

 MN \  \

 $ H

 MN \  \

 $ H





 MN \  \



 $ H

 MN \  \

H H H

 MN \  \

 MN \  \

 MN [ [



 MN [ [



 MN [ [



E 

'DPSLQJDW-RLQWV

)LJ3DUWLFOHPRWLRQLQDYLVFRXVPHGLXPDGMDFHQWWRDYLEUDWLQJSODWH

8SRQVXEVWLWXWLQJ E LQWR  WKHUHODWLRQV N[  N \  

$  $

N \  N [  $  $

$  $

ZU    $ N [   MZQ F



UHVXOWIRUWKHVRIDUXQNQRZQDPSOLWXGHV7KHODWWHUDUHREWDLQHGIURPWKH ERXQGDU\ FRQGLWLRQV DW WKH SODQH \   ,Q WKLV FDVH WKH QRUPDO YHORFLW\ FRPSRQHQWPXVWHTXDOWKHSODWHYHORFLW\ZKLOHWKHWDQJHQWLDOYDQLVKHVLQD YLVFRXVPHGLXPLH YÖ \ [ 

 MN [ YÖ S H S  YÖ[ [ 

F 



NS SOS LVWKHZDYHQXPEHURIWKHSODWHZDYHDQG YÖ S LWVDPSOLWXGH8SRQ FRPELQLQJWKHH[SUHVVLRQVLWLVIRXQGWKDW YÖ \ [ \ SÖ [ \

YÖ S N \N \   N S YÖ S

N

N H

 MN \  \

\ \ 

N\

  MZQ  F N \N \   N S

 N S H

 MN \  \

H

 MN S [



G  H

 MN \  \  MN S [

H



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

'DPSLQJ

7R FDOFXODWH WKH GDPSLQJ LW LV QHFHVVDU\ WR GHYHORS WKH SRZHU WKDW LV WUDQVPLWWHGIURP WKHSODWHWRWKHDPELHQWPHGLXP,QJHQHUDOWKHSRZHULV JLYHQE\ :

 ª 5H SÖ [ YÖ \ [ G[ º  ¼  ¬³

³ 5H ª¬ S [ W º¼ 5H ª¬Y [ W º¼ G[ \

H 

7KHLQWHJUDWLRQLVWREHWDNHQRYHUWKHWRWDOOHQJWKRIWKHSODWHOS$QDYHU DJLQJRYHUWLPHLVLQGLFDWHGE\WKHRYHUEDU8SRQVXEVWLWXWLQJWKHH[SUHV VLRQLQ G LQWR(T H LVREWDLQHG ª º Z  N S « »  ZU O S YÖ S « 5H ª N \ º  Q N S ,P ª N \  º  Q N S F » ª º N ,P ¬ ¼ Z Z ¬ \ ¼ »  §Z ·« ¬ ¼ Z  N   ¨   N S ¸ « S » F ¼ © F ¹¬ 

:

I 

ZKHQ WKH VPDOO WHUPV QZF DUH QHJOHFWHG %HFDXVH WKH ZDYHOHQJWK LQ WKHSODWHOSLVVXEVWDQWLDOO\ODUJHUWKDQWKHERXQGDU\OD\HUWKLFNQHVVGLQDOO FDVHVRILQWHUHVWLQSUDFWLFH,P>N\@ ,P>N\@7KLVPHDQVWKDWWKHILUVW WHUP LQ I UHSUHVHQWLQJWKHUDGLDWHGVRXQGDQGWKHVHFRQGUHSUHVHQW LQJ WKH HQHUJ\ FRQYHUVLRQ WR KHDW UHPDLQV ZKHUHDV WKH WKLUG LV LQVLJQLIL FDQW7KHORVVIDFWRUFDQEHREWDLQHGE\LQVHUWLQJ I LQWKHGHILQLWLRQ   REVHUYLQJ WKDW WKH GLVVLSDWHG HQHUJ\ SHU SHULRG 7 LV JLYHQ E\ (Q : ˜ 7  S:Z 0RUHRYHU DSSUR[LPDWLQJ WKH UHYHUVLEOH HQHUJ\ E\  (5 U K YÖ S O S   LWIROORZVWKDW :

K

ZU K YÖ S





J 

+HUHLQ U LV WKH GHQVLW\ RI WKH SODWH DQG K LWV WKLFNQHVV )RU WKLQ SODWHV _N\_!!NS!!ZFVXFKWKDW5H>N\@|DQG 

:|

ZU  K YÖ S O S 

ŸK |

U G  U K

K 

6LQFH WKH ERXQGDU\ OD\HU WKLFNQHVV G DV PHQWLRQHG LV YHU\ VPDOO DQG GLPLQLVKHVLQYHUVHO\SURSRUWLRQDOWRWKHVTXDUHURRWRIIUHTXHQF\WKHORVV IDFWRU RI D V\VWHP DV GHSLFWHG LQ )LJ  LV DOVR VPDOO DQG SOD\V D UROH RQO\ IRU WKLQ IROLRV &RUUHVSRQGLQJ H[SHULPHQWV VKRZ WKDW K  ZHOO GHVFULEHVVXFKUHDOL]DWLRQV 7KH VLWXDWLRQ LV VOLJKWO\ GLIIHUHQW IRU GRXEOH SODWHV DV VKRZQ LQ )LJ 7KHSUHYLRXVDQDO\VLV DK KRZHYHUDSSOLHVZKHQDFFRXQWLV WDNHQRIWKHIDFWWKDWIRUVXFKV\VWHPVWKHUHDUHDOVRZDYHVLQWKHQHJDWLYH

'DPSLQJDW-RLQWV

\GLUHFWLRQ 7KLV PHDQV WKDW WHUPV LQYROYLQJ H MN \  DQG H MN \  PXVW EH LQ FOXGHGLQ E :LWKWKHDGGLWLRQDOERXQGDU\FRQGLWLRQV \

YÖ \ [ 

 YÖ SH

YÖ[ [ 



MN S  [

 YÖ \ [ G

 YÖ S  H

 YÖ[ [ G



MN S  [

\



DJDLQ UHVXOWV D VROYDEOH VHW RI OLQHDU HTXDWLRQV 7KH H[SOLFLW DQDO\VLV LV RPLWWHG IRU WKLV FDVH VLQFH WKH RXWFRPH LV UHDGLO\ XQGHUVWRRG IURP (T K 

)LJ0RWLRQDWWLPHVWDQGWIRUDGRXEOHSODWHZLWKDIOXLGLQWHUOD\HU

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

K|

U G § O S · ¨ ¸  U K © S G ¹

 

IRU D GRXEOH SODWH V\VWHP IRU ZKLFK WKH JDV SXPSLQJ RU VTXHH]H ILOP GDPSLQJLVWKHJRYHUQLQJORVVPHFKDQLVP,Q)LJLVVKRZQDIHZH[

'DPSLQJ

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

)LJ  0HDVXUHG DQG FDOFXODWHG ORVV IDFWRU IRU D VTXHH]H ILOP GDPSHGPP DOXPLQLXPSODWH

'DPSLQJE\5HODWLYH0RWLRQ7DQJHQWLDOWRWKH,QWHUIDFH )RU EXLOWXS V\VWHPV FRQVLVWLQJ RI WKLFN VXEVWUXFWXUHV VXFK DV HQJLQHV RU LQGXVWULDOPDFKLQHVWKHPRWLRQSHUSHQGLFXODUWRDMXQFWLRQRULQWHUIDFHLV XVXDOO\YHU\VPDOODQGKHQFHGRHVQRWOHDGWRVLJQLILFDQWGDPSLQJ,QVWHDG WKHWDQJHQWLDOUHODWLYHPRWLRQEHFRPHVWKHJRYHUQLQJ :KHQ DQLQWHUIDFHHQFRPSDVVHVD WKLQYLVFRXV ILOPHJRLOGXVWHWF 6HFWUHJDUGLQJPXOWLOD\HUHGV\VWHPVDSSO\LHWKHLQWHUOD\HUDFWVLQ

'DPSLQJDW-RLQWV

VKHDUDQGFRQYHUWVDIUDFWLRQRIWKHVWUXFWXUHERUQHVRXQGHQHUJ\ LQWRKHDW RZLQJ WR LWV GLVVLSDWLRQ$WKHRUHWLFDODVVHVVPHQWRIWKHORVVIDFWRUVLVLQ WKLVFDVHLQSULQFLSOHSRVVLEOHIRUKDUPRQLFPRWLRQE\UHSODFLQJWKHYLVFRV LW\E\DFRPSOH[VKHDUVWLIIQHVV (PSOR\LQJWKHGHILQLWLRQVRIWKHYLVFRVLW\ Q WYDQGWKHVKHDUPRGXOXV *  W[ DQG XVLQJ WKH UHODWLRQ YÖ MZ[Ö  IRU KDUPRQLF PRWLRQ LW LV IRXQG WKDW *

D 

MZQ 

ZKLFK LV SXUHO\ LPDJLQDU\ $Q LQWURGXFWLRQ RI WKLV VKHDU PRGXOXV LQ WKH H[SUHVVLRQVRI6HFWWKHUHIRUHLPSOLHVWKHIROORZLQJVXEVWLWXWLRQ * K J G

MZQ   *cc K *c ZQ 

 ( G G  N 

ZQ   

ZQ  §   ·  ¸  ¨ G  N © (G ( G  ¹

E 

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

 S1



ZKHUH S1 LV WKH QRUPDO SUHVVXUH LQ 1PP DQG WKH FRQVWDQW VXFK WKDW*GKDVWKHSURSHUGLPHQVLRQRI1PP6RPHPHDVXUHPHQW UHVXOWV DUH VKRZQ LQ )LJ E ,W LV H[SHFWHG WKDW WKH LQWHUOD\HU VWLIIQHVVLQFUHDVHVZLWKLQFUHDVLQJQRUPDOSUHVVXUH7KHORVVIDFWRU OLHV EHWZHHQ  DQG  DQG WHQGV WR GHFUHDVH ZLWK LQFUHDVLQJ

'DPSLQJ

QRUPDOSUHVVXUH:LWKWKHVXUIDFHURXJKQHVVHVFRPPRQO\IRXQGLQ PDFKLQH PDQXIDFWXULQJ WKH LQWHUOD\HU VWLIIQHVVHV DQG ORVV IDFWRUV REVHUYHG IRU GU\ MRLQWV EHWZHHQ PHWDOV DUH UHPDUNDEO\ VLPLODU ,Q PHDVXUHPHQWVRILQWHUOD\HUVWLIIQHVVDQGORVVIDFWRUVZDVREVHUYHG RQO\DQHJOLJLEOHUHGXFWLRQZKHQWKHDYHUDJHURXJKQHVVZDVDOWHUHG IURPPPWRPP

)LJ&RQGLWLRQVDWGU\MRLQWVZLWKWDQJHQWLDOUHODWLYHPRWLRQ D ,GHDOL]HGLQ WHUIDFHDQGPHFKDQLFDOPRGHO E 6WLIIQHVVDQGORVVIDFWRUYHUVXVQRUPDOSUHVVXUH

5HIHUHQFHV >@0H\HU2(=XU7KHRULHGHULQQHUHQ5HLEXQJ-RXUQDOIUUHLQHXQG DQJHZDQGWH0DWKHPDWLNS >@ 9RLJW :hEHULQQHUH5HLEXQJIHVWHU.|USHULQVEHVRQGHUHGHU0HWDO OH$QQDOHQGHU3K\VLNS >@%ROW]PDQQ/=XU7KHRULHGHUHODVWLVFKHQ1DFKZLUNXQJ$QQDOHQGHU 3K\VLN(UJ%GS >@0D[ZHOO -&  2Q WKH G\QDPLFDO WKHRU\ RI JDVHV 3KLO 7UDQV 5R\DO 6RFS >@ =HQHU&(ODVWLFLW\DQG$QHODVWLFLW\LQ0HWDOV8QLYHUVLW\RI&KLFDJR 3UHVV&KLFDJR0, >@7DPP.DQG:HLVV28QWHUVXFKXQJHQEHUSHULRGLVFKH:HOOHQH[ SRQHQWLHOOH XQGNRPSOH[H1DKIHOGHULPEHJUHQ]WHQ)HVWN|USHU$FXVWLFD S >@:HLVV2hEHUGLH6FKDOODXVEUHLWXQJLQ YHUOXVWEHKDIWHWHQ0HGLHQPLW NRPSOH[HP6FKXEXQG.RPSUHVVLRQVPRGXO$FXVWLFDS >@&UHPHU/9LHUSROGDUVWHOOXQJXQG5HVRQDQ]NXUYHQEHLVFKZLQJHQGHQ 6WlEHQ 6LW]XQJVEHULFKWH GHU 3UHXVVLVFKHQ $NDGHPLH GHU :LVVHQVFKDIWHQ SK\VPDWK.O,S >@ 0DVRQ :3  3K\VLFDO DFRXVWLFV DQG WKH SURSHUWLHV RI VROLGV 9DQ 1RVWUDQG1HZ@2EHUVW+DQG)UDQNHQIHOG.hEHUGLH'lPSIXQJGHU%LHJHVFKZLQ JXQJHQ GQQHU %OHFKHGXUFKIHVWKDIWHQGH%HOlJH$FXVWLFD%HLKHIWH1R S >@ 2EHUVW +%HFNHU*:DQG)UDQNHQIHOG.hEHUGLH'lPSIXQJGHU %LHJHVFKZLQJXQJHQGQQHU%OHFKHGXUFKIHVWKDIWHQGH%HOlJH,,$NXVWLVFKH %HLKHIWH+HIW LQ$FXVWLFD S >@ 1LMPDQ(-0 DQG%HOOXVVL*10+HOLXPDVJRRGDVYDFXXP7KH SRRUPDQ¶VDSSURDFKIRUDFFXUDWHPDWHULDOGDPSLQJPHDVXUHPHQWV$SSOLHG $FRXVWLFVS >@ 7UXHOO 5 DQG (OEDXP &  +LJK IUHTXHQF\ XOWUDVRQLF VWUHVV ZDYHV +DQGEXFKGHU3K\VLN 6)OJJHHG 6SULQJHU9HUODJ%HUOLQ >@ /RYH $(+  $ WUHDWLVH RQ WKH PDWKHPDWLFDO WKHRU\ RI HODVWLFLW\ S'RYHU3XEOLFDWLRQV1HZ@%RUGRQL3*'LVORFDWLRQUHOD[DWLRQDWKLJKIUHTXHQFLHV1XRYR&L PHQWR6XSSOS >@ :HJHO 5DQG:DOWHU+,QWHUQDOGLVVLSDWLRQLQVROLGVIRUVPDOOF\FOLF VWUDLQV3K\VLFVS >@=HPDQHN-U-DQG5XGQLN,$WWHQXDWLRQDQGGLVSHUVLRQRIHODVWLF ZDYHVLQ DF\OLQGULFDOEDU-RXUQDORIWKH$FRXVWLFDO6RFLHW\RI$PHULFD S >@%HQQHZLW].DQG5|WJHU+hEHUGLHLQQHUH5HLEXQJIHVWHU.|USHU $EVRUSWLRQVIUHNYHQ]HQ YRQ 0HWDOOHQ LP DNXVWLVFKHQ *HELHW 3K\VLNDOLVFKH =HLWVFKULIWS >@%RUGRQL3*1XRYR0DQG9HUGLQL/5HOD[DWLRQRIGLVORFDWLRQVLQ FXELF IDFHFDQWHUHG PHWDOV 3URF UG ,QWHUQDWLRQDO &RQJUHVV RQ $FRXVWLFV 6WXWWJDUW9ROS(OVHYLHU$PVWHUGDP   >@ )|UVWHU ) DQG .|VWHU :  (ODVWL]LWlWVPRGXO XQG 'lPSIXQJ LQ $E KlQJLJNHLWYRP:HUNVWRII]XVWDQG0HWDOONXQGHS >@%HFNHU*:DQG2EHUVW+hEHUGDVG\QDPLVFKHODVWLVFKH9HUKDO WHQOLQHDUYHUQHW]WHUXQGJHIOOWHU.XQVWVWRIIH.ROORLG=HLWVFKULIWS >@/LQKDUGW)DQG2EHUVW+ hEHUGLH7HPSHUDWXUDEKlQJLJNHLWVFKZLQ JXQJVGlPSIHQGHU.XQVWVWRIIH$FXVWLFDS >@:LOOLDPV0//DQGHO5)DQG)HUU\-'0HFKDQLFDOSURSHUWLHVRI VXEVWDQFHVRIKLJKPROHFXODUZHLJKW7KHWHPSHUDWXUHGHSHQGHQFHRIUH OD[DWLRQ PHFKDQLVPV LQ DPRUSKRXV SRO\PHUV DQG RWKHU JODVVIRUPLQJ OLT XLGV-RXUQDORIWKH$PHULFDQ&KHPLFDO6RFLHW\S >@:DWWHUV%*7UDQVPLVVLRQORVVRIVRPHPDVRQU\ZDOOV-RXUQDORI WKH$FRXVWLFDO6RFLHW\WRI$PHULFDS

'DPSLQJ >@.XKO:DQG.DLVHU+$EVRUSWLRQRIVWUXFWXUHERUQHVRXQGLQ EXLOG LQJPDWHULDOVZLWKRXWDQGZLWKVDQGILOOHGFDYLWLHV$FXVWLFDS >@ +HFNO 0DQG1XWVFK-7DVFKHQEXFKGHU7HFKQLVFKHQ$NXVWLNQG HG&K 0+HFNO+$0OOHUHGV 6SULQJHU9HUODJ%HUOLQ >@ .XUW]H *  .|USHUVFKDOOGlPSIXQJ GXUFK N|UQLJH 0HGLHQ 9', %HULFKWHS >@ .LUVWHLQ 7DQG0OOHU+:%HULFKWHGHV%HLUDWVIU%DXIRUVFKXQJ EHLP %XQGHVPLQLVWHU IU :RKQXQJVEDX +  .|USHUVFKDOO LQ *HEl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|VHU 0  'LH GlPSIHQGH :LUNXQJ GLFNHU (QW GU|KQVFKLFKWHQDXI3ODWWHQ=HLWVFKULIWIU/lUPEHNlPSIXQJS >@6FKPLGW+'LH6FKDOODXVEUHLWXQJLQN|UQLJHQ6XEVWDQ]HQ$FXVWLFD S >@ 0DLGDQLN *  (QHUJ\ GLVVLSDWLRQ DVVRFLDWHG ZLWK JDVSXPSLQJ LQ VWUXFWXUDOMRLQWV-RXUQDORIWKH$FRXVWLFDO6RFLHW\RI$PHULFDS >@7URFKLGLV$.|USHUVFKDOOGlPSIXQJPLWWHOV*DVRGHU)OVVLJNHLWV VFKLFKWHQ$FXVWLFDS >@ &KRZ /& DQG 3LQQLQJWRQ 5-  3UDFWLFDO LQGXVWULDO PHWKRG RI LQ FUHDVLQJVWUXFWXUDOGDPSLQJLQPDFKLQHU\,6TXHH]HILOPGDPSLQJZLWKDLU -RXUQDORI6RXQGDQG9LEUDWLRQS >@ &KRZ /& DQG 3LQQLQJWRQ 5-  3UDFWLFDO LQGXVWULDO PHWKRG RI LQ FUHDVLQJVWUXFWXUDOGDPSLQJLQPDFKLQHU\,,6TXHH]HILOPGDPSLQJZLWKDLU -RXUQDORI6RXQGDQG9LEUDWLRQS

5HIHUHQFHV >@ 2WW '8QWHUVXFKXQJHQ]XU6FKZLQJXQJVGlPSIXQJGXUFK+\VWHUHVH LQ6FKUDXEXQG1LHWYHUELQGXQJHQ9',%HULFKWH >@0LQGOLQ5'DQG'HUHVLHZLF]+(ODVWLFVSKHUHVLQ FRQWDFW XQGHU YDU\LQJREOLTXHIRUFHV7UDQV$60(-RXUQDORI$SSOLHG0HFKDQLFVS  >@ *DXO /2QWKHGDPSLQJDQGWUDQVPLVVLRQRIIOH[XUDOZDYHVDWVWUXF WXUDOMRLQWV,QJHQLHXU$UFKLYS >@ 3HWXHOOL *  7KHRUHWLVFKH XQG H[SHULPHQWHOOH %HVWLPPXQJ GHU 6WHL ILJNHLWV XQG 'lPSIXQJVHLJHQVFKDIWHQ QRUPDOEHODVWHWHU )JHVWHOOHQ 'LVV 5:7+$DFKHQ >@ 6FKREHU 8  8QWHUVXFKXQJHQ GHU .|USHUVFKDOOGlPSIXQJ GXUFK ) JHVWHOOHQLQ0RWRUHQ3URFRI'$*$:LHQS

,PSHGDQFHDQG0RELOLW\

'HILQLWLRQV ,Q WKH SUHYLRXV WZR FKDSWHUV LV VWXGLHG WKH ZDYH SURSDJDWLRQ LQ WKH DE VHQFH DQG SUHVHQFH RI GDPSLQJ )RFXV LQ WKLV FKDSWHU LV RQ WKH SURFHVV SUHFHGLQJWKHSURSDJDWLRQQDPHO\WKHH[FLWDWLRQRIVWUXFWXUHERUQHVRXQG ,QSUDFWLFHH[FLWDWLRQRIVWUXFWXUHERUQHVRXQGFDQEHGXHWRIRUH[DP SOH PDFKLQHU\ LQVWDOOHG RQ IRXQGDWLRQV LQ EXLOGLQJV VKLSV YHKLFOHV HWF EXWDOVRZDONLQJNQRFNLQJDQGPDJQHWLFIRUFHVLQHOHFWULFDOGHYLFHVFRQ VWLWXWHV H[FLWDWLRQ PHFKDQLVPV )OXLG IOXFWXDWLRQV PRUHRYHU VXFK DV DLU ERUQHVRXQGRUSUHVVXUHSXOVDWLRQVLQOLTXLGILOOHGSLSHVJLYHULVHWRVWUXF WXUHERUQH VRXQG H[FLWDWLRQV &RPPRQ IRU DOO WKHVH PHFKDQLVPV LV DQ H[HUWHG IRUFH RU D SUHVFULEHG YHORFLW\ ZKLFK JLYHV ULVH WR D SURSRUWLRQDO PRWLRQ RU IRUFH UHVSHFWLYHO\ ZKHQ H[WUHPH ORDGV DUH H[FOXGHG /LPLWLQJ WKH GLVFXVVLRQ WR KDUPRQLF PRWLRQV VLQFH DOO RWKHU WLPH KLVWRULHV FDQ EH V\QWKHVL]HGIURPWKHIRUPHULWLVPHDQLQJIXOWRLQWURGXFHWKHSRLQWLPSHG DQFHRUSRLQWPRELOLW\DVLQPDQ\RWKHUDUHDVRISK\VLFV 7KHEDVLFGHILQLWLRQVIRUWKHPHFKDQLFDOLPSHGDQFHDQGPRELOLW\UHDG =

)Ö  < YÖ

YÖ )Ö

 

UHVSHFWLYHO\FI>@+HUHLQ )Ö LVWKHFRPSOH[DPSOLWXGHRIWKHIRUFHDQG YÖ WKDWRIWKHYHORFLW\ERWKDWWKHSRLQWRIH[FLWDWLRQ7KHSRLQWLPSHGDQFH DQGPRELOLW\DUHLQJHQHUDOFRPSOH[IXQFWLRQVRIIUHTXHQF\VLQFHWKHUDWLRV RIWKHILHOGYDULDEOHVYDU\ZLWKIUHTXHQF\ 7KHSRLQWTXDQWLWLHVGHILQHGLQ  FRQWUDU\WRSDUWLFOHPHFKDQLFVDUH QRW XQDPELJXRXV  VLQFH WKH ³SRLQW´ RI H[FLWDWLRQ RIWHQ LV D VL]HDEOH DUHD RYHUZKLFKWKHIRUFHDQGYHORFLW\PD\YDU\7KHGHILQLWLRQVPRUHRYHUGR QRWLQYROYHWKHDUHDRIH[FLWDWLRQ1RQHWKHOHVVWKHUHFDQEHVXEVWDQWLDOGLI IHUHQFHVLIIRUH[DPSOHDWKLQSODWHLVH[FLWHGRYHUDVPDOOSRLQWOLNHDUHD RUWKHH[FLWDWLRQVLVGLVWULEXWHGRYHUDODUJH7RFLUFXPYHQWWKLVDPELJXLW\ WKHUHIRUH DSUHFLVHVSHFLILFDWLRQ PXVW EH PDGHRIWKHH[FLWDWLRQGLVWULEX WLRQDQGSRVLWLRQ

'HILQLWLRQV

7KH VLPSOHVW DQG PRVW FRPPRQO\ XVHG VSHFLILFDWLRQ LV WKDW WKH H[FLWD WLRQLVFRQFHQWUDWHGWRD³SRLQW´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a

)Ö   @ GXH WR WKH OLPLWHG VWLIIQHVV RI WKH SLH]RHOHFWULFFU\VWDOUHJLVWHULQJWKHIRUFHVHH)LJ7KHPDJQLWXGHDQG SKDVHDUHUHJLVWHUHGIRUWKHWZRHOHFWULFDOVLJQDOVSURSRUWLRQDOWRWKHIRUFH DQG WKH YHORFLW\ UHVSHFWLYHO\ 7KH UDWLR RI WKH WZR VLJQDOV QHHG RQO\ EH PXOWLSOLHGE\ DFDOLEUDWLRQIDFWRUZKLFKLQWXUQLVREWDLQHGIURPDFDOLEUD WLRQ PHDVXUHPHQW XVLQJ D ZHOO GHILQHG UHIHUHQFH PRELOLW\ IRU LQVWDQFH D PDVV

0HDVXUHPHQWRI0RELOLWLHV ,PSHGDQFHV 

)LJ3ULQFLSDOGHVLJQRIDQLPSHGDQFHKHDG

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­ YÖ ½ ® ¾ ¯ ZÖ ¿

ª @ -RKQVRQ ./  &RQWDFW 0HFKDQLFV &K  &DPEULGJH 8QLYHUVLW\ 3UHVV1HZ@ /MXQJJUHQ 6  *HQHUDWLRQ RI ZDYHV LQ DQ HODVWLF SODWH E\ D YHUWLFDO IRUFHDQGE\ DPRPHQW LQWKHYHUWLFDOSODQH-RXUQDORI6RXQGDQG9LEUDWLRQ S >@0|VHU0+HFNO0DQG*LQWHUV.+=XU6FKDOODXVEUHLWXQJLQIOV VLJNHLWVJHIOOWHQNUHLV]\OLQGULVFKHQ5RKUHQ$FXVWLFDS

5HIHUHQFHV >@ 3HWHUVVRQ %$7  6WUXFWXUDO DFRXVWLF SRZHU WUDQVPLVVLRQ E\ SRLQW PRPHQWDQGSRLQWIRUFHH[FLWDWLRQ3DUW  %HDPDQGIUDPHOLNHVWUXFWXUHV -RXUQDORI6RXQGDQG9LEUDWLRQS >@(LFKOHU(3ODWHHGJHDGPLWWDQFHV-RXUQDORIWKH$FRXVWLFDO6RFLHW\ RI$PHULFDS >@ 3HWHUVVRQ %$7  6WUXFWXUDO DFRXVWLF SRZHU WUDQVPLVVLRQ E\ SRLQW PRPHQW DQG SRLQW IRUFH H[FLWDWLRQ 3DUW  3ODWHOLNH VWUXFWXUHV -RXUQDO RI 6RXQGDQG9LEUDWLRQS >@6WHQ]HO+DQG%URV]H2/HLWIDGHQ]XU%HUHFKQXQJYRQ6FKDOOYRU JlQJHQ6SULQJHU9HUODJ%HUOLQ >@+DPPHU3DQG3HWHUVVRQ%$76WULSH[FLWDWLRQ3DUW,,8SSHUDQG ORZHUERXQGVIRUSRZHUWUDQVPLVVLRQ-RXUQDORI6RXQGDQG9LEUDWLRQ S >@3HWHUVVRQ%$7DQG+DPPHU36WULSH[FLWDWLRQRIVOHQGHUEHDPV -RXUQDORI6RXQGDQG9LEUDWLRQS >@&RXUDQW5DQG+LOEHUW'0HWKRGVRI0DWKHPDWLFDO3K\VLFV9RO, -RKQ:LOH\ 6RQV1HZ@5D\OHLJK-:67KHRU\RIVRXQG QGHG 9ROV'RYHU3XEOL FDWLRQV1HZ@*|VHOH.5HLKHU+DQG-HNOH56FKDOOWHFKQLVFKH8QWHUVXFKXQJHQ DQ+RO]EDONGHFNHQ%HULFKWHDXVGHU%DXIRUVFKXQJ+HIW (UQVW 6RKQ %HUOLQ >@0RUVH30DQG)HVKEDFK+0HWKRGVRIWKHRUHWLFDO SK\VLFV9RO  &KV0F*UDZ+LOO%RVWRQ0$ >@ :HFN0DQG+XPSHUW5*HUlXVFKHPLVVLRQYRQ:HUN]HXJPDVFKL QHQELHGHU%HDUEHLWXQJ,,6FKULIWHQUHLKH%XQGHVDQVWDOWIU$UEHLWVFKXW])E :LUWVFKDIWVYHUODJ%UHPHUKDYHQ >@0F,QW\UH0(6FKXPDFKHU57 DQG:RRGKRXVH-2QWKHRVFLOOD WLRQVRIPXVLFDOLQVWUXPHQWV-RXUQDORIWKH$FRXVWLFDO 6RFLHW\ RI$PHULFD S >@ )LQJEHUJ8(LQ0RGHOOI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

,PSHGDQFHDQG0RELOLW\ >@7LPRVKHQNR63DQG@3HWHUVVRQ%3URFHGXUHVIRUVRXQGSDWKTXDQWLILFDWLRQGHILQLWLRQRI VWDWHRIWKHDUW3DUW6WUXFWXUHERUQHVRXQGWUDQVPLVVLRQEHWZHHQPDFKLQ HU\ DQG IRXQGDWLRQV UHYLVLWHG LQ SDUWLFXODU ULJLG SDWKV ,QVWLWXWH RI $SSOLHG 3K\VLFV7KH1HWKHUODQGV73'6$537 >@ 6\NHV $2  'HYHORSPHQW DQG DSSOLFDWLRQ RI OLQHDU PXOWLWKHUPDO QHWZRUN WKHRU\ WR YLEUDWLRQ SUREOHPV 'LVFV 7KH &DWKROLF 8QLYHUVLW\ RI $PHULFD:DVKLQJWRQ'& >@ 0RQGRW -0 DQG 3HWHUVVRQ %  &KDUDFWHUL]DWLRQ RI VWUXFWXUH ERUQH VRXQG VRXUFHV 7KH VRXUFH GHVFULSWRU DQG WKH FRXSOLQJ IXQFWLRQ -RXUQDO RI 6RXQGDQG9LEUDWLRQS >@%HUDQHN/1RLVHDQG9LEUDWLRQ&RQWURO0F*UDZ+LOO1HZ@0RRUKRXVH$7DQG*LEEV%07RZDUGVDSUDFWLFDOFKDUDFWHUL]D WLRQIRUVWUXFWXUHERUQHVRXQGVRXUFHVEDVHGRQPRELOLW\WHFKQLTXHV-RXUQDO RI6RXQGDQG9LEUDWLRQS >@ 5 $ )XOIRUG DQG % 0 *LEEV  6WUXFWXUHERUQH VRXQG SRZHU DQG VRXUFHFKDUDFWHULVDWLRQLQPXOWLSRLQWFRQQHFWHGV\VWHPVSDUW&DVHVWXG LHVIRUDVVXPHGIRUFHGLVWULEXWLRQV-RXUQDORI6RXQGDQG9LEUDWLRQS 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

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

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

HVVHV WKLV PHDQV WKHVLPXOWDQHRXVVROXWLRQRIDVHWRIOLQHDU HTXD WLRQV)RUPRVWFDVHVLQSUDFWLFHWKHDVVXPSWLRQRIOLQHDUSURFHVVHV LVMXVWLILHG2QHH[FHSWLRQLVWKHWKLQSODWHXQGHUJRLQJODUJHDPSOL WXGHYLEUDWLRQVDWORZIUHTXHQFLHV H  %\ PHDQVRIWKHDPSOLWXGHVGHWHUPLQHGLQG WKHHQHUJ\EDVHGUH IOHFWLRQDQGWUDQVPLVVLRQHIILFLHQFLHVDUHFDOFXODWHG 6LQFH WKH DWWHQXDWLRQ RI VWUXFWXUHERUQH VRXQG PRVWO\ LV IUHTXHQF\ GH SHQGHQWWKHDQDO\VHVDUHFDUULHGRXWZLWKZDYHVDWDIUHTXHQF\)RUEUHY LW\ PRUHRYHU WKH RSHUDWLRQ 5H >«HMZW@ LV RPLWWHG DV LV WKH XQGHU EDU GHQRWLQJDFRPSOH[ TXDQWLW\ $WWHQXDWLRQRI/RQJLWXGLQDO:DYHV 7KH VXEVHTXHQW DQDO\VLV ZKLFK LV FDUULHG RXW IRU URGV LV DOVR YDOLG IRU SODWHVDQGQRUPDOLQFLGHQFHXSRQWKHGLVFRQWLQXLW\7KHRQO\PRGLILFDWLRQ LV WKDW = @ Y 



 : 

  5H > =  @ Y   

 

WKHWUDQVPLVVLRQHIILFLHQF\ W ::LVREWDLQHGDV W

= =

§  = · ¨ ¸ © =  =  ¹







=  =   =   =







 

7KLV H[SUHVVLRQ FDQ DOVR EH GHULYHG IURP WKH UDWLR YY REWDLQHG E\ HOLPLQDWLQJYZKLFKGHILQHVWKHUHIOHFWLRQFRHIILFLHQWUFI6HFWVDQG 7KHUHE\

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG =  =   =  = 

U

 

7KHVTXDUHRIWKLVH[SUHVVLRQZKLFKUHSUHVHQWVWKHUDWLRRIUHIOHFWHGSRZHU WRWKHLQFLGHQWLVWHUPHGWKHUHIOHFWLRQHIILFLHQF\ U

 



U 

6LQFH WKH QRQUHIOHFWHG SRZHU PXVW HTXDO WKDW WUDQVPLWWHG LQ WKH SUHVHQW FDVHWKHWUDQVPLVVLRQHIILFLHQF\FDQEHREWDLQHGDOVRDV W

 U

 



 U 

ZKLFKVXEVHTXHQWO\OHDGVWR   7KHUHODWLRQLQ  LVQRWYDOLGIRUDOODWWHQXDWLRQSUREOHPVKRZHYHU )RUGLVFRQWLQXLWLHVLQYROYLQJGLVVLSDWLYHVWUXFWXUDOSDUWVWKHQRQUHIOHFWHG SDUWRIWKHSRZHUHQFRPSDVVHVERWKWKDWGLVVLSDWHGDQGWKDWWUDQVPLWWHG 7KH DWWHQXDWLRQ RIIHUHG E\ D VLQJOH FURVVVHFWLRQDO DUHD FKDQJH LV QRW SDUWLFXODUO\ ODUJH :LWK WKH DUHD UDWLR GHQRWHG E\ '  66 IRU EHDPV RU ' KKIRUSODWHVLQWURGXFHGLQ   



D 

5  ORJ  W G%

E 

W

'

 

 ' 



7KHWUDQVPLVVLRQORVVGHILQHGE\

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

0DWHULDODQG&URVV6HFWLRQDO&KDQJHV

ULVHV IURP WZR WR IRXU ,Q D FDVH ZLWK SXUH EHQGLQJ ZDYH WUDQVPLVVLRQ DFURVVDGLVFRQWLQXLW\QRWRQO\WKHWUDQVYHUVHYHORFLWLHVDQGIRUFHVPXVWEH FRQVLGHUHG EXW DOVR WKH URWDWLRQV DQG PRPHQWV DV LOOXVWUDWHG LQ )LJ  7KLVPHDQVWKDWDWWKHLQWHUIDFH Y \

Y\ 

 )\

)\  

 

Z] 

 0 ]

0 ]

 

DVZHOODV Z]

7RVDWLVI\WKHVHIRXUERXQGDU\FRQGLWLRQVWKHVROXWLRQDVVXPHGPXVWFRQ VLVWRIIRXUXQNQRZQDPSOLWXGHV,Q6HFWWKHFRPSOHWHUHIOHFWLRQRI EHQGLQJZDYHZDVDQDO\VHG%HVLGHVWKHSURSDJDWLQJZDYHVDSSHDUHGDOVR HYDQHVFHQW ZDYHV TXDVLVWDWLRQDU\ QHDUILHOGV  DV VROXWLRQV $FFRUGLQJO\ WKHYHORFLW\ILHOGVPXVWWDNHWKHIRUP Y \

Y ª¬ H  MN [  UH MN [  UM HN [ º¼  [ d 

Y\

Y ª¬WH  MN [  W M H  N [ º¼

[t

 

7KHILUVWFRQGLWLRQLQ  WKXVPHDQVWKDW U  UM  W  W M

D 



1H[WVWHSLQWKHDQDO\VLVFRQVLVWVRIGHWHUPLQLQJWKHURWDWLRQDOYHORFLW\ Z] WKHPRPHQW 0]DQGWKHFURVVVHFWLRQDOIRUFH)\$VVKRZQLQ6HFW WKHVHTXDQWLWLHVDUHGHULYDWLYHVRIWKHWUDQVYHUVHYHORFLW\JLYHQE\ Z]

wY \ w[

 0]



 % w Y\  )\ MZ w[ 

 % w Y\ MZ w[

 

8SRQ VXEVWLWXWLQJ   DQG FRQVLGHULQJ WKH FRQGLWLRQV LQ   DQG  LWLVIRXQGWKDW MNU  NUM  MN W  N W M

MN 

%N U  %N UM  % NW  % N W M  M%N U  %N UM  M% NW  % NW M

 %N   M%N 

E  F  G 

IRUWKHURWDWLRQDOYHORFLWLHVWKHPRPHQWVDQGWKHIRUFHVUHVSHFWLYHO\

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

)LJ  7UDQVPLVVLRQ RI EHQGLQJ ZDYHV DW D FURVVVHFWLRQDO DUHD FKDQJH RI D EHDP

:KLOVWWKHSRZHUWUDQVPLVVLRQIRUORQJLWXGLQDOZDYHVFDQEHFKDUDFWHU L]HGE\DVLQJOHFRQVWDQWWKHLPSHGDQFHUDWLR==WKHUHDUHDOZD\V WZR VXFK UDWLRV QHFHVVDU\ LQ WKH EHQGLQJ ZDYH FDVH )URP WKH VHW RI (TV  WKHVHFDQEHLGHQWLILHGDV N

N N



Pc % Pc%

O  O

DQG \

N % N %

Pc %  Pc%  

,QWKHODWWHUUDWLRWKHTXDQWLWLHV Pc% FDQEHVDLGWRUHSUHVHQWWKHFKDU DFWHULVWLFLPSHGDQFHZLWKUHVSHFWWREHQGLQJZDYHVDQGZLOOEHWHUPHGWKH EHQGLQJZDYHLPSHGDQFHV = %c

Pc%

 = %c

Pc % 

7KH DQDORJ\ WR WKH LPSHGDQFH IRU ORQJLWXGLQDO ZDYHV LV LQFRPSOHWH KRZHYHUVLQFHWKHEHQGLQJZDYHLPSHGDQFHVGRQRWUHSUHVHQWWKHUDWLREH WZHHQWKHIRUFHDQGYHORFLW\ RIDSRVLWLYHO\WUDYHOOLQJZDYHEXWLVUDWKHUD JHRPHWULFPHDQRIWKHIRUFHDQGPRPHQWLPSHGDQFHV :LWKWKHLQWURGXFWLRQRIWKHUDWLRVNDQG\LQ  DQGGHYHORSLQJWKH QXPHUDWRUDQGGHQRPLQDWRUGHWHUPLQDQWVFRUUHVSRQGLQJWRWKHFRHIILFLHQWV VRXJKWRQHREWDLQVWKHUHIOHFWLRQFRHIILFLHQWIRUSURSDJDWLQJZDYHVDV

0DWHULDODQG&URVV6HFWLRQDO&KDQJHV

U

\   N   MN   \



N   \  \   N  



 

DQGWKDWZLWKUHVSHFWWRUHIOHFWHGHYDQHVFHQWZDYHVDV UM

N   \   MN   \  N   \  \   N  



 

6LPLODUO\ WKH WUDQVPLVVLRQ FRHIILFLHQW IRU SURSDJDWLQJ ZDYHV LV IRXQG WR EHJLYHQE\ W

   N   \ N   \  \   N  



 

ZKHUHDVWKDWZLWKUHVSHFWWRWKHQHDUILHOGLVJLYHQE\ WM

   \  M N   \ N   \  \   N  



 

2ISULPDU\ LQWHUHVWLVRIFRXUVHWKHWUDQVPLVVLRQHIILFLHQF\,WFDQHLWKHU EH IRXQG IURP ±_U_ RU IURP PXOWLSO\LQJ W E\ WKH UDWLR Pc F  PcF N\ VLQFHWKHSRZHULVJLYHQE\WKHSURGXFW PcF IRUDJLYHQ YHORFLW\DPSOLWXGHFI(T  ,UUHVSHFWLYHRIZD\WKHWUDQVPLVVLRQHI ILFLHQF\LVREWDLQHGDV 

W

ª  N\   N   \ º « »  « N   \   \   N  » ¬ ¼

 

,QWKLVFRQWH[WWKHGLVFXVVLRQZLOOEHOLPLWHGWRWKHFDVHRIHTXDOPDWH ULDOSURSHUWLHVRQHLWKHUVLGHRIWKHGLVFRQWLQXLW\EXWZLWKDFKDQJHRIWKH FURVVVHFWLRQDODUHDRUSODWHWKLFNQHVVUHVSHFWLYHO\%\LQWURGXFLQJWKHSD UDPHWHU'DVGHILQHGLQFRQMXQFWLRQZLWK(T D NEHFRPHV'DQG\ LV'7KLVPHDQVWKDWWKHWUDQVPLVVLRQHIILFLHQF\VLPSOLILHVWR 

W

ª '     '     '     '    º «  »       '   '    ¼ ¬'    '

 

ZKLFKDJDLQLVUHFLSURFDOO\V\PPHWULF ,Q )LJ  WKH WUDQVPLVVLRQ HIILFLHQF\ IRU WKLV FDVH LV VKRZQ WRJHWKHU ZLWK WKDW IRU ORQJLWXGLQDO ZDYHV :KLOVW WKH EHQGLQJ ZDYH WUDQVPLVVLRQ RFFXUV DOPRVW XQLPSHGHG IRU VPDOO DUHD RU WKLFNQHVV FKDQJHV LW LV VHHQ WKDW WKH LQFUHDVH LQ DWWHQXDWLRQ IRU ODUJH DUHD RU WKLFNQHVV UDWLRV LV ODUJHU

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

IRUEHQGLQJZDYHVWKDQIRUORQJLWXGLQDO7KHRUHWLFDOO\WKHWUDQVPLVVLRQHI ILFLHQF\DPRXQWVWR 5 G% IRUDUDWLRRI' 6XFKDQDUHD FKDQJHKRZHYHUZKLFKLVKDUGO\UHDOLVWLFIURPDSUDFWLFDOSRLQWRIYLHZ ZRXOGSUREDEO\LPSO\DYLRODWLRQRIWKHDVVXPSWLRQVXQGHUO\LQJWKHWKHRU\ DQGLQWHUIDFHFRUUHFWLRQVZRXOGEHUHTXLUHG)RUHTXDOPDWHULDOSURSHUWLHV DUHDRUWKLFNQHVVUDWLRVODUJHUWKDQDUHKDUGO\FRQFHLYDEOHIXUQLVKLQJD PD[LPXPDWWHQXDWLRQRIG%

)LJ  7UDQVPLVVLRQHIILFLHQF\RIDFURVVVHFWLRQDODUHDFKDQJHDVIXQFWLRQRI WKHDUHDUDWLR  EHQGLQJDQG  ORQJLWXGLQDOZDYHV

7KH IRUPXOD LQ   KDV EHHQ H[SHULPHQWDOO\ FRUURERUDWHG IRU VXIIL FLHQWO\ORQJEHDPV>@,WKDVDOVREHHQVXFFHVVIXOO\DSSOLHGLQWKHHVWL PDWLRQ RI IODQNLQJ WUDQVPLVVLRQ LQ EXLOGLQJ DFRXVWLF PHDVXUHPHQW IDFLOL WLHVZKHUHWKHWHVWZDOOLVSODFHGLQDQRSHQLQJRIDWKLFNSDUWLWLRQLQJZDOO >@

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV

,QFLGHQW%HQGLQJ:DYH 2IHYHQJUHDWHUSUDFWLFDOLPSRUWDQFHWKDQWKHSUHYLRXVO\GLVFXVVHGGLVFRQ WLQXLWLHV HVWDEOLVKHG E\ FKDQJHV LQ PDWHULDO SURSHUWLHV RU FURVVVHFWLRQDO DUHDDUHWKRVHHVWDEOLVKHGE\VWUXFWXUDOFRPSRQHQWVWKDWPHHWDWULJKWDQ JOHV6XFKDFRQILJXUDWLRQLVUHDOL]HGIRUH[DPSOHE\DGMDFHQWZDOOVRID URRP WKH FHLOLQJ RU IORRU DQG WKH VXSSRUWLQJ ZDOOV DQG EHDPV LQ D WUXVV VHH )LJ  7R JDLQ VRPH LQVLJKW LQWR WKH FRUUHVSRQGLQJ SKHQRPHQD LV

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV

FRQVLGHUHGDVLPSOHFRUQHUIRUPHGE\WZREHDPVRUSODWHVSRVVLEO\ ZLWK GLIIHULQJ PDWHULDO SURSHUWLHV DQG WKLFNQHVVHV )RU D ULJLG FRQQHFWLRQ VXFK WKDWDEHQGLQJPRPHQWLVWUDQVPLWWHGLWIROORZVWKDW 0 ]

0 ] 

D 

DVZHOODVFRQWLQXLW\LQURWDWLRQDOYHORFLW\ Z]

Z]  

E 

7KXV WKHVH WZR ERXQGDU\ FRQGLWLRQV UHPDLQ WKH VDPH DV LQ WKH SUHYLRXV VHFWLRQ:LWKUHVSHFWWRWKHRWKHUERXQGDU\FRQGLWLRQVLWPXVWEHREVHUYHG WKDWQRWRQO\IRUFHVDQGYHORFLWLHVDUHSUHVHQWZKLFKDFWSHUSHQGLFXODUWR WKHQHXWUDOOD\HUVRIWKHDGMRLQLQJVWUXFWXUDOFRPSRQHQWVEXWDOVRWKRVHWKDW DFWSDUDOOHOWKHUHWR7KLVPHDQVWKDW ORQJLWXGLQDOZDYHVDULVHZLWKWKHDV VRFLDWHGWUDQVPLVVLRQDQGUHIOHFWLRQDVGLVFXVVHGLQ6HFW

)LJ7UDQVPLVVLRQDWDIOH[XUDOO\ULJLGFRUQHU

:LWKWKHQRWDWLRQLQWURGXFHGLQ)LJWKHILHOGDVVXPHGIRUWKHKRUL ]RQWDOSDUWFDQEHGHVFULEHGE\ Y% [

Y% H  MN [  UH MN [  UM H N [ 

Y/ [

Y% U%/ H MN / [ 

DQGIRUWKHYHUWLFDOE\

D 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG Y% \

Y% WH MN \  W M H  N \ 

Y/ \

Y% W %/ H MN / \ 

E 

,WLVKHUHREVHUYHGWKDWWKHLQFLGHQWZDYHLVDIOH[XUDOZDYHRIDPSOLWXGH Y% DQGWKDWWKHORQJLWXGLQDOZDYHVDULVLQJDUHSURSDJDWLQJIURPWKHFRUQHU ZLWKDPSOLWXGHV Y% U%/ DQG Y% W%/ UHVSHFWLYHO\ %HVLGHVWKHDOUHDG\LQ DE LQWURGXFHGERXQGDU\ FRQGLWLRQVWKHIRO ORZLQJUHODWLRQVPXVWKROG 



)/

 )% 



 )%

)/ 

 Y/

 Y% 

 Y%

Y%  

FI 

ZKHUHWKHVLJQLVJLYHQE\WKHFKRVHQGLUHFWLRQVVHH)LJ7KHVLJQVDUH VHOHFWHGVXFKWKDWE\VLPSO\URWDWLQJWKHYHUWLFDOSDUWƒWKHGLUHFWLRQVIRU WKHKRUL]RQWDODUHSUHVHUYHG 7RGHWHUPLQHWKHXQNQRZQUHIOHFWLRQDQGWUDQVPLVVLRQIDFWRUVWKHILHOGV LQ  DUHVXEVWLWXWHGLQWR  DQGLQWR  UHJDUGLQJWKHORQJLWXGL QDOZDYHV%\HPSOR\LQJWKHERXQGDU\FRQGLWLRQVLQ  WKHIROORZLQJ VHWRIOLQHDUHTXDWLRQVLVREWDLQHG %N   U  UM % N

W  W M 

N  M  MU  UM N

 MW  W M 



( 6N / U%/ Z



%N M  MU  UM MZ U%/

% N MW  W M  MZ 

( 6  N /  W %/  Z

 W  W M 

  U  UM

W%/ 

:LWKWKHDEEUHYLDWLRQVGHILQHGDERYH \

% N % N

 N

N  N

DQGDGGLWLRQDOO\ ( 6N / % N ( 6  N /  %N

PcF/Z N PcF/   F/Z Pc N Pc F%  Pc F/    PcF% E 

  E

 

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV

WKHHTXDWLRQVIRUWKHUHIOHFWLRQVDQGWUDQVPLVVLRQVIDFWRUVUHGXFHWR     U  UM W  W M   M  MU  UM  MW  W M  \ N    W  W M W  MW M    U  MUM   U  UM  E E

 

ZKHQ U%/ DQG W%/ DUH HOLPLQDWHG $IWHU VRPH IXUWKHU PDQLSXODWLRQV LW LV IRXQGWKDW E  E   M   E E   1 \    E   E E   N    E  E E   U 1 \   E  E E   N   E   E E   M 1

W



UM

  E   U   E     ME

WM

  E W   M E

DE 

FG 

ZKHUHLQ 1

\   E   E   E E   N    E  E   E E   M \  N   E E  

%\REVHUYLQJWKDWIRUWKHSRZHUSURSDJDWHGE\IOH[XUDOZDYHVDQGE\ ORQ JLWXGLQDOZDYHV PcF% Y%

:%

 :/

 

PcF/ Y/ 

UHVSHFWLYHO\WKHUHIOHFWLRQDQGWUDQVPLVVLRQHIILFLHQFLHVDUHGHULYHGDV U %%

:% :%

U %/

:/ :%

W %/

:/  :%

U



 W %%

PcF/  U%/ c PF% Pc F/   W %/ PcF%

:%  :%

Pc F%   W PcF%

 F/  W M  F%



N\ W 

 

    U  UM  E 

$V H[SHFWHG IURP DQ HQHUJ\ FRQVHUYDWLRQSRLQWRIYLHZDQGDVLVUHDGLO\ GHPRQVWUDWHGE\VRPHVPDOOFDOFXODWLRQ

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG U %%  W %%  U %/  W %/

 

 

,Q)LJLVH[HPSOLILHGWKHWUDQVPLVVLRQHIILFLHQF\ W%%DQGWKHYHORF LW\UDWLR_Y%__Y%_7KHIRUPHUTXDQWLW\LVDSRZHUUDWLRZKHUHDVWKHODWWHU LVDUDWLREHWZHHQWZRPHDQVTXDUHYHORFLWLHV7KHYHORFLW\UDWLRFRQVWLWXWH DQ HDVLO\ PHDVXUHG TXDQWLW\ ZKHQ WKH WZR VWUXFWXUDO SDUWV DUH VHYHUDO ZDYHOHQJWKVORQJDQGWKHUHFHLYHULVPRGHUDWHO\GDPSHG

)LJ  0HDQ VTXDUH YHORFLW\ UDWLR DQG WUDQVPLVVLRQ HIILFLHQF\ IRU D EHQGLQJ ZDYH LQFLGHQW XSRQ D FRUQHU PDGH XS E\ EHDPV RI WKH VDPH PDWHULDO «  K I  F/     K I  F/  DQG ±±±  K I  F/ 

:LWK HTXDO WKLFNQHVV IRU WKH WZR SDUWV WKH GLIIHUHQW UHIOHFWLRQ DQG WUDQVPLVVLRQHIILFLHQFLHVDUHSORWWHGDVIXQFWLRQVRIQRUPDOLVHGIUHTXHQF\ LQ )LJ  ,W LV VHHQ WKDW IRU ORZ IUHTXHQFLHV WKH FRXSOLQJ LV VPDOO EH WZHHQ EHQGLQJ DQG ORQJLWXGLQDO ZDYHV U%/ DQG W%/ 7KH FRUUHVSRQGLQJ DV\PSWRWLFYDOXHVIRUU%%DQGW%%DUHJLYHQE\ U %% |

\  N 

\  N



 W %% |

N\

\  N

ZKLFKDUHYDOLGIRUEDQGEOHVVWKDQ





 

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV

)LJ (QHUJ\ GLVWULEXWLRQIRUDEHQGLQJZDYHLQFLGHQWXSRQDFRUQHUPDGHXS E\WZRHTXDOEHDPV K K 

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

Y/ UH MN [  UM H N [ 

Y/ [

Y/ H  MN/ [  U// H MN/ [ 

Y%  \

Y/ WH  MN \  W M H N [ 

Y/  \

Y/ W// H  MN / \ 

 

)RUWKHXVXDOO\LQWHUHVWLQJFDVHZLWKHTXDOWKLFNQHVVRQHLWKHUVLGHRIWKH FRUQHUIRUZKLFK\ N DQGE E EWKHUHIOHFWLRQDQGWUDQVPLV VLRQHIILFLHQFLHVUHDG

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

W /%

:%  :/

E  E    E  E 

U /%

:% :/

E   E  E 

W //

:/  :/

E  W /%    E  E 

U //

:/ :/

  U /%    E  E 

W %/  U %/ 

 

7KH UHODWLRQV W/%  W%/ DQG U/%  U%/ DUH IRXQG IURP D FRPSDULVRQ ZLWK  RUDVDFRQVHTXHQFHRIWKHUHFLSURFLW\SULQFLSOH7KHIUHTXHQF\GH SHQGHQFLHVRIW/%U/%DQGW//DUHVKRZQLQ)LJ

)LJ (QHUJ\ GLVWULEXWLRQIRUDORQJLWXGLQDOZDYHLQFLGHQWXSRQDFRUQHUPDGH XSE\WZRHTXDOEHDPV K K 

5LJKW$QJOHG%UDQFKHVZLWK,QFLGHQW%HQGLQJDQG /RQJLWXGLQDO:DYHV 7KH DQDO\VLV RI VWUXFWXUHERUQH VRXQG WUDQVPLVVLRQ DW GLVFRQWLQXLWLHV ZDV GHWDLOHG LQ WKH WZR SUHYLRXV VHFWLRQV +HUH 7 DQG  MXQFWLRQV ZLOO EH WUHDWHGVLPXOWDQHRXVO\$7MXQFWLRQFDQEHFRQVLGHUHGDVDFURVVLQJZKHUH RQHEUDQFKLVPLVVLQJ :LWK WKH QRWDWLRQV GHILQHG LQ )LJ  WKH ILHOGV LQYROYHG FDQ EH DV VXPHGDV

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV Y% [

Y% $% H MN [  UH MN [  UM H N [ 

Y/ [

Y% $/ H  MN/ [  U%/ H MN/ [ 

Y%  \

Y% W H  MN \  W M H  N \  Y/  \

Y%  [

Y% WH  MN [  W M H  N [  Y/  [

Y%  \

Y%

W H

 MN \



 Y \

 W  M H  N \

/

Y% W %/ H MN/  \ 

 

Y% W %/ H  MN/  [  Y% W %/ H MN/  \ 

,QWKHVHWRIHTXDWLRQVDERYHDEHQGLQJZDYHRIDPSOLWXGH$%LVLQFLGHQW IURP [  f DV ZHOO DV D ORQJLWXGLQDO ZDYH RI DPSOLWXGH $/ ,I RQO\ RQH ZDYHW\SHLVSUHVHQWWKHDPSOLWXGHRIWKHRWKHUYDQLVKHVY%LVDXQLWYH ORFLW\ZKLFKVXEVHTXHQWO\FDQFHOV,QDQDORJ\ZLWK DI WKHERXQG DU\FRQGLWLRQVLQWKHFDVHUHDG 0

0  0  0 

Z

Z

 Z

)/

 )%   )/   )% 

Y/

 Y% 

Y%

Y/ 

 Y/  Y%

Z

 Z Y/ 

Y% 

Z 

 )%  Y/  Y%

)/   )%   )/    Y%   Y/  

)LJ7UDQVPLVVLRQDWDULJLGEHDP SODWH FURVVLQJ

 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

,Q WRWDO WKHUH DUH  HTXDWLRQV IRU WKH  XQNQRZQV LQ WKH VHW   ZKLFKDIWHUVRPHHYLGHQWVLPSOLILFDWLRQVOHDGVWRWKHOLQHDUVHWRIHTXDWLRQV  

E 

$W

EHVLGHV WKH UHODWLRQV W %/ W %/ $%  U  UM  DQG W%/ FLHQWPDWUL[$FRPSULVHVWKHIROORZLQJHOHPHQWV D

  \ 

 D

D

\ 

D

M  D

D

M N

D

M  D

D

  %  M%  %

D

%   M  D

D

M/  /  M/  

 D

 \ 

\ 

 D

 N

  D

\  

\  \  

  M N 

  D

 D

 D

 D

N  

N    M 

 D

N    M  D

 D

$/  U%/  7KH FRHIIL

N  

 M  %  M%  % 

/   M 

D

 /   M 

DQGWLVDFROXPQYHFWRUZLWKWKHHOHPHQWVW7 ^U UM W W W U%/`$OVRE LVDFROXPQYHFWRUZKHUHWKHHOHPHQWVDUH E

$%   \   $/ \   \   E

E

$% M  N   E

E

$%   %  M%  %  E

M$%  $/N  

M$%  $/N   $/   M/  /  M/ 

7KHDEEUHYLDWLRQVXVHGUHDG \ N / %

% N %N

 \

% N %N

 \

N N N  N  N  N N N PcF/  Pc F%   /  / PcF/ PcF/ PcF%  Pc F/   %  % c PF% PcF%

% N  %N

Pc F%   PcF/ Pc F/   PcF%

 

)LJXUHVDEVKRZWKHUHVXOWVIRUVRPHH[DPSOHVSDUWLDOO\GLVSOD\HG ORJDULWKPLFDOO\DVWUDQVPLVVLRQORVV7KHUHLQWKHSUDFWLFDOO\ FRPPRQFRQ ILJXUDWLRQ LV LQYHVWLJDWHG IRU ZKLFK WKH VWUXFWXUDO PHPEHUV  DQG  DUH HTXDODVDUHDQGLISUHVHQW,QWKHOHJHQGWR)LJEDUHDOVRVXSSOLHG VRPHDV\PSWRWLFH[SUHVVLRQVZKLFKDUHYDOLGIRUDQLQFLGHQWEHQGLQJZDYH $/  DQGK IF/

5LJKW$QJOHG&RUQHUVDQG%UDQFKHV

,Q )LJ  DUH DGGLWLRQDOO\ VKRZQ VRPH PHDVXUHPHQWV UHVXOWV >@ REWDLQHG IRU SHUVSH[ EHDPV 7R UHGXFH WKH LQIOXHQFH RI UHIOHFWLRQV LQ PHPEHUVDQGWKHEHDPHQGVZHUHHPEHGGHGLQDPL[RIVDQGDQG VDZGXVWRYHUDOHQJWKRIP

)LJ D 7UDQVPLVVLRQ ORVV IRU D ZDOOFURVVLQJ WRS  DQG D 7MXQFWLRQ ERWWRP ZLWK DOO EUDQFKHV RI WKH VDPH PDWHULDO ,QFLGHQW EHQGLQJ ZDYH $/    «  K I  F/     K I  F/  DQG ±±±  K I  F/ 

)LJE(QHUJ\GLVWULEXWLRQIRUEHQGLQJZDYHVLQFLGHQWXSRQDZDOOFURVVLQJ ZLWK DOO EUDQFKHV RI WKH VDPH PDWHULDO 7RS KK   %RWWRP KK   W%/ | U%/ |  $V\PSWRWLFH[SUHVVLRQVIRUDZDOOFURVVLQJ

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG 

W %%

 N  \     \   N  

W %%

   \   N   \   N  

W %%



W %%

  \



DQGD7MXQFWLRQ N  \   \   N 





 



 N   \   N  

)LJ&RPSDULVRQRIFDOFXODWHG  DQGPHDVXUHG R UHIOHFWLRQDQGWUDQV PLVVLRQHIILFLHQFLHV

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

(ODVWLF,QWHUOD\HUV

)LJ7UDQVPLVVLRQORVVDWDZDOOFURVVLQJZLWKPDVVHVSHUXQLWDUHDRIPDQG PUHVSHFWLYHO\$IWHU>@

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³KDUG´ RQO\ DFRXVWLFDOO\ ³VRIW´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

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

$WWHQXDWLRQRI/RQJLWXGLQDO:DYHV 7KHPHWKRGFKRVHQKHUHWRDQDO\VHWKHWUDQVPLVVLRQWKURXJKSODQHSDUDOOHO OD\HUVLVEDVHGRQ6HFW,QYLHZRIWKHPDWKHPDWLFDOODERULWLVQRW WKHPRVWHOHJDQW7KHUHE\WKHHODVWLFOD\HULVVHHQDVDGRXEOHFKDQJHLQ PDWHULDOZKHUHWKHUHVXOWLQJPXOWLSOHUHIOHFWLRQVDUHWDNHQLQWRDFFRXQW )URP (T  DQLQFLGHQWORQJLWXGLQDOZDYHRIDPSOLWXGHYJLYHVULVH WRDZDYHLQWKHLQWHUOD\HUZLWKWKHYHORFLW\ Y f

= Y =  = 

 

ZKHUHE\WKHLQGH[f GHQRWHVWKDWWKHH[SUHVVLRQLVYDOLGRQO\IRUDQLQIL QLWHO\ H[WHQGHG UHFHLYLQJ PHGLXP :KHQ WKLV SULPDU\ ZDYH UHDFKHV WKH RWKHUHQGRIWKHLQWHUOD\HUDWUDQVPLWWHGZDYHDULVHVDQGZLWKWKHUROHVRI =DQG=QRZLQWHUFKDQJHGWKHYHORFLW\LV Yf

=  Y f =  = 

 = = 

=  = 



Y

 

7KH WUDQVPLVVLRQ HIILFLHQF\ IRU VXFK D WUDQVPLVVLRQ WKURXJKDQLQWHUOD\HU FRUUHVSRQGLQJWRDFDVFDGLQJRIWZRVLPLODUGLVFRQWLQXLWLHVZRXOGWKXVEH HTXDO WR WKH VTXDUH RI WKH WUDQVPLVVLRQ HIILFLHQF\ IRU RQH GLVFRQWLQXLW\ JLYHQ E\ (T   7KLV UHDVRQLQJ KRZHYHU LV QRW FRUUHFW DV FDQ EH H[ SHFWHG IURP WKH IDFW WKDW WKH OHQJWK O RI WKH LQWHUOD\HUGRHVQRWDSSHDU LQ  5DWKHUPXVWEHFRQVLGHUHGWKHIDFWWKDWWKHSULPDU\ZDYHLQWKHLQ WHUOD\HUZKLFKLVUHIOHFWHGRYHUDQGRYHUDJDLQDWWKHWZRHQGVZLWKWKHUH IOHFWLRQIDFWRU U

=   =  =  = 

 

ORVHVHQHUJ\DWWKRVHUHIOHFWLRQV 7KHUHVXOWDQWZDYHSURSDJDWLQJLQWKHSRVLWLYH[GLUHFWLRQKHQFHLVFRP SRVHGRIDQLQILQLWHQXPEHURIFRPSRQHQWVZKHUHWKHLUUHODWLYHSKDVHVDUH GHWHUPLQHGE\WKHWRWDO³URXQGWULS´GLVWDQFHOWKHSKDVHVSHHGFRIWKH LQWHUOD\HUPDWHULDODQGDQ\SKDVHFKDQJHVSURGXFHGDWWKHUHIOHFWLRQ$F FRUGLQJO\ Y 

Y f   U  H  M  N O  U  H M  NO  

 

6LQFHUWKLVJHRPHWULFVHULHVFRQYHUJHVWR Y 

Yf    U  H  M  N O

 

(ODVWLF,QWHUOD\HUV

%\ DSSO\LQJ(T  WRWKLVUHVXOWDQWZDYHWKDWLPSLQJHVDWWKHIDUHQGRI WKH LQWHUOD\HU WKH ZDYH WUDQVPLWWHG WR WKH UHPRWH SDUW RI WKH VWUXFWXUH LV REWDLQHGDV Y  H  MNO

Y

=  =  = 

 

ZKHUH=DQG=DUHLQWHUFKDQJHGWRDGDSWWRWKHSUHVHQWFDVH 8SRQFRPELQLQJ(TV  DQG  WR  WKHWUDQVPLVVLRQHIIL FLHQF\VRXJKWLVIRXQGWREHJLYHQE\ W

Y Y





 == H  MN O

=  = 



 

 =   = H  MN O



§ = = · FRV  N O  ¨    ¸ VLQ  NO  © =  = ¹

 

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

  SK =  = 

 

7KLV LV YDOLG SURYLGHG WKH PDWHULDO SURSHUWLHV DUH LQGHSHQGHQW RI IUH TXHQF\DFRQGLWLRQWKDWLVIDUIURPDOZD\VIXOILOOHG 0RVW RIWHQ WKH HODVWLF LQWHUOD\HUV DUH VR WKLQ WKDW WKH DSSUR[LPDWLRQV FRVNO |± NO DQGVLQNO | NODUHYDOLG7KLVPHDQVWKDWWKHWUDQV PLVVLRQHIILFLHQF\LQ  WDNHVWKHIRUP 

W



  N O



§ = = ·   ¨    ¸ N O  © =  = ¹



 

$VLVWREHH[SHFWHGWKHWUDQVPLVVLRQHIILFLHQF\GHFUHDVHVZLWKLQFUHDVLQJ OHQJWKRUWKLFNQHVVRIWKHLQWHUOD\HU6LQFHHYHU\GLPHQVLRQLQDZDYHILHOG VKRXOGEHFRPSDUHGZLWKWKHJRYHUQLQJZDYHOHQJWKDQGWKHZDYHOHQJWKRI WKH ORQJLWXGLQDO ZDYH LQ WKH SUHVHQW FDVHLVLQYHUVHO\ SURSRUWLRQDOWRIUH

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

TXHQF\ WKH WKLFNQHVV GHSHQGHQFH LV WKH VDPH DV WKH IUHTXHQF\ GHSHQG HQFH 0RUHRYHU LW PDNHV QR GLIIHUHQFH ZKHWKHU = LV JUHDWHU RU VPDOOHU WKDQ=RULIWKHPDWHULDOVDUHWKHVDPHZKHWKHUWKHFURVVVHFWLRQDODUHDRI WKHLQWHUOD\HULVODUJHURUVPDOOHUWKDQWKDWRIWKHSULPDU\VWUXFWXUH )RUDQHODVWLFOD\HURUDFURVVVHFWLRQDOUHGXFWLRQWRUHDOL]HDSUDFWLFDOO\ DSSUHFLDEOH DWWHQXDWLRQ = LV DOZD\V WKDW VPDOO FRPSDUHG ZLWK = WKDW  FDQEHIXUWKHUVLPSOLILHGLQWR W

ª § = N O · º «  ¨   ¸ » «¬ ©  =  ¹ »¼



ª § Z = O · º  «  ¨ ¸ »  ( 6 «¬ ©   ¹ »¼





ª § Z = · º «  ¨  ¸ »  «¬ ©  V ¹ »¼

 

)URP WKH ODVW H[SUHVVLRQ LW LV FOHDU WKDW WKH VWLIIQHVV V  (6O RI WKH WKLQLQWHUOD\HUFRQWUROVWKHWUDQVPLVVLRQ 2QHFRXOGKDYHREWDLQHGWKHH[SUHVVLRQLQ  PRUHVLPSO\ E\ FRQ VLGHULQJ i SULRUL RQO\ WKH VWLIIQHVV RI WKH LQWHUOD\HU 7KH IRUFHV RQ HLWKHU VLGHRIWKHLQWHUOD\HUPXVWWKHQEHHTXDO )

)  )

)

 

DQGUHODWHGWRLWVFRPSUHVVLRQYLDLWVVWLIIQHVVDV Y  Y MZ

 )  6

 

%\QRWLQJWKDW Y

Y  Y

 )  ) =

 

)  =

 

DQG Y

(T  FDQEHUHZULWWHQDV )  )

§ MZ = · ) ¨  ¸ 6 ¹ ©

ZKLFK WRJHWKHU ZLWK   IXUQLVKHV D UHODWLRQ EHWZHHQ WKH LQFLGHQW DQG WUDQVPLWWHG SDUWV IURP ZKLFK WKH WUDQVPLVVLRQ HIILFLHQF\ LQ   LV UH FRYHUHG $VSRLQWHGRXWWKHDSSURDFKRIVXPPLQJWKHPXOWLSO\UHIOHFWHGZDYHV LQ  LVQRWWKHPRVWHOHJDQW$OWHUQDWLYHO\RQHPD\DVVXPHWKHIRUFH DQGYHORFLW\ILHOGV

(ODVWLF,QWHUOD\HUV Y

$H  MN [  %H MN [ 

)

=  $H  MN [  %H MN [

 

IRU WKH LQWHUOD\HU DQG DSSO\ H[SUHVVLRQV FRUUHVSRQGLQJ WR (TV   DQG   IRU WKH ORQJLWXGLQDO ZDYHV EHIRUH DQG DIWHU WKH OD\HU 7KH UHVXOWLQJ IRXUXQNQRZQFRHIILFLHQWVDUHWKHQGHWHUPLQHGE\ WKHFRQGLWLRQVRIHTXDO LW\LQIRUFHDQGYHORFLW\DWWKHLQWHUIDFHVRQHLWKHUVLGHRIWKHHODVWLFOD\HU ,QVWHDG RI   LW ZRXOG KDYH EHHQ SRVVLEOH WR ZULWH WKH YHORFLW\ DQG IRUFHDV )  VLQ N [

Y [

Y  FRV N [  M

) [

 M=  Y  VLQ N [  )  FRV N [

=



 

+HUHLQY  DQG)  DUHWKHYHORFLW\DQGIRUFHDWWKHLQFLGHQWVLGHRI WKH HODVWLF OD\HU UHVSHFWLYHO\ 7KLV VR FDOOHG IRXUSROH UHSUHVHQWDWLRQ KDV VRPHDGYDQWDJHVZKHQVHYHUDOILQLWHURGVDQGHODVWLFOD\HUVDUHFRPELQHG )RUWKHDLUERUQHVRXQGDQDORJXHRIUHDFWLYHVLOHQFHUWKHWUDQVPLVVLRQWKH RU\RIDFRXVWLFILOWHUVLVIDUGHYHORSHG>@ $WWHQXDWLRQRI%HQGLQJ:DYHV 7KH ILQGLQJ WKDW DQ HODVWLF LQWHUOD\HU FDQ EH UHJDUGHG DV D SDUWLWLRQLQJ VSULQJ IRU ORQJLWXGLQDO ZDYHV DW QRW WRR KLJK IUHTXHQFLHV FDQ EH WUDQV IHUUHG WR WKH FDVH RI LQFLGHQW EHQGLQJ ZDYHV 7KH VLPSOH EHQGLQJ ZDYH WKHRU\ LVYDOLGRQO\IRUVXFKIUHTXHQFLHV)RUDQHODVWLFLQWHUOD\HURIOHQJWK FRPSDUDEOHZLWKWKHIOH[XUDOZDYHOHQJWKWKHDQDO\VLVFDQEHSHUIRUPHGDV GHVFULEHGLQFRQMXQFWLRQZLWK(TV  DQG  7KLVPHDQVWKDWWKH ZDYHILHOGVLQWKHVWUXFWXUHVRQHLWKHUVLGHRIWKHHODVWLFOD\HUDUHDVVXPHG DQDORJRXVWRWKRVHLQ  DQGWKHILHOGLQWKHOD\HULVGHVFULEHGE\   DQG  RU  )URPFRQWLQXLW\LQPRWLRQDQGHTXLOLEULXPRIIRUFHV DW WKH WZR HQGV RI WKH OD\HU WKH XQNQRZQ FRHIILFLHQWV FDQ EH VROYHG ,W VKRXOGEHSRLQWHGRXWKRZHYHUWKDWDOLPLWLQJYDOXHSURFHVVZLWKUHVSHFW WR VKRUW RU WKLQ OD\HUV IXUQLVK DQ LQFRUUHFW UHVXOW EHFDXVH VKHDU VWLIIQHVV DQG URWDWLRQDO LQHUWLD DUH QRW HQFRPSDVVHG E\ WKH VLPSOH EHQGLQJ WKHRU\ )RUVXFKDOLPLWLQJYDOXHSURFHVV7LPRVKHQNRWKHRU\PXVWEHHPSOR\HG $V LQ WKH FDVH RI D VLQJOH FURVVVHFWLRQDO DUHD FKDQJH WKH ILUVW WZR ERXQGDU\FRQGLWLRQVIRUDWKLQPDVVOHVVHODVWLFOD\HUDUH )\

)\ 

 0 ]

0 ]  

D 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

VLQFHVXFKDOD\HUGRHVQRWDIIHFWWKHWUDQVIHURIIRUFHVDQGPRPHQWV7KH NLQHPDWLF YDULDEOHV RQ WKH WZRVLGHVRIWKHHODVWLFOD\HUGLIIHUE\WKHGH IRUPDWLRQV RI WKH OD\HU WKH ODWWHU RI ZKLFK DUH SURSRUWLRQDO WR WKH IRUFHV DQGPRPHQWVPHDQLQJWKDW Y \  Y \ 

MZ )  .

 Z]  Z] 

E 

MZ 0   & 

,Q WKHVH UHODWLRQV . LV WKH VKHDU VWLIIQHVV DQG & WKH URWDWLRQDO VWLIIQHVV ZKLFKE\PHDQVRI(TV    DQG  FDQEHREWDLQHGYLD )\ 

Y \  Y \ 

.

MZ



)\ 

Z]  Z]  MZ

'O]

0 ]]

J O]

³ V \G6

 W G6 .³

.

V O] ŸV (

 *J G6 .³

*J 6  .

Z]  Z]  ( J  MZ O]

Z]  Z]  (  \ G6 MZ O] ³

Z]  Z]  (,  MZ O]

D 

E 

F 

+HUHLQJLVWKHVKHDUDQJOHWWKHVKHDUVWUHVV'O]WKHFKDQJHLQOHQJWKDW WKH FURVVVHFWLRQDO FRRUGLQDWH \ DQG V WKH QRUPDO VWUHVV DOVR DW WKLV FR RUGLQDWH)XUWKHUPRUH(LVWKH@ ZKLFK GHPRQVWUDWH D JRRGDJUHHPHQWEHWZHHQWKHRU\DQGH[SHULPHQW $VLPLODUDSSUR[LPDWLRQIRUORQJLWXGLQDOZDYHVLVUHDGLO\REWDLQHGIURP  DV 5/

§ ORJ ¨ ©W/

· ª § KI · º ¸ | «  ORJ H  ORJ ¨ ¸ » G% ¹ ¬ © F/ ¹ ¼

 

ZKLFKLVYDOLGIRUWKHUDQJHLQZKLFK5/!G%6LQFHKIF/LQWKHSUHVHQW DQDO\VLV LV FRQVLVWHQWO\ PXFK OHVV WKDQ XQLW\ WKH WUDQVPLVVLRQ ORVV LQ   LV DOZD\V VPDOOHU WKDQ WKDW RI   $FFRUGLQJO\ ORQJLWXGLQDO ZDYHVDUHPRUHHDVLO\DWWHQXDWHGE\PHDQVRIHODVWLFLQWHUOD\HUVWKDQEHQG LQJZDYHV

%ORFNLQJ0DVVHV

)LJ  %HQGLQJ ZDYH WUDQVPLVVLRQ ORVV IRU VRPH GLIIHUHQW HODVWLF LQWHUOD\HUV « H   H DQG  H RRRPHDVXUHGYDOXHVIURP>@

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

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

$WWHQXDWLRQRI/RQJLWXGLQDO:DYHV 7KH HIIHFWV RI EORFNLQJ PDVVHV RQ ORQJLWXGLQDO ZDYHV FDQ EH VWUDLJKWIRU ZDUGO\ GHWHUPLQHG IURP (T   DSSOLFDEOH IRU WUDQVPLVVLRQ WKURXJK DUELWUDU\ PDWHULDO RU DUHD FKDQJHV %\ OHWWLQJ = EHFRPH ODUJH FRPSDUHG ZLWK=  UHGXFHVWR W

ª § = N O · º «  ¨   ] ¸ » «¬ ©  = ¹ »¼





ª § Z P · º «  ¨ ¸ »  «¬ ©  = ¹ »¼

 

U 6O] 

 

+HUHLQ P

=  O]  F

LVWKHPDVVRIWKHLQWHUOD\HUQRZDFWLQJDVDEORFNLQJPDVV7KHWUDQVPLV VLRQ HIILFLHQF\ DOVR FRXOG KDYH EH GHULYHG IURP (T   VXEMHFW WR WKH ERXQGDU\FRQGLWLRQV Y

Y

 )  )

MZ PY 

 

7KH VXEVHTXHQW DQDO\VLV DPRXQWV WR WKH VROXWLRQ RI D VHW RI OLQHDU HTXD WLRQV 6RPHZKDWGLVFRXUDJLQJKRZHYHUWKHWUDQVPLVVLRQHIILFLHQF\ IRUORQJL WXGLQDO ZDYHV DFFRUGLQJ WR   LV FORVH WR XQLW\ HYHQ IRU VXEVWDQWLDO PDVVHV $WWHQXDWLRQRI%HQGLQJ:DYHV±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

%ORFNLQJ0DVVHV Y \

Y\ 

 Z=   Z= 

)\  )\ 

MZ PY \ 

 0 ]  0 ] 

MZ4 Z] 

 

7KLV PHDQV WKDW WKH PRWLRQ RQ HLWKHU VLGH RI WKH ULJLG PDVV LV WKH VDPH ZKHUHDVWKHIRUFHVDQGPRPHQWVGLIIHUE\WKHWUDQVODWLRQDODQGURWDWLRQDO LQHUWLDUHVSHFWLYHO\ %\ HPSOR\LQJ  DQG  WKHDSSOLFDWLRQRIWKH ERXQGDU\FRQGLWLRQVDJDLQ\LHOGVDVHWRIOLQHDUHTXDWLRQVIURP ZKLFKWKH UHIOHFWLRQ DQG WUDQVPLVVLRQ FRHIILFLHQWV DUH VROYHG $OWHUQDWLYHO\ XVH FDQ EHPDGHRIWKHFRQYHUVLRQV 4 o  .  P o  &  Pc Y \ o  M0 ]  Z] M)\ 

U 6 o  %

ZLWK ZKLFK   LV WXUQHG LQWR   DQG WKH UHVXOWV RI WKH DQDO\VLV LQ 6HFWFDQEHWUDQVIHUUHGGLUHFWO\,UUHVSHFWLYHRIWKHDSSURDFKWKHUH VXOWFDQEHZULWWHQDV U UM W WM 1

P  - N  - N  1  ª - N  - N   M N  - N  º 1  ¬ ¼  M   N - N 1  - N  MN 1  N  - N  M   N  - N  - N   

























 















0RUHRYHUDUHLQWURGXFHGWKHDEEUHYLDWLRQV N

N

P  Pc

P 4  Pc P

 

FI(TV  DQG   7KHPRVWLPSRUWDQWIDFWRUVLQWKLVFRQWH[WUDQGWFDQEHVKRZQWRSRV VHVVWKHIROORZLQJFKDUDFWHULVWLFV D (QHUJ\ FRQVHUYDWLRQ 

U W



 

D 

E 2UWKRJRQDOLW\ U W

 

E 

F 9DQLVKLQJ DWWHQXDWLRQ DW ORZ IUHTXHQFLHV DQG IRU VPDOO PDVVHV SUR YLGHGDOVRWKHVTXDUHRIWKHUDGLXVRIJ\UDWLRQ4PLVVPDOOZKLFKLV W\SLFDOO\WKHFDVH)RU N  NP  Pc o  DQGWKXV Z o  

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG W o   MN   

F 

G $WWHQXDWLRQSURSRUWLRQDOWR Z DWKLJKIUHTXHQFLHVVLQFH W o   N   Z

G 

IRUODUJHN H ([LVWHQFH RI WRWDO WUDQVPLVVLRQ VLQFH WKH UHIOHFWLRQ FRHIILFLHQW YDQ LVKHVIRU N   N 

  -  

H 

I ([LVWHQFH RI WRWDO UHIOHFWLRQ VLQFH WKH WUDQVPLVVLRQ FRHIILFLHQW YDQ LVKHVIRU   N  - N 

 

I 

$Q DSSUR[LPDWLRQ IRU WKH URRWV WR WKLV HTXDWLRQ RU D YDOLG LQLWLDO YDOXHIRUDQLWHUDWLYHVROXWLRQLVJLYHQE\ N WD |    - 

J 

$IWHUVXEVWLWXWLRQRINDQG- J FRUUHVSRQGVWRDZDYHOHQJWKRI OWD

S 4  P 

K 

J :LWKWKHEHDPVLPSO\VXSSRUWHGDWDSRLQWWKLVFRUUHVSRQGVWRDFRQ ILJXUDWLRQIRUZKLFK - o  DQG N o f 7KHQFH W

 M   M

L

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

%ORFNLQJ0DVVHV

7KHWUDQVPLVVLRQORVVLVZHOODSSUR[LPDWHGE\ §N · 5% |  ORJ ¨ ¸ ©  ¹

§ P    ORJ ¨  © UK

· § K I · ¸ ¸   ORJ ¨ ¹ © F/ ¹

 

LQ WKH UDQJH URXJKO\ WZR RFWDYHV DERYH WKH IUHTXHQF\ RI WRWDO UHIOHFWLRQ )RUFRPSDULVRQ(T  OHDGVWRWKHWUDQVPLVVLRQORVV 5/

ª § S K I P · º »  ORJ «  ¨   ¸ «¬ © F/ UK ¹ »¼

 

IRUORQJLWXGLQDOZDYHV,QWKHVHH[SUHVVLRQVWKHEORFNLQJPDVVHVDUHWDNHQ SHUXQLWZLGWK

)LJ  %HQGLQJ ZDYH DWWHQXDWLRQ IRU V\PPHWULFDOO\ DWWDFKHG UHFWDQJXODU EORFNLQJPDVVHVRIWKHVDPHPDWHULDODVWKHEHDP   KE  ˜ KE  « KE  ±±±± KE  ±±± KE DQG ˜˜ ORQJLWXGLQDOZDYHDWWHQXDWLRQ

,WVKRXOGEHSRLQWHGRXWWKDWWKHDWWHQXDWLRQLVGUDVWLFDOO\UHGXFHGZKHQ WKHPDVVHVRUWKHLUDWWDFKPHQWWRWKHEHDPDUHQRWSHUIHFWO\ULJLG,QVXFKD FDVH DOO PDWHULDO SRLQWV GR QRW SDUWLFLSDWH IXOO\ LQ WKH URWDWLRQDO PRWLRQ 8OWLPDWHO\ WKH HIIHFW RI D EORFNLQJ PDVV UHOLHV RQ WKH EHDP EHLQJ ³FODPSHG´DWDSRVLWLRQZKLFKLVSRVVLEOHRQO\ LIEHDP DQGPDVVFRQVWL WXWHDXQLW7KHFODPSLQJUHIHUVRIFRXUVHWRERWKWKHWUDQVODWLRQDODQGWKH

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

URWDWLRQDOPRWLRQ,WLVLQVXIILFLHQWIRULQVWDQFHZLWKDYDQLVKLQJWUDQVOD WLRQEXWDILQLWHURWDWLRQDOVWLIIQHVVRIWKHDWWDFKPHQW7KLVLVVKRZQE\WKH VSHFLILFFDVHZLWKDSLYRWHGEORFNLQJPDVVIRUZKLFK4 DQGWKXV- | 8QGHUVXFKFRQGLWLRQV(T  \LHOGV W |M

 N  ŸW N  M  N

W

 !   N    N 

7KH FRUUHVSRQGLQJ WUDQVPLVVLRQ ORVV LV RQO\ DERXW  G% D UHVXOW PRUH RYHUDOVRIRXQGIRUDFURVVLQJRIDVWLIIDQGDIOH[LEOHEHDPVHH5%%LQ )LJD )LJXUH  GLVSOD\V WZR DGGLWLRQDO PHDVXUHPHQWV UHVXOWV 7KH ILUVW UHSUHVHQWV D EORFNLQJ PDVV DV ULJLGO\ FRQQHFWHG DV SRVVLEOH ZKHUHDV WKH VHFRQGLVFRQQHFWHGZLWKDILQLWHURWDWLRQDOVWLIIQHVVDOORZLQJIRUDGLIIHU HQFHLQURWDWLRQDOYHORFLW\EHWZHHQWKHPDVVDQGWKHEHDP

)LJ  0HDVXUHG DQG FDOFXODWHG WUDQVPLVVLRQ ORVV IRU V\PPHWULFDOO\ DWWDFKHG EORFNLQJPDVVHV RRRR ULJLGFRQQHFWLRQDQG [[[[ IOH[LEOHFRQQHFWLRQ

7KHLQIOXHQFHRIWKHILQLWHURWDWLRQDOVWLIIQHVVFDQUHDGLO\EHWDNHQLQWR DFFRXQWLQWKHDQDO\VLVE\JHQHUDOL]LQJWKHERXQGDU\FRQGLWLRQVLQ   WR HQFRPSDVV DUELWUDU\ IRUFH DQG PRPHQW LPSHGDQFHV IRU WKH EORFNLQJ HOHPHQW7KHWZRODVWHTXDWLRQVLQ  DUHWKHUHE\UHSODFHGE\ )\  )\ 

= )Y Y \ 

 0 ]  0 ] 

= 0Z Z] 

 

7KHDOWHUDWLRQKDVQRHIIHFWRQWKHILQDO(TV  SURYLGHGWKDW N

M

N = )Y Z Pc

M

= )Y PcF%

 -

MZ

Pc = )Y

= 0Z = )Y

 

%ORFNLQJ0DVVHV

)RU WKHIOH[LEOHFRQQHFWHGPDVVRIPDVVPRPHQWRILQHUWLD4WKHPR PHQWLPSHGDQFHFDQEHZULWWHQ = 0Z

0 Z

MZ4

   Z  Z 

 



ZKHUHZ VZ4LVWKHVTXDUHRIWKHURWDWLRQDOUHVRQDQFHIUHTXHQF\ZLWK VZEHLQJWKHURWDWLRQDOVWLIIQHVVRIWKHFRQQHFWLRQ8SRQVXEVWLWXWLQJ   LQWR  WKHGDVKHGFXUYHLQ)LJLVREWDLQHGZKHUHWKHUHVRQDQFH IUHTXHQF\LVWDNHQWREHZ S%\DSSURSULDWHO\VHOHFWLQJWKHURWD WLRQDO VWLIIQHVV RIWKHFRQQHFWLRQWKH IUHTXHQF\RIWRWDO UHIOHFWLRQFDQEH VKLIWHG VXEVWDQWLDOO\ GRZQZDUGV LQ IUHTXHQF\ 6XFK D VKLIW KRZHYHU LV DOZD\VRQEHKDOIRIWKHEDQGZLGWKDQGWKHKLJKIUHTXHQF\DWWHQXDWLRQ $WWHQXDWLRQRI%HQGLQJ:DYHV±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

F/ Y/   F% Y% 

 

ZKHUHDVWKHHTXDOO\ODUJHWUDQVPLVVLRQHIILFLHQF\IRUDSULPDU\ORQJLWXGL QDOZDYHUHDGV W /%

F% Y%   F/ Y/ 

 

,WIROORZVWKDWWKHPHDVXUHGYHORFLW\OHYHOGLIIHUHQFHVDUHJLYHQE\

> ' /Y  @%/ DQG

 ORJ

Y%  G% Y / 

5%/   ORJ

F/ G% F%

D 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

> ' /Y  @/%

 ORJ

Y/  G% Y % 

5/%   ORJ

F/ G% F%

E 



F 

LQWKHWZRVLWXDWLRQVUHVSHFWLYHO\ZKHUHE\ 5%/

 ORJ

 W %/

5/%

 ORJ

 W /%

:KLOVW WKH VHFRQGDU\ ORQJLWXGLQDO YHORFLW\ FDQ EH ORZHU WKDQ WKH EDFN JURXQGQRLVHLQWKHILUVWVLWXDWLRQWKHSULPDU\ ORQJLWXGLQDODQGVHFRQGDU\ EHQGLQJYHORFLWLHVDUHRIWKHVDPHRUGHURIPDJQLWXGH$OLNHPDQ\RWKHU WKLVH[DPSOHVKRZVWKHJUHDWYDOXHRIWKHUHFLSURFLW\SULQFLSOHIRUH[SHUL PHQWV

)LJ D 0RWLRQRIDQHFFHQWULFDOO\SRVLWLRQHGEORFNLQJPDVVRQDORQJLWXGL QDOO\H[FLWHGURG E ([SHULPHQWDOGHPRQVWUDWLRQRIZDYHFRQYHUVLRQ$IWHU>@

$OVR IRU WKH DQDO\VLV WKH VHFRQG VLWXDWLRQ SURYHV PRUH WUDQVSDUHQW )LUVWWKHORQJLWXGLQDOPRWLRQDWWKHFRQQHFWLRQSRLQWPXVWREH\

%ORFNLQJ0DVVHV Y

Y  Y

Y 

D 

Y 

2ZLQJWRWKHHFFHQWULFLW\DURWDWLRQZRFFXUVDWWKHFRQQHFWLRQSRLQW:LWK UVGHQRWLQJWKHGLVWDQFHIURPWKHSRLQWRIPRWLRQ FHQWUHRIURG WRWKHFHQ WUHRIJUDYLW\RIWKHPDVVWKHYHORFLW\DWWKHODWWHUSRLQWLV YV

E 

Y  UV Z 

6HFRQGWKHGLIIHUHQFHLQIRUFHVEHWZHHQWKHWZRURGSDUWVPXVWHTXDOWKH LQHUWLDIRUFHRIWKHEORFNLQJPDVV )  )

)  )  )

F 

MZ PYV 

ZKLFK LV H[HUWHG DW WKH FHQWUH RI JUDYLW\ 7KH ORQJLWXGLQDO IRUFHV PRUH RYHUDUHUHODWHGWRWKHYHORFLWLHVE\WKHLPSHGDQFHV )

Y = )Y

Y = )Y

 )

 )

Y = )Y 

G 

ZKHUHE\LWLVREVHUYHGWKDWWKHGLUHFWLRQ)LVLQWKHQHJDWLYH[GLUHFWLRQ 8SRQ VXEVWLWXWLRQ LQ F  DQG HOLPLQDWLRQ RI Y DQG YV E\ PHDQV RI D DQG E  § MZ P · MZ P UV Z ¨ ¸ Y  = = )Y )Y ¹ ©

Y

 

7KHWKLUGUHODWLRQQHFHVVDU\EHWZHHQYDQGZDULVHVIURPWKHPRPHQW 0

MZ PUV Y 

D 

ZKLFKPXVWHTXDOWKDWJLYHQE\WKHPRPHQWLPSHGDQFHPXOWLSOLHGE\WKH URWDWLRQDOYHORFLW\ MZ PUV Y

= 0Z Z

E 

7KHPRPHQWLPSHGDQFHLQWKLVFDVHLVFRPSRVHGRIWKHURWDWLRQDOLQHUWLDRI WKHHFFHQWULFPDVVDQGWKHPRPHQWLPSHGDQFHRIWKHEHDP LQIOH[XUHGLV FXVVHGLQ6HFW = 0Z

MZ4 

%   M  F%

F 

7KLVPHDQVWKDWYLQ(T  FDQEHHOLPLQDWHGHVWDEOLVKLQJ Y Z

§ MZ P · = 0Z MZ P  UV  ¨  ¸ ©  = )Y ¹ MZ PUV  = )Y

 

,WVKRXOGEHQRWHGWKDWWKHPDVVPRPHQWRILQHUWLDUHIHUVWRWKHSRLQWRIUR WDWLRQLHWKHFHQWUHRIWKHURGDQGQRWWRWKHFHQWUHRIJUDYLW\+HQFH

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG K

4



³ \ GP

D 

K

ZKHUHWKHRULJLQLVDWWKHFHQWUHRIWKHURG)RUDUHFWDQJXODUEORFNLQJPDVV DVGHSLFWHGLQ)LJRQHREWDLQV 4

P K  KK  K    UV

K  K   

E 

,QRUGHUWRDUULYHDWWKHWUDQVPLVVLRQHIILFLHQF\LWPXVWEHREVHUYHGWKDWWKH WUDQVPLWWHGEHQGLQJZDYHSRZHULHKDOIRIWKHWRWDOSRZHUWUDQVPLWWHGIRU DQH[FLWDWLRQE\PHDQVRIDSUHVFULEHGURWDWLRQDOYHORFLW\ ZLVJLYHQE\ : % 

 :% 

  5H > = 0Z @ Z  

D 

7KHLQFLGHQWORQJLWXGLQDOSRZHULV :/ 

  = )Y Y  

E 

7KLVPHDQVWKDWWKHWUDQVPLVVLRQHIILFLHQF\LVILQDOO\REWDLQHGDV W /%

MZ PUV  ª = 0Z º § MZ P · = 0Z 5H «  ¸ » ¨   ¬ = )Z ¼ ©  = )Y ¹ MZ PUV  = )Y





 

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³ORQJLWXGLQDO SHULRGLF V\VWHP´ FRQVLVWV RID EHDP ZLWK SHULRGLF FKDQJHV LQ WKH FURVVVHFWLRQ DV GHSLFWHG LQ )LJ  $VORQJDVWKHZDYHOHQJWKO/LVODUJHFRPSDUHGZLWKWKHOHQJWKVOPDQGOV

3HULRGLF6WUXFWXUHV

RIWKHGLIIHUHQWSDUWVWKRVHZLWKWKHFURVVVHFWLRQVDFWVDVPDVVHVZKHUHDV WKRVHZLWKVPDOODVVSULQJV,WLVWKXVSRVVLEOHWRUHSUHVHQWWKHSHULRGLFV\V WHPLQWHUPVRIPHFKDQLFDOFLUFXLWHOHPHQWVDVDVHULHVDUUD\ RIPDVVHVDQG VSULQJV

)LJ3HULRGLFPDVVVSULQJV\VWHPZLWKWKHQRWDWLRQXVHG

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

MZ PYQ

 

DQGWKHIRUFH±YHORFLW\UHODWLRQIRUDVSULQJZLWKVWLIIQHVVVDV YQ   YQ

MZ )Q  V

 

$V IRU WKH ZDYH WUDQVPLVVLRQ SUREOHP WKH VROXWLRQ PD\ EH WDNHQ LQ WKH H[SRQHQWLDOIRUPZKLFKPHDQVWKDWWKHUDWLRRIWKHYDOXHRIDILHOGYDUL DEOHDWWKHLQSXWSRLQWRIDQHOHPHQWWRWKDWDWWKHRXWSXWLVWKHVDPHIRUDOO HOHPHQWVLH   7KLVUHGXFHV(TV  DQG  WRWZROLQHDUHTXDWLRQVLQWHUPVRI)Q DQGYQ )Q 

)Q H J

 YQ 

YQ H J 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

  H ) J

 MZ PYQ

Q



MZ )Q  H  J   YQ V



 



ZKLFKKDYHQRQWULYLDOVROXWLRQVIRUDYDQLVKLQJGHWHUPLQDQW 'HW

H J  H J   

Z P V



7KLV FRQGLWLRQV IRU WKH LQ JHQHUDO FRPSOH[ SURSDJDWLRQ FRQVWDQW J FDQ DOVREHZULWWHQLQWKHIRUP FRVK J

  Z   Z O

D 

ZKHUHLQWKH³OLPLWLQJIUHTXHQF\´ ZO

S I O

 VP

E 

KDVEHHQLQWURGXFHG 7KHPHDQLQJRIZOEHFRPHVFOHDUZKHQWKHUHDODQGLPDJLQDU\SDUWVRIJ DUHGLVWLQJXLVKHG J

D  ME 

D 

ZKHUHE\ D LV WKH ³DWWHQXDWLRQ FRQVWDQW´ UHQGHULQJ DQ H[SRQHQWLDO DPSOL WXGH GHFD\ DQG E LV WKH ³SKDVH FRQVWDQW´ 7KHQFH (T D  FDQ EH UH ZULWWHQDV FRVK D FRV E  M VLQK D VLQ E   Z   Z O 

E 

6LQFHWKHULJKWKDQGVLGHLVSXUHO\UHDOWKHLPDJLQDU\SDUWRQWKHOHIWKDQG VLGHPXVWYDQLVKLHHLWKHUWKHDWWHQXDWLRQFRQVWDQWLV D



D 

RUWKHSKDVHFRQVWDQWWDNHVWKHYDOXHV E

QS

 Q

 

E 

7KHILUVWFRQGLWLRQ LVRQO\SRVVLEOHIRUIUHTXHQFLHVEHORZDQGWKHVHFRQG RQO\IRUIUHTXHQFLHVDERYHWKHOLPLWLQJIUHTXHQF\ ,QWKHILUVWUHJLRQZKLFKPD\EHFDOOHG³WUDQVPLVVLRQUHJLRQ´WKHUHRF FXUVRQO\DSKDVHFKDQJHEHWZHHQFRQVHFXWLYHHOHPHQWVDVLVIRXQGIURPD VXEVWLWXWLRQRI D LQ E  E

§ Z  · DUFFRV ¨   ¸ ZO ¹ ©

§Z ·  DUFVLQ ¨ ¸  © ZO ¹

D 

3HULRGLF6WUXFWXUHV

)RUIUHTXHQFLHVZHOOEHORZWKHOLPLWLQJIUHTXHQF\WKLVH[SUHVVLRQUHGXFHV WRWKHOLQHDUUHODWLRQ E



Z ZO

Z

P  V

E 

$SKDVHFRQVWDQWWKDWYDULHVOLQHDUO\ZLWKIUHTXHQF\VLPSO\ LPSOLHVDFRQ VWDQWSKDVHVSHHGDQGWKXVDFRQVWDQWSURSDJDWLRQVSHHG F

Z OP  OV ZO OP  OV   E

F 

V OP  O V  P

1HDUWKHOLPLWLQJIUHTXHQF\WKHSKDVHFRQVWDQWLQFUHDVHVPRUHUDSLGO\ XQ WLOLWUHDFKHVWKHYDOXHS DWWKHOLPLWLQJIUHTXHQF\7KHUHDERYHLWUHPDLQV DWSLPSO\LQJWKDWWKHILHOGYDULDEOHVRIDGMDFHQWSHULRGLFHOHPHQWVDUHDO ZD\VRIRSSRVLWHSKDVHFI)LJ

)LJ$WWHQXDWLRQDQGSKDVHFRQVWDQWVRIDORZSDVVILOWHU

,QFRQWUDVWWKHDWWHQXDWLRQFRQVWDQWDLQFUHDVHVUDSLGO\ZLWKLQFUHDVLQJ IUHTXHQF\ LQ WKH DWWHQXDWLRQ UHJLRQ DV LV VHHQ E\ LQVHUWLQJ (T E  LQ E  D

§ Z  · DUFFRVK ¨    ¸ Z © O ¹

§Z DUFVLQK ¨ © ZO

· ¸ ¹

G 

%HFDXVH WKH DWWHQXDWLRQ LV ODUJH DW IUHTXHQFLHV DERYH WKH OLPLWLQJ IUH TXHQF\ EXW VPDOO EHORZ D V\VWHP OLNH WKDW GHSLFWHG LQ )LJ  PD\ EH FDOOHGD³ORZSDVV´V\VWHPDVLWVHOHFWULFDODQDORJ

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

:LWK UHVSHFW WR DWWHQXDWLRQ LW LV EHQHILFLDO ZLWK DQ DV ORZ DV SRVVLEOH OLPLWLQJ IUHTXHQF\ ,IWKH PDWHULDORIWKHPDVVHV DQG VSULQJVLV WKH VDPH WKHQ WKLV GHVLUDEOH FRQGLWLRQ FDQ RQO\ EH DFKLHYHG E\ PHDQV RI ODUJH FKDQJHV LQ WKH FURVVVHFWLRQDO DUHD RU ODUJH HOHPHQW OHQJWKV :LWK P U 6P OP DQG V (6 V  OV (T E FDQEHZULWWHQDV ZO

F/

6V 6P

 OP O V



D 

ZKLFKLQWKHFDVHRIDF\OLQGULFDOHOHPHQWVRIGLDPHWHUV'DQGGUHVSHF WLYHO\VLPSOLILHVWR ZO

F/

G   ' OP O V

E 

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



7 4

§G · F7 ¨ ¸ ©'¹



 OP O V

 F

7 OP  OV  4

 

$WWHQXDWLRQRI/RQJLWXGLQDO:DYHV )RU WKH VLPSOHVW FRQWLQXRXV SHULRGLF V\VWHP RQO\ TXDVLORQJLWXGLQDO ZDYHV DUH FRQVLGHUHG RU ZKHQ URWDWLRQDO PRWLRQV WRUVLRQDO ZDYHV 7KH ZDYHILHOGLQHDFKSDUWPXVWFRQVLVWRIRQHSRVLWLYHO\ DQGRQHQHJDWLYHO\ SURSDJDWLQJZDYHZLWKZDYHQXPEHUVN ZF/DQGN ZF/UHVSHFWLYHO\

3HULRGLF6WUXFWXUHV

7KHUHDUHGLIIHUHQWPHWKRGVWRDQDO\VHWKHSURSDJDWLRQLQWKHFRQILJXUD WLRQ GHSLFWHG LQ )LJ  )LUVW D PHWKRG LV HPSOR\HG LQ ZKLFK WKH SURSDJDWLRQLVH[SUHVVHGDVDIXQFWLRQRIWKHWUDQVPLVVLRQFRHIILFLHQWVGH ILQHGLQ6HFWVDQG,WLVDVVXPHGWKDWWKHSRVLWLYHO\SURSDJDWLQJ ZDYHKDVLWVRULJLQDWWKHOHIWHQGZKHUHLWKDVWKHDPSOLWXGHYQDQGY Q± UHVSHFWLYHO\)RUWKHQHJDWLYHO\SURSDJDWLQJZDYHWKHRULJLQLVDVVXPHGDW WKHULJKWHQGKDYLQJWKHDPSOLWXGHYQDQGY Q± $FFRUGLQJO\ YQ [

Y Q H  MN [ OQ/  Y Q H MN [  OQ5 

YQ  [

Y Q H  J H

 MN [  OQ  /

 Y Q H  J H

MN [  OQ  5



 

ZKHUHWKHSHULRGLFLW\UHODWLRQLQ  LVXVHG 6LQFH WKH DPSOLWXGH YQ LV FRPSRVHG RI WKDW RI WKH ZDYH WUDQVPLWWHG IURPWKHHOHPHQW Q± RIDPSOLWXGHWY Q±  WYQ HJDQGWKDWRIWKHUH IOHFWHGZDYHKDYLQJWKHDPSOLWXGHUYQLWFDQEHH[SUHVVHGDV Y Q H  J WH

Y Q

 MN OQ/  OQ /

 Y Q UH

MN OQ/  OQ5



D 

6LPLODUO\WKHDPSOLWXGHRIWKHUHIOHFWHGZDYHLQWKHHOHPHQW Q± FDQEH ZULWWHQDV Y Q 

Y Q H  J UH

Y Q H

 Y Q WH

O S DQG OQ/  OQ5

2EVHUYLQJWKDW OQ/  OQ / Y Q

 MN OQ/  OQ /

Y Q H  J WH J

 MNO S

J

Y Q H UH

 Y Q UH  MNO S

MN OQ/  OQ5



E 

O S LWIROORZVWKDW  MNO S

 Y Q WH



 MNO S



 

7KLV KRPRJHQHRXV VHW RI OLQHDU HTXDWLRQV KDV D QRQWULYLDO VROXWLRQ IRU D YDQLVKLQJGHWHUPLQDQWZKLFKPHDQVWKDW FRVK J

H 

MNO S

 W   U  H W

 MNO S



)LJ3HULRGLFORQJLWXGLQDOZDYHJXLGHZLWKWKHQRWDWLRQXVHG

 

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

)RUFRQVHUYDWLYHGLVFRQWLQXLWLHV(T  FDQEHVLPSOLILHGE\LQYRN LQJHQHUJ\FRQVHUYDWLRQ,IDZDYHRIDPSOLWXGH$LVLQFLGHQWRQDGLVFRQ WLQXLW\WKHQ $





$U  $W

D 



DVGHPRQVWUDWHGLQ6HFWVDQG:LWKDZDYH%LPSLQJLQJVLPXOWD QHRXVO\ RQ WKH GLVFRQWLQXLW\ IURP WKH RWKHU VLGH WKH LQ ERWK GLUHFWLRQV HPDQDWLQJZDYHVDUH$U%WDQG$W%UUHVSHFWLYHO\)ROORZLQJHQHUJ\ FRQVHUYDWLRQ 

$ %







E 

$U  %W  $W  %U 

DQGZLWK D DQGDFRUUHVSRQGLQJH[SUHVVLRQIRUWKH%ZDYHLQWURGXFHG RQWKHOHIWKDQGVLGHLWLVIRXQGWKDW 





$U  $W  %U  %W





F 



$U  %W  $W  %U 

%\ VHSDUDWLQJ WKH JHQHUDOO\ FRPSOH[ UHIOHFWLRQ DQG WUDQVPLVVLRQ FRHIIL FLHQWLQUHDODQGLPDJLQDU\SDUWVLH U U c  MU cc DQG W W c  MW cc LWIROORZV IURP F WKDW U cW c  U ccW cc

G 



,Q WKH FRPSOH[ SODQH WKHUHIRUH WKH YHFWRUV U DQG W DUH SHUSHQGLFXODU WR HDFKRWKHUDVZDVVHHQDOVRLQ(TV E DQG E +HQFHLWLVDOVR SRVVLEOHWRZULWH W

W H MM

U H MM  MS  

 U



W  U





W H  MM  U H  MM

H  MM

H 

DQGZKHQVXEVWLWXWHGLQ  LWLVIRXQGWKDW FRVK J

H 

MNO S

 H  MM H W

 MNO S

 MNO MNO ª H S H S º « MM  »  «¬ W H W H MM »¼

ª H MNO S 5H « «¬ W

º »  »¼

 

8VH LV KHUH PDGH RI WKH IDFW WKDW WKH VXP RI D FRPSOH[ QXPEHU DQG LWV FRQMXJDWHEHFRPHV WZLFH WKH UHDO SDUW $V DQ H[DPSOH RI WKH DSSOLFDWLRQ RI   )LJ  VKRZV FRVK J DV ZHOO DV WKH DWWHQXDWLRQ FRQVWDQW D DQG WKH SKDVH FRQVWDQW E IRU D VSHFLILF FDVH,QDGGLWLRQLVLQFOXGHGWKHFXUYHDUFFRVK_W_ZKLFKLVH[SHFWHGWRUH DOL]HWKHXSSHUOLPLWIRUWKHDWWHQXDWLRQ)URP(T  DQG)LJWR JHWKHUZLWKVRPHFDOFXODWLRQVRIRWKHUH[DPSOHVWKHIROORZLQJFRQFOXVLRQ FDQEHGUDZQIRUWKHEHKDYLRXURISHULRGLFV\VWHPVFDUU\LQJORQJLWXGLQDO RUWRUVLRQDOZDYHV

3HULRGLF6WUXFWXUHV

D  &RQWUDU\ WR SHULRGLF V\VWHPV PDGH RI OXPSHG VXEV\VWHPV D VHTXHQFH RISDVVDQGVWRSEDQGVDSSHDU E  ,Q WKH VWRS EDQGV WKH SKDVH LV FRQVWDQW ZKHUHDV LW JURZV LQ WKH SDVV EDQGV F  7KH PD[LPXP DPSOLWXGH UDWLR IRU DGMDFHQW VXEV\VWHPV LV JLYHQ E\ FRVKJ _W_ G  2QDYHUDJHWKHIUHTXHQF\VHSDUDWLRQEHWZHHQFRQVHFXWLYHRQVHWVRIWKH VWRSEDQGVLVVHWE\ 'NOS  'ZF OS SDVVRFLDWHGZLWKWKHSHULRGLFLW\ RIWKHH[SRQHQWLDOIXQFWLRQLQ  

)LJ$WWHQXDWLRQDQGSKDVHFRQVWDQWVIRUDURGZLWKVSDWLDOO\SHULRGLFPDVVHV FDUU\LQJDORQJLWXGLQDOZDYH

$QDOWHUQDWLYHPHWKRGIRUWKHDQDO\VLVRIWKHDWWHQXDWLRQDQGSKDVHFRQ VWDQWVFRQVLVWVRIXVLQJWKHIRUFHDQGYHORFLW\ILHOGVRIWZRDGMDFHQWVXE V\VWHPVDQGLQYRNLQJWKHERXQGDU\FRQGLWLRQV

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

)URP(T  WKHIRUFHVDUHREWDLQHGDV U F6 ªY Q H Q/  Y Q H Q5 º  ¬ ¼ ª  J  MN [  O Q  /  J MN [  O Q  5 º )Q  [ U F6 «Y Q H H  Y Q H H »¼  ¬  MN [  O

)Q [

MN [  O

 

7KHERXQGDU\FRQGLWLRQVDWWKHGLVFRQWLQXLW\ [ OQ/ O Q± 5 RIIHUWKH QHFHVVDU\ UHODWLRQV WR GHYHORS WKH VHW RI OLQHDU HTXDWLRQV )RU EORFNLQJ PDVVHVPIRULQVWDQFHFRQWLQXLW\LQYHORFLW\DQGHTXLOLEULXP LQIRUFHVUH TXLUH YQ 

YQ   

)Q    )Q 

D 

MZ PYQ  

:LWKWKHFKRVHQRULJLQO Q± / OSDQGOQ5 OSVXFKWKDWZKHQ  DQG  DUHLQVHUWHGLQ D WKHV\VWHPRIHTXDWLRQEHFRPHV Y Q  Y Q H Y Q H  J H

 MNO S

 MNO S

Y Q H  J H

 MNO S

 Y Q H  J 

 Y Q H  J  Y Q  Y Q H

 MNO S

MZ P  MNO Y Q  Y Q H S  U F6





E 

)URPDYDQLVKLQJGHWHUPLQDQWLWLVIRXQGWKDW FRVK J

FRV NO S 

ZP VLQ NO S   U F6

F 

%\ HPSOR\LQJ WKH WUDQVPLVVLRQ FRHIILFLHQW DVVRFLDWHG ZLWK WKLV VWUXFWXUDO FRQILJXUDWLRQ W

§ MZ P · ¨  ¸ U F6 ¹  ©



LQ(T  LWLVUHDGLO\DVFHUWDLQHGWKDW F DQG  DUHLGHQWLFDO @%\ VXE

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

VWLWXWLQJ±O$IRU[ LQ(T  IRUDEHDPRIOHQJWKO$DVGHSLFWHGLQ)LJ WKHHLJKWSROHGHVFULSWLRQLVREWDLQHGDV ­ Y$ ½ °Z ° ° $° ® ¾ °0 $ ° °¯ )$ °¿

§ &$ ¨ ¨ NV $ ¨  M= F% F $ ¨¨ ©  M= F% N6 $

6 $  N &$ M= F% V $  N M= F% F $

F$  M= F% N6 $  M= F% &$ NV $

 V $  M= F% N · ­ Y% ½ ¸ F $  M= F% ¸ °° Z% °° ® ¾ 6 $  N ¸ °0 % ° ¸¸ ° &$ ¹ ¯ )% °¿

 

& $ > FRVK NO $  FRV NO $ @    F $ > FRVK NO $  FRV NO $ @      VLQK N O  VLQ NO   V VLQK NO  VLQ NO   = F% %P DQGNLVWKH > > $ $@ $ $ $@

+HUHLQ 6$

EHQGLQJZDYHQXPEHU:KHQDOVRWKHVHFRQGEXLOGLQJEORFNRIWKHVXEV\V WHP LV D EHDP SRVVLEO\ RI DQRWKHU PDWHULDO DQG ZLWK D GLIIHUHQW FURVV VHFWLRQ WKH VHFRQGHLJKWSROHH[SUHVVLRQ LVVLPLODUWRWKDWLQ  WKH GLIIHUHQFHEHLQJWKDWWKHPDWUL[HOHPHQWVDUHPRGLILHG

)LJ 3HULRGLFEHQGLQJZDYHJXLGH7RSRQHSHULRGLFHOHPHQWFRQVLVWLQJRI WZR GLIIHUHQW EHDPV %RWWRP RQH SHULRGLF HOHPHQW FRQVLVWLQJ RI D EHDP DQG D EORFNLQJPDVV

$QRWKHU LPSRUWDQW FRQILJXUDWLRQ LQ SUDFWLFH LV WKDW ZKHUH WKH ZDYH JXLGHFRQVLVWVRIDEHDPZLWKSHULRGLFDOO\VSDFHGPDVVHV,QVXFKDFDVH WKH UHODWLRQV EHWZHHQ WKH ILHOG YDULDEOHV RQ HLWKHU VLGH RI D PDVV FDQ EH ZULWWHQDV ­ Y% ½ °Z ° ° %° ® ¾ °0 % ° °¯ )% °¿

§ ¨ ¨ ¨ ¨¨ ©

   MZ P

  MZ4 

   

 · ­ Y& ½ ¸  ¸ °° Z& °° ® ¾  ¸ °0 & ° ¸  ¸¹ °¯ )& °¿

 

LQDFFRUGDQFHZLWK  8SRQGHQRWLQJWKHFROXPQYHFWRUVDQGPDWULFHV Y$Y%Y&DQG0$0%UHVSHFWLYHO\WKHFKDUDFWHULVWLFHTXDWLRQLVREWDLQHG DV

3HULRGLF6WUXFWXUHV H J Y&

0 $ 0 % Y& 

 

LQDQDORJ\ZLWK  ,Q)LJDUHVKRZQWKHUHVXOWVIRUWZRGLIIHUHQW FRQILJXUDWLRQV

)LJ&DOFXODWHGDWWHQXDWLRQSHUVXEV\VWHP RIDSHULRGLFIOH[XUDOZDYHJXLGH D  6LPSOH VXSSRUWV DQG E  V\PPHWULF EORFNLQJ PDVVHV RI PDVV  NJRQDVWHHO EHDP ±±± FDOFXODWHGIURP    IURP  DQG RR PHDVXUHG

)URPWKHVHUHVXOWVDQGWKRVHRIVRPHDGGLWLRQDOFDOFXODWLRQVDVZHOODV WKHIRUPRI  VRPHJHQHUDOILQGLQJVFDQEHQRWHG x 7ZRDWWHQXDWLRQFRQVWDQWVH[LVWIRUSHULRGLFIOH[XUDOZDYHJXLGHVRQH RIZKLFKLVWRRODUJHWREHSUDFWLFDOO\LQWHUHVWLQJ x $V IRU WKH SHULRGLF ZDYHJXLGH FDUU\LQJ ORQJLWXGLQDO ZDYHV WKH SDVV DQGVWRSEDQGDOWHUQDWH x :LWKWKHSURSHUWLHVIRUWKHEHQGLQJZDYHVXEVWLWXWHG(T  UHSUH VHQWVWKHHQYHORSHWRWKHDWWHQXDWLRQ 2QO\ WKHILUVWSDVVDQGVWRSEDQG PXVWEHGHWHUPLQHGIURPWKHPRUHFRPSOHWHH[SUHVVLRQ   x )RU D EHDP ZKLFK LV SHULRGLFDOO\ VLPSO\ VXSSRUWHG RU HTXLYDOHQWO\ P Æ fDQG4 ÆLQ  DVWRSEDQGH[LVWVXSWRNO$ S)RUWKH  XSSHU VWRS EDQGV WKH PD[LPXP DWWHQXDWLRQ LV JLYHQ E\ FRVK D VLQFHW M ±M LQYLHZRI L FI)LJD

$WWHQXDWLRQRI6WUXFWXUH%RUQH6RXQG

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



wY \ w[

w0 ] w[

w0 ] wW



)\



%

w)\ w[

wZ]  w[

Pc

wY \ wW

D 

E 



7RWDNHWKHLQIOXHQFHRIWKHGLVFRQWLQXLWLHVLQWRDFFRXQWDUHDGGHGWRWKH G\QDPLF UHODWLRQV LQ E  WKH ³VPHDUHG´ PRPHQWV 0'O$ DQG IRUFHV )'O$UHVXOWLQJIURPWKHGLVFRQWLQXLWLHV7KHUHE\ LWLVQDWXUDOWRGLVWULEXWH WKHDFWXDOO\SRLQWZLVHDFWLQJPRPHQWVDQGIRUFHVRYHUWKHSHULRGLFOHQJWK O$ ZKHUHE\ WKHUHODWLRQVLQ E DUHPRPHQWVDQGIRUFHVSHUOHQJWK%\ PHDQVRISKDVRUQRWDWLRQDQGLPSHGDQFHVIRUWKHPRPHQWVDQGIRUFHVLWLV IRXQGWKDW 

w0 ] w[

)\ 

0' O$

)\ 

= 0Z Z] O$

 

w)\ w[

MZ PcY \ 

= )Y Y \  O$

F 

)XUWKHUPRUHDFRPELQDWLRQRI D DQG F \LHOGVWKHHTXDWLRQRI PRWLRQIRUWKH³VPHDUHG´SHULRGLFEHQGLQJZDYHJXLGHFI6HFW %

w Y\ w[ 

 Z  PcY \  MZ

 = 0Z w Y \ =  MZ )Y Y \  O $ w[ O$

 

G 

8SRQHPSOR\LQJWKHXVXDOZDYHILHOGWKHGLVSHUVLRQHTXDWLRQEHFRPHV ª  § ·º = )Y = 0Y   N  ¸» Y\ « %N  Z Pc ¨   MZ PcO $ MZ PcO $ ¹ »¼ © ¬«

 

H 

+HQFHWKHEHQGLQJZDYHQXPEHUVNFDQEHGHWHUPLQHGDQGWKHQFHWKHJHQ HUDOO\ FRPSOH[ SURSDJDWLRQ FRQVWDQW MJ  NO$ IRU WKH ZDYH JXLGHV DW ORZ IUHTXHQFLHVLHIRU,P>NO$@

3HULRGLF6WUXFWXUHV

,QPRVWFDVHV=0Z N=)YVXFKWKDW N |

= )Y · Z  Pc § ¨  ¸ % © MZ PcO $ ¹

UHVXOWVDSSUR[LPDWHO\)RUGLVFRQWLQXLWLHVFRQVLVWLQJRIVSULQJV=)Y VMZ ZKLFKPHDQVWKDWIRU Z   V  Pc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

Z )Q  MN [  [  N [  [Q  MH H  %N 





@ /\RQ 5+ DQG GH-RQJ 5'  6WDWLVWLFDO (QHUJ\ $QDO\VLV QG HG  %XWWHUZRUWK+HLQHPDQ%RVWRQ0$ >@+HFNO0DQG6HLIHUW.8QWHUVXFKXQJHQEHUGHQ(LQIOXVVGHU(L JHQUHVRQDQ]HQGHU0HVVUlXPHDXIGLH(UJHEQLVVHYRQ6FKDOOGlPPPHVVXQ JHQ$FXVWLFDS

6RXQG5DGLDWLRQIURP6WUXFWXUHV

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



9  S  U F 7

 

6RXQG5DGLDWLRQIURP6WUXFWXUHV

ZKHUH W\SLFDOO\ ILYH WR WHQ PHDVXUHPHQW SRVLWLRQV DUH DYHUDJHG U DQG F DUHWKHGHQVLW\DQGZDYHVSHHGRIWKHPHGLXP$VVRFLDWHGZLWKWKHSRZHU LVWKHSRZHUOHYHOGHILQHGDV /:

 ORJ

: G% :UHI

 

ZLWKWKHUHIHUHQFH:UHI S:8VHRIWKHSRZHUOHYHOLVVRPHWLPHVFRQ YHQLHQWLQFRQMXQFWLRQZLWKWKHVRXQGSUHVVXUHOHYHOGHILQHGE\ /S

S  G%   S UHI

 ORJ

 

ZKHUH WKH UHIHUHQFH LV VWDQGDUGL]HG DV S UHI  P3D DQG WKH WLOGH GHQRWHV WKHUPVYDOXH 7KHDFFXUDF\ZLWKZKLFKWKHVRXQGSRZHUFDQEHPHDVXUHGHPSOR\LQJ WKHPHWKRGGHVFULEHGDERYHGHSHQGVRQWKHYROXPHVKDSHDQGUHYHUEHUD WLRQ WLPH RI WKH UHYHUEHUDQW URRP WKH QXPEHU RI PHDVXUHPHQW SRVLWLRQV DQG SDUWLFXODUO\ RQ WKH FKDUDFWHU RI WKH VRXQG )RU VWDWLRQDU\ EURDGEDQG QRLVHWKHWKLUGRFWDYHRURFWDYHSRZHUFDQEHGHWHUPLQHGZLWKLQDERXWr FRUUHVSRQGLQJWRrG%,QVLWXDWLRQVZLWKQRQVWDWLRQDU\ RUVWURQJ WRQDOFRQWULEXWLRQVIRUH[DPSOHIURPURWDWLQJPDFKLQHU\WKHHUURUPLJKW EHFRQVLGHUDEO\ODUJHU $VHFRQGSRVVLELOLW\WRPHDVXUHWKHSRZHUUDGLDWHGIURPDQREMHFWFRQ VLVWVRISODFLQJ WKHREMHFWLQIUHHILHOGFRQGLWLRQVVXFKDVWKRVHRIDQDQ HFKRLF FKDPEHUDQGPHDVXULQJWKHVRXQGSUHVVXUH SQDWPDQ\SRLQWVDWD GLVWDQFH57KLVGLVWDQFHPXVWEHVLJQLILFDQWO\ JUHDWHUWKDQWKHGLPHQVLRQV RIWKHREMHFW7KHUDGLDWHGSRZHULVWKHQREWDLQHGIURP :

 UF

¦S

 Q

6Q 

 

ZKHUH6QLVWKHVXUIDFHHOHPHQWDVVRFLDWHGZLWKWKHQWKPHDVXUHPHQWSRLQW 7KH VXP RI DOO VXUIDFH HOHPHQWV PXVW HTXDO WKDW RI D KHPLVSKHUH RU D VSKHUH GHSHQGLQJ RQ ZKHWKHU WKH REMHFW LV ORFDWHG RQ D VRXQGUHIOHFWLYH VXUIDFH RU QRW ,Q WKH VSHFLDO FDVH WKDW WKH PHDVXUHPHQW SRLQWV DUH XQL IRUPO\GLVWULEXWHGRYHUDKHPLVSKHULFDOVXUIDFH : 

S 5  UF 1

1

¦S

Q

Q 



S 5  S UF

D 

ZKHUH S UHSUHVHQWVWKHDYHUDJHPHDQVTXDUHSUHVVXUH ,WLVHYLGHQWWKDWIUHHILHOGPHDVXUHPHQWVDUHPRUHWHGLRXVWKDQWKRVHLQ D UHYHUEHUDWLRQ FKDPEHU SDUWLFXODUO\ LI WKH UDGLDWLRQ YDULHV VWURQJO\ LQ

'HILQLWLRQDQG0HDVXUHPHQWRI5DGLDWLRQ(IILFLHQF\

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

D 

SY 

)URPWKLVGHILQLWLRQWKHSRZHULVREWDLQHGDV :

v³ -G6 | ¦ - ' 6 H Q

V

Q Q



E 

ZKHUHHQLVWKHQRUPDOXQLWYHFWRURIWKHVXUIDFHHOHPHQW6Q7KHLQWHQVLW\ LVPRVWRIWHQPHDVXUHGQRUPDOWRWKHVXUIDFHHOHPHQWDQGDFFRUGLQJO\WKH YHFWRUQRWDWLRQLQ E FDQEHRPLWWHG ,QSULQFLSOHWKHLQWHQVLW\ PHWKRGFDQEHHPSOR\HGDOVRLQWKHSUHVHQFH RI DGMDFHQW QRLVH VRXUFHV DQG LQ DUELWUDU\ ILHOG FRQGLWLRQV VXFK DV LQ WKH QHDUILHOG,QWKLVFRQWH[WWKHµLQSULQFLSOH¶LVLPSRUWDQWVLQFHLQSUHVHQFH RIQRLVHVRXUFHVDVLQVWURQJQHDUILHOGVWKHDFFXUDF\LVPDUNHGO\ UHGXFHG >@ $QRIWHQRYHUORRNHGSUREOHPLQFRQMXQFWLRQZLWKSRZHUPHDVXUHPHQWV RISDUWVRIDVRXUFHLVWKHLQIOXHQFHRIDGMDFHQWVRXUFHSDUWV,WLVUHDGLO\ VKRZQ HJ >@  WKDW WKH SRZHU PHDVXUHG RQ D VXUIDFH HQFORVLQJ WKH VRXUFHSDUWRILQWHUHVWFDQYDU\DOPRVWDUELWUDULO\:KHUHDVVXFKDYDULD WLRQLVHDVLO\IRXQGIRUDQDUURZIUHTXHQF\EDQGLWLVUDWKHUXQOLNHO\RYHU EURDGUDQJH

'HILQLWLRQDQG0HDVXUHPHQWRI5DGLDWLRQ(IILFLHQF\ ,Q DGGLWLRQ WR WKHUDGLDWHG SRZHUDOVRWKHUHODWLRQ EHWZHHQWKH VWUXFWXUDO YLEUDWLRQVDQGWKHSRZHULVXVXDOO\RILQWHUHVW,QWKLVSXUVXLWWKHUDGLDWLRQ

6RXQG5DGLDWLRQIURP6WUXFWXUHV

HIILFLHQF\ V LVDXVHIXOEXWGDQJHURXVTXDQWLW\LQPDQ\FDVHV7KHGHILQL WLRQUHDGV>@ V

: 



D 

U F6 Y  

ZKHUH:LVWKHSRZHUUDGLDWHGIURPDVWUXFWXUHZLWKVXUIDFHDUHD6DQGYL  EUDWLQJZLWKWKHVSDWLDOO\DYHUDJHGPHDQVTXDUHYHORFLW\ Y  )RU D ULJLG YLEUDWLQJ VXUIDFH VXFK DV D SLVWRQ ZLWK GLPHQVLRQV PXFK ODUJHUWKDQWKHZDYHOHQJWKLQWKHDPELHQWIOXLGWKHIOXLGFDQQRWHVFDSHODW HUDOO\EXWLWVSDUWLFOHYHORFLW\PXVWHTXDOWKDWRIWKHVXUIDFHHYHQRXWVLGH LWV LPPHGLDWH YLFLQLW\ 6LQFH WKH UDGLDWLRQ WKHQ RFFXUV QRUPDO WR WKH VXU IDFHWKHVRXQGSUHVVXUHSURGXFHGLQWKHIOXLGLVJLYHQE\ S U FY E  

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

)LJ6RPHH[SHULPHQWDOUDGLDWLRQHIILFLHQFLHVRI D DPPFRQFUHWHFHLO LQJ E YDULRXVGLHVHOHQJLQHEORFNV F PPSODVWHUERDUGRQDODWKJULGDQG G  DVWHHOGXFWRIPPGLDPHWHUDQGPPZDOOWKLFNQHVV

5DGLDWLRQ/RVV)DFWRU

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

5DGLDWLRQ/RVV)DFWRU %HVLGHV WKH UDGLDWLRQ HIILFLHQF\ WKH UDGLDWLRQ FKDUDFWHULVWLFV RIVWUXFWXUHV FDQDOVREHGHVFULEHGE\RWKHUTXDQWLWLHV2QHVXFKDOWHUDWLYHLVUHDOL]HGE\ WKH UDWLR RI UDGLDWHG SRZHU LQWR WKH DPELHQW IOXLG :5DG WR WKDW IHG WR WKH VWUXFWXUH ]

:UDG  :6

 

8QIRUWXQDWHO\ WKLV IURP D SK\VLFV SRLQW RI YLHZ HDVLO\ LQWHUSUHWDEOH TXDQWLW\ LV QRW HDVLO\ PHDVXUHG 7KLV LV EHFDXVH WKH LQSXW SRZHU XVXDOO\ FDQQRWEHUHDGLO\PHDVXUHG7KHUHIRUHWKLVUDWLRZKLFKHYLGHQWO\ FDQQRW H[FHHGXQLW\ZLOOEHGLVFXVVHGRQO\IRUSRLQWH[FLWDWLRQ $QRWKHUTXDQWLW\ RILQWHUHVWLVWKHUDGLDWLRQORVVIDFWRU KUDGZKLFKLQGL FDWHVWRZKDWH[WHQWWKHYLEUDWLRQRIDVWUXFWXUHDUHGDPSHGGXHWRUDGLD WLRQWRDQDPELHQWIOXLG7KHUDGLDWLRQORVVIDFWRULVGHILQHGDV KUDG

(UDG  S (6

 

LQFRPSOHWHDQDORJ\ZLWKWKHRUGLQDU\ORVVIDFWRULQ  DQGZKHUH(UDG LVWKHUDGLDWHGHQHUJ\ZLWKLQRQHSHULRGDQG(6WKHUHYHUVLEOHVWUXFWXUDOHQ HUJ\%\FRQVLGHULQJWKDW(UDGHTXDOVWKHUDGLDWHGSRZHU:UDGPXOWLSOLHGE\ WKHSHULRG7 I  FDQEHUHZULWWHQDV

6RXQG5DGLDWLRQIURP6WUXFWXUHV

KUDG

:UDG  Z (6

 

)RUWKHVWUXFWXUHVRISULPDU\LQWHUHVWLQWKLVFRQWH[WQDPHO\EHDPVSODWHV  DQGVKHOOVYLEUDWLQJLQIOH[XUH (6 Pcc6 Y   &RPELQLQJWKLVDQG D  DUHODWLRQFDQEHGHYHORSHGEHWZHHQWKHUDGLDWLRQORVVIDFWRUDQGWKHUDGLD WLRQHIILFLHQF\DV K6

U FV  Z Pcc

 

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

)LJ  ([DPSOHV RI PHDVXUHG UDGLDWLRQ ORVV IDFWRUV D  PP VWLIIHQHG VWHHO SODWHUDGLDWLQJWRZDWHUDQG E PPVWLIIHQHGVWHHOSODWHUDGLDWLQJWRDLU

(OHPHQWDU\5DGLDWRUV

%\ PHDQV RI WKH UDGLDWLRQ ORVV IDFWRU WKH SRZHU UDWLR HIILFLHQF\  GH ILQHGLQ  FDQEHGHWHUPLQHG7KHWRWDOLPSDUWHGVWUXFWXUHERUQHVRXQG SRZHUREYLRXVO\PXVWHTXDOWKHVXPRIUDGLDWHGSRZHU:UDGDQGWKHSRZHU WUDQVIRUPHGLQWRKHDW:K:LWKDORVVIDFWRUKKUHSUHVHQWLQJWKDWIUDFWLRQRI HQHUJ\WKDWLVFRQYHUWHGLQWRKHDW  JLYHV ]

:UDG :UDG  :K

   ZKK (6  :UDG

   KK  KUDG

 

7KH HIILFLHQF\ ] DFFRUGLQJO\ LV GHWHUPLQHG E\ WKH ORVV IDFWRU UDWLR $V H[SHFWHG  WHQGVWRXQLW\IRUYHU\ VPDOOLQQHUORVVHV KK KUDG7KLV PHDQVWKDWDOOWKHSRZHULQMHFWHGLVUDGLDWHGDVIOXLGERUQHVRXQG)RUDLU ERUQHVRXQGWKLVFRQGLWLRQQHYHURFFXUV EXWLWLVUDWKHURIWHQHQFRXQWHUHG IRUUDGLDWLRQLQWRZDWHU

(OHPHQWDU\5DGLDWRUV 6SKHULFDO5DGLDWRUV 7KHVLPSOHVWFDVHRIVRXQGUDGLDWLRQLVWKDWRIDVSKHUHZKLFKFKDQJHVLWV YROXPHZLWKWLPHLQDQXQERXQGHGXQGLVWXUEHGKRPRJHQHRXVIOXLG6XFK DÄEUHDWKLQJVSKHUH³RUVRXUFHRI]HURRUGHULVUDUHO\HQFRXQWHUHGLQSUDF WLFH EXW LV RI JUHDW WKHRUHWLFDO LPSRUWDQFH 7KLV LV EHFDXVH FRPSOLFDWHG VRXQG VRXUFHV FDQ RIWHQ EH FRQVLGHUHG DV PDGH XS RI D ODUJH QXPEHU RI VSKHULFDORUKHPLVSKHULFDOVRXUFHV :LWKWKHUDGLDOPRWLRQDVIXQFWLRQRIWLPHRIDEUHDWKLQJVSKHUHDWDIUH TXHQF\ ZJLYHQE\ D  5H ª¬[ÖD H MZ W º¼ WKHYHORFLW\RIWKHVSKHULFDOVXUIDFHEH FRPHV YD W

5H ª¬ MZ[Ö D H MZ W º¼

5H ª¬YÖD H MZ W º¼ 

 

,QWKHIROORZLQJWKHYDULRXVV\PEROVLQGLFDWLQJSKDVRUVZLOOEHRPLWWHG 7KLVYHORFLW\FOHDUO\PXVWEHLGHQWLFDOWRWKDWRIWKHDPELHQWIOXLGDWWKHLQ WHUIDFHU D

6RXQG5DGLDWLRQIURP6WUXFWXUHV

)LJ5DGLDWLRQIURPDVSKHULFDOVRXUFH

0DWKHPDWLFDOO\WKHSUREOHPFRQVLVWVRIILQGLQJDVROXWLRQWRWKHZDYH HTXDWLRQ ’  S  N S



 

VXEMHFWWRWKHERXQGDU\FRQGLWLRQWKDWWKHUDGLDOYHORFLW\RIWKHIOXLGHTXDOV WKDW DW WKH VXUIDFH RI WKH VSKHUH ,QWKHDERYHH[SUHVVLRQWKH YLVFRVLW\ RI WKH IOXLG LV QHJOHFWHG DQG N  ZF LV WKH ZDYHQXPEHU ,Q VSKHULFDO FR RUGLQDWHV U-M WKHZDYHHTXDWLRQUHDGV>@  w §  wS ·  w § wS · ¨U ¸ ¨ VLQ ¸ U  wU © wU ¹ U  VLQ - w- © w- ¹  w S     N S  U VLQ - wM 

D 

7KH JHQHUDO UHODWLRQVKLS EHWZHHQ SUHVVXUH DQG YHORFLW\ LV REWDLQHG IURP WKHHTXDWLRQRIFRQVHUYDWLRQRIPRPHQWXPDV Y



 ’S  MZU 

 

IURPZKLFKWKHUDGLDOFRPSRQHQWLVJLYHQE\ YU





wS MZU  wU

D 

)RUWKHSUHVHQWSUREOHPUDGLDOV\PPHWU\H[LVWVVXFKWKDWWKHILHOGTXDQWL WLHVDUHLQGHSHQGHQWRIWKHDQJOHV -DQG M+HQFHWKHFRUUHVSRQGLQJGLI IHUHQWLDOV LQ (T D  YDQLVKHV DQG RQH PD\ DVVXPH D VSDWLDO SUHVVXUH ILHOGLQWKHIRUP

(OHPHQWDU\5DGLDWRUV

SU

&  MN U  H U

 

7KLV SUHVVXUH ILHOG VDWLVILHV D  DQG UHSUHVHQWV DQ RXWZDUGV SURSDJDW LQJZDYHLQWKHUGLUHFWLRQ7KHFRQVWDQW&LVGHWHUPLQHGIURPWKHERXQG DU\FRQGLWLRQDWU D)URP D WKHUDGLDOFRPSRQHQWRIWKHYHORFLW\ FDQEHGHYHORSHGDV YU



& §  MN   MZU  ¨© U  U

·  MN U  ¸H ¹

 

7KLVPHDQVWKDW YD

&   MN D MZU  D 

H  MN D Ÿ &

MZU  D  YD H MN D    MN D

 

VRWKDWWKHYHORFLW\DQGSUHVVXUHILHOGVFDQEHVXPPDUL]HGDV D    MN U  MN U  D H    MN D U 

YU

YD

SU

MZU  D  H MN U  D  YD   MN D U

 

)URPWKHVHWKHUDGLDWHGSRZHUFDQEHIRXQGWREHJLYHQE\ :

º  ª 5H « ³ SU YU G6 »  ¬6 ¼





S D  YD U  F

N D     N D

 

7KH LQWHJUDWLRQ LQYROYHG LV WDNHQ RYHU D VSKHULFDO VXUIDFH DW DQ DUELWUDU\ UDGLXVU DQGFDQVLPSO\EHUHSODFHGE\DPXOWLSOLFDWLRQE\SURZLQJWR V\PPHWU\ 7KHUDGLDWLRQHIILFLHQF\IRUDEUHDWKLQJ VSKHUHFDQQRZEHGHWHUPLQHG GLUHFWO\IURP  DQG  WREH 

V

N D     N D

 

7KXV WKH UDGLDWLRQ HIILFLHQF\ RI VSKHULFDO UDGLDWRUV LQFUHDVHV ZLWK WKH VTXDUH RI IUHTXHQF\ IRU ND   LH IRU ORZ IUHTXHQFLHV DQG VPDOO UDGLL )RU KLJK IUHTXHQFLHV DQG ODUJH UDGLDWRUVWKHUDGLDWLRQ HIILFLHQF\WHQGVWR XQLW\ 7KHVSHFLDOFDVHRIYHU\VPDOOVSKHULFDOUDGLDWRUVIRUZKLFKNDLV RISDUWLFXODULPSRUWDQFHZLWKUHVSHFWWRWKHDQDO\VLVWKDWIROORZV)RUVXFK

6RXQG5DGLDWLRQIURP6WUXFWXUHV

DUDGLDWRUZKLFKLVFDOOHGDÄSRLQWUDGLDWRU³RUPRQRSROHRQHILQGVIURP (T  WKDW S

MZU 

YD S D   MN U H S U

MZU  T

H  MN U  N D   S U

 

+HUHLQTUHSUHVHQWVWKHVRFDOOHGYROXPHYHORFLW\LHWKHSURGXFWRIYH ORFLW\ DQG VXUIDFH DUHD RI WKH UDGLDWRU 7KH LQWURGXFWLRQ RI WKLV VRXUFH VWUHQJWKKDVWKHDGYDQWDJHWKDWWKHUHVXOWFDQEHDSSOLHGWRSRLQWUDGLDWRUV RI DUELWUDU\ VKDSH :LWK UHVSHFW WR WKH VRXQG UDGLDWLRQ WKH VKDSH RI WKH VPDOOUDGLDWRULVXQLPSRUWDQW5DWKHULWLVWKHGLVSODFHGYROXPHWKDWPDW WHUVDQGKHQFHT$QLQIOXHQFHRIWKHVKDSHLVIRXQGRQO\LQWKHLPPHGLDWH YLFLQLW\RIVPDOOUDGLDWRUVDQGWKXVFDQEHQHJOHFWHG 'LSROH5DGLDWRUVDQG5DGLDWLRQIURP)RUFHV $QRWKHU YHU\ LPSRUWDQW HOHPHQWDU\ VRXUFH LV D IOXFWXDWLQJ IRUFH GLUHFWO\ DSSOLHGWRDKRPRJHQHRXVLQILQLWHIOXLG6XFKGLSROHVRXUFHVDUHUHTXLUHG IRUWKHJHQHUDOWUHDWPHQWRIUDGLDWLRQVFDWWHULQJDQGLQIORZDFRXVWLFV 7KHGLSROHUDGLDWRUFDQEHFRQVLGHUHGDVDULJLGVSKHUHRVFLOODWLQJIRU ZDUGVDQGEDFNZDUGVDWDIUHTXHQF\ Z DVLOOXVWUDWHGLQ)LJ7KHSRVL WLRQRIWKHVSKHULFDOVXUIDFHRUWKHUDGLDOYHORFLW\FRPSRQHQWDUHJLYHQE\ D  5H ª¬[ÖD FRV - H MZ W º¼  YD -

YÖD FRV - 

D 

,Q WKHVH H[SUHVVLRQV DSSHDUV RQO\ WKH DQJOH - VLQFH WKH PRWLRQ DQG WKH SUHVVXUHDUHLQGHSHQGHQWRIWKHDQJOHLQWKHKRUL]RQWDOSODQHM

)LJ5DGLDWLRQIURPDQRVFLOODWLQJULJLGVSKHUH

(OHPHQWDU\5DGLDWRUV

7KHDQDO\VLVLVIXOO\DQDORJRXVWRWKDWLQWKHSUHFHGLQJVHFWLRQ$YDOLG VROXWLRQWRWKHZDYHHTXDWLRQLQWKHDEVHQFHRIMGHSHQGHQFHLV S U -

§  MN · & ¨    ¸ FRV - H  MNU  U ¹ ©U

E 

DVUHDGLO\FDQEHYHULILHGE\VXEVWLWXWLRQLQ D %\PHDQVRI D  WKH UDGLDO FRPSRQHQW RI WKH YHORFLW\ FDQ EH GHWHUPLQHG DQG E\ FKRRVLQJ WKH FRQVWDQW & VXFK WKDW WKH YHORFLW\ HTXDOV WKDW RI WKH VRXUFH VXUIDFH YD - WKHSUHVVXUHDQGYHORFLW\ILHOGVDUHREWDLQHGDV S U - Y U -

ZKHUH &D JLYHQE\ :

§  MN · &D ¨    ¸ FRV - H MN U  D  U ¹ ©U &D FRV - §   MN N ·  MN U  D  ¨    ¸H MZU  © U  U ¹ U



MZU  YÖD D      MN D  N D  5H ª¬ S U  - YU U - º¼ G6  ³6



F 

 7KH UHVXOWLQJ UDGLDWHG SRZHU LV 

U  F S D   N D  YÖD      N D

G 

ZKHUHWKHLQWHJUDWLRQLVPDGHRYHUWKHFRPSOHWHVSKHULFDOVXUIDFH7KHVXU IDFHHOHPHQWG6LVGHWDLOHGLQ)LJ ,Q WKH GHULYDWLRQ RI F  WKH WDQJHQWLDO YHORFLW\ FRPSRQHQW LV QRW FRQVLGHUHG7KLVPHDQVWKDWWKHYHORFLW\RIWKHIOXLGLQSULQFLSOHFRXOGEH GLIIHUHQWIURPWKDWRIWKHVSKHULFDOVXUIDFH7KLVFRQWUDGLFWVWKHH[SHULHQFH EXW KDV DOPRVW QR LQIOXHQFH RQ WKH VRXQG UDGLDWLRQ 7KH LQ SUDFWLFH RE VHUYHGHTXDOLW\RIWKHWDQJHQWLDOFRPSRQHQWV QRVOLSFRQGLWLRQ LVUHODWHG WRWKHYLVFRVLW\HIIHFWVZKLFKZHUHRPLWWHGLQWKHZDYHHTXDWLRQLQ   $QLQFOXVLRQRIWKHYLVFRVLW\HIIHFWVKRZHYHULVKDUGO\ ZRUWKZKLOHVLQFH WKH YLVFRXV IRUFHV DUH ORFDOL]HG WR WKH YHU\ WKLQ DFRXVWLF ERXQGDU\ OD\HU 6WRNHV OD\HU  >@ MXVW RXWVLGH WKH VSKHUH 7KLV ERXQGDU\ OD\HU KDV WKH WKLFNQHVV G

Q  Z

 

FI(TV  7KHNLQHPDWLFYLVFRVLW\ QLVDSSUR[˜PVIRUDLU DQG  IRU ZDWHU ZKLFK PHDQV WKDW G LQ JHQHUDO LV VPDOOHU WKDQ PP DQGKHQFHXQLPSRUWDQWIRUUDGLDWLRQSUREOHPV ,QWUHDWLQJ WKHPRQRSROHWKHH[SUHVVLRQVIRUWKHSUHVVXUHDQGYHORFLW\ ILHOGV ZHUH IXUWKHUHG IRU WKH FDVH RI YHU\ VPDOO +HOPKROW] QXPEHUV ND 7KLVZDVGRQHE\LQWURGXFLQJWKHYROXPHYHORFLW\VXFKWKDWDUEL

6RXQG5DGLDWLRQIURP6WUXFWXUHV

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

)L  )U

PL MZ YD  ³ S D- FRV - G6  6

D 

ZKHUHPL  USDLVWKHPDVVRIWKHHQFORVHGIOXLGDQGMZ YDWKHDFFHO HUDWLRQS D - LVWKHVRXQGSUHVVXUHRQWKHVSKHULFDOVXUIDFHZKLFKH[ HUWVDIRUFHS D - FRV - G6 RQWKHVXUIDFHHOHPHQWG6LQWKHGLUHFWLRQRI PRWLRQ 8SRQ VXEVWLWXWLQJ WKH SUHVVXUH ILHOG LQ E  LQWR D  LW LV IRXQGWKDW )

MZU  YD D   S D  ³ FRV  - S D  VLQ - G  D § S D  S D  ·   U MZ YD ¨ U  ¸ MZU  YD S D    ¹ © MZU  YD

E 

IRU VR VPDOO VSKHUHV WKDW DOO WHUPV LQYROYLQJ ND LQ F  FDQ EH QH JOHFWHG7KLVPHDQVWKDW E LVYDOLGRQO\IRUVSKHUHVRIUDGLXVVPDOOHU WKDQDVL[WKRIWKHZDYHOHQJWK $V LV VHHQ IURP E  WKH UHDFWLRQ IRUFH GXH WR WKH VRXQG SUHVVXUH WKH LQWHJUDO  DSSHDUV DV DQ DGGLWLRQDO PDVV ,Q WKLV FDVH ZLWK D VSKHULFDO VRXUFHWKHK\GURG\QDPLFPDVVRUDGGHGPDVVLVSUHFLVHO\ KDOIWKDWRIWKH HQFORVHGIOXLGYROXPH)RUDVPDOOGLVFZLWKWKLFNQHVVKDQGUDGLXVDWKH DGGHGPDVVDPRXQWVWR PD

 ª Kº U  D  «    S »   ¬ D ¼

F 

(OHPHQWDU\5DGLDWRUV

ZKLFKLVVOLJKWO\ODUJHUWKDQWKDWIRUDVSKHUHZKHQWKHGLVFWKLFNQHVVYDQ LVKHV)URP E LWIROORZVWKDWMZ U YD D )S%\VXEVWLWXWLQJWKLV LQWR F DQG G RQHREWDLQVIRUVPDOO+HOPKROW]QXPEHUV  MNU § ·H ) FRV - ¨  MN ¸  ©U ¹ S U §   MN · H MNU ) FRV - ¨    N ¸  U ©U ¹ S U

S U - YU U  - :

G 

) Z    SU  F

7KHUHE\ UHODWLRQVKLSV DUH HVWDEOLVKHG EHWZHHQ D IRUFH DFWLQJ RQ D YHU\ VPDOODUHDRIWKHIOXLGDQGWKHUHVXOWLQJVRXQGILHOG ,QILQLWH3ODWHV $OLNHWKHSRLQWVRXUFHDQLQILQLWHO\H[WHQGHGSODWHYLEUDWLQJWUDQVYHUVHO\ UHSUHVHQWVDQHOHPHQWDU\FDVHZKLFKHQDEOHVWKHDQDO\VLVRIUDGLDWLRQIURP PRUHFRPSOLFDWHGV\VWHPVE\PHDQVRIDSSURSULDWHVXPPDWLRQVRULQWHJUD WLRQV )RUDSODWHDVVNHWFKHGLQ)LJZKLFKKDVDYHORFLW\ILHOGJLYHQE\ Y [

Y H  MN% [ 

D 

RQHPD\H[SHFWWKHVRXQGSUHVVXUHWREHLQWKHIRUP S [ \

S H MN% [ H

 MN \ \



E 

,Q WKH DERYH H[SUHVVLRQV N% LV WKH ZDYHQXPEHU RI WKH IOH[XUDO YLEUDWLRQ DQG WKH ZDYHOHQJWK LV O%  SN% 7KH H[SUHVVLRQ IRU WKH VRXQG SUHVVXUH PXVWVDWLVI\WKHZDYHHTXDWLRQLQ  DQGWKHQRUPDOFRPSRQHQWRIWKH YHORFLW\REWDLQHGIURP  ZLWK E VXEVWLWXWHGPXVWHTXDOWKDWRI WKHSODWHVXUIDFH7KHILUVWFRQGLWLRQOHDGVWR S  N %  N \ H MN% [ H

 MN \ \

 S N H MN% [ H

 MN \ \



ZKLFKLVVDWLVILHGLI N \

N  N % 

D 

ZKHUH DJDLQ N LV WKH ZDYHQXPEHU RI WKH IOXLG 7KH VHFRQG FRQGLWLRQ \LHOGV

6RXQG5DGLDWLRQIURP6WUXFWXUHV  wS MZU  w\

S N \  MN [ Ÿ S H ZU  %

\ 

Y U  F N  N\

E 

7KLVPHDQVWKDWWKHVRXQGSUHVVXUHLQWKHSRVLWLYHKDOIVSDFHEHFRPHV S [ \

Y U  F  %

 N  N

 

H  MN% [ H

 M N  N % \



F 

)LJ5DGLDWLRQIURPDQLQILQLWHSODWH

(TXDWLRQ F FDQEHFODULILHGSK\VLFDOO\E\UHZULWLQJLWLQWHUPVRIWKH DQJOHDWZKLFKWKHVRXQGLVUDGLDWHG7KLVDQJOHPXVWEHVXFKWKDWWKHSODWH YLEUDWLRQVDQGWKHVRXQGZDYHKDYHWKHVDPH[GHSHQGHQFHLHWKHWUDFH RIWKHDFRXVWLFZDYHDORQJWKHSODWHPXVWKDYHWKHVDPHZDYHOHQJWKDVWKH SODWHZDYH$VLVVHHQIURP)LJWKLVHTXDOLW\ RFFXUVLI OO% VLQ - ZKLFKLVHTXLYDOHQWWRN%N VLQ-7KXV(T F FDQEHUHZULWWHQDV S [ \

Y U  F  MN% [  MN \ FRVH H FRV -

 O% ! O 

D 

7KLV H[SUHVVLRQ RQO\ H[LVWV ZKHQ WKH ZDYHOHQJWK LQ WKH IOXLG LV VPDOOHU WKDQ WKDW RI WKH SODWH 2WKHUZLVH WKHUH H[LVWV QR UHDO DQJOH DW ZKLFK WKH VRXQG LV UDGLDWHG ,QVWHDG D VRXQG SUHVVXUH LV REWDLQHG ZKLFK GHFUHDVHV H[SRQHQWLDOO\ZLWKWKHGLVWDQFHIURPWKHSODWHDQGREH\V S [ \

MY U  F

N %  N



H MN% [ H 



N %  N  

 O%  O

E 

LQ YLHZ RI F  $FFRUGLQJO\ WZR IXQGDPHQWDOO\ GLIIHUHQW UHJLPHVIRU WKHUDGLDWLRQIURPLQILQLWHSODWHVPXVWEHGLVWLQJXLVKHG,IWKHZDYHOHQJWK RIWKHSODWHZDYHLVODUJHUWKDQWKDWLQWKHIOXLGWKHQSODQHZDYHVDUHUDGL DWHGLQWRWKHIOXLGDWDQDQJOHGHWHUPLQHGE\WKHUDWLRRIWKHZDYHOHQJWKV 7KHVRXQGSUHVVXUHLQWKHLPPHGLDWHYLFLQLW\RIWKHSODWHLVLQSKDVHZLWK

(OHPHQWDU\5DGLDWRUV

WKHSODWHYHORFLW\$VPD\UHDGLO\EHGHWHUPLQHGWKHUDGLDWLRQHIILFLHQF\LV JLYHQE\ V

 FRV -

N  

N  N %

 N %  N 

 

,IRQWKHRWKHUKDQGWKHZDYHOHQJWKLQWKHSODWHLVVPDOOHUWKDQWKDWLQWKH IOXLG D QHDUILHOG LV IRUPHG 7KH ODUJHU WKH GLIIHUHQFH LQ ZDYHOHQJWKV WKH IDVWHULVWKHGHFD\ ZLWKGLVWDQFH,QWKLVUHJLPHWKHSODWHYHORFLW\DQGWKH VRXQGSUHVVXUHDUHSKDVHVKLIWHGƒDQGKHQFHQRVRXQGSRZHULVUDGLDWHG 7KHUDGLDWLRQHIILFLHQF\WKHUHIRUHLV]HURDVVKRZQLQ)LJ7KHGHFD\ LVUDSLG)RUH[DPSOHIRUN%  OWKHVRXQGSUHVVXUHDWDGLVWDQFHRI KDOI D SODWH ZDYHOHQJWK IURP WKH VXUIDFH WKH SUHVVXUH LV RQO\  RI LWV YDOXHDWWKHVXUIDFHZKLFKFRUUHVSRQGVWRDERXWG%6LQFHWKHIOH[XUDO ZDYHOHQJWKVRISUDFWLFDOLQWHUHVWDUHRIWKHRUGHURIRQHPHWHURUOHVVWKH QHDUILHOG FDQ EH QHJOHFWHG QRW RQO\ LQ FRQMXQFWLRQ ZLWK VRXQG SRZHU EXW DOVRZLWKUHVSHFWWRWKHSUHVVXUH

)LJ5DGLDWLRQHIILFLHQF\IRUDQLQILQLWHSODWH

$V LQ WKH FDVH RI WKH RVFLOODWLQJ VSKHUH LV QRW FRQVLGHUHG WKH WDQJHQWLDO PRWLRQ ZKLFK GLIIHUV DW WKH SODWH DQG MXVW RXWVLGH $JDLQ RQO\ LQ D YHU\ WKLQERXQGDU\ OD\HUWKHYLVFRXVHIIHFWVHVWDEOLVKLQJWDQJHQWLDOIRUFHVFDQ EHREVHUYHGFI(T   $WWKHXSWRQRZH[FOXGHGSRLQWZKHUHN% NDQGZKHUHWKHVWUXFWXUDO DQG IOXLG ZDYHOHQJWK DUH HTXDO WKH WKHRU\ IXUQLVKHV D UDGLDWLRQ DQJOH RI

6RXQG5DGLDWLRQIURP6WUXFWXUHV

ƒ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

MN % Y  %

N N Y H

 

H  MN% [ H

   MN % [  N %  N \

H

 N %  N \



D  

7KHVHFRPSRQHQWVDFFRUGLQJO\DUHSKDVHVKLIWHGƒ7KLVPHDQVWKDWWKH DLUSDUWLFOHVJHQHUDOO\XQGHUJRHOOLSWLFPRWLRQVDVGHSLFWHGLQ)LJ7KH HOOLSWLF RUELWV PD\ EH LQWHUSUHWHG DV WKH DLU SDUWLFOHV HVFDSH FRPSUHVVLRQ 7KHFRPSUHVVLRQQHFHVVDU\IRUJHQHUDWLQJDVRXQGZDYHLVWKXVDYRLGHGE\ PRYLQJ ODWHUDOO\ VR WKDW HIIHFWLYHO\ DOO WKDW UHPDLQV LV D EDFN DQG IRUWK VORVKLQJRIDLUIURPZDYHFUHVWWRWKURXJKDK\GURG\QDPLFVKRUWFLUFXLW

)LJ3DUWLFOHPRWLRQVLQIDUILHOGDQGQHDUILHOGUDGLDWLRQ

(OHPHQWDU\5DGLDWRUV

&\OLQGULFDO5DGLDWRUV 7RGHWHUPLQHWKHUDGLDWLRQIURPYLEUDWLQJSLSHVWXEHVDQGGXFWVLWLVKHOS IXO WR NQRZ WKH UDGLDWLRQ SURSHUWLHV RI F\OLQGHUV &RQVLGHU DQ LQILQLWHO\ ORQJF\OLQGHUDVDQLGHDOL]DWLRQRIDYLEUDWLQJSLSH7KHF\OLQGHUYLEUDWHVDW DIUHTXHQF\ ZLQSKDVHDOODORQJWKHZKROHOHQJWKDQGWKHFLUFXPIHUHQWLDO YHORFLW\GLVWULEXWLRQLV Y M

YQ FRV QM 



DVGHSLFWHGOHIWLQ)LJ7KHWDVNDJDLQLVWRILQGDIXQFWLRQZKLFK x VDWLVILHVWKHZDYHHTXDWLRQLQF\OLQGULFDOFRRUGLQDWHVLH  w § wS ·  w  S  N S ¨U ¸   U wU © wU ¹ U wM

x

D 

\LHOGVDGHSHQGHQFHLQWKHIRUP S

x



 U

H MN U

 U of

E 

DW ODUJH GLVWDQFHV DQG KHQFH GHVFULEHV RXWZDUGV SURSDJDWLQJ F\OLQ GULFDOZDYHVDQG VDWLVILHVWKHERXQGDU\FRQGLWLRQ MZU  Y M

wS wU

F  U D

DWWKHVXUIDFHRIWKHUDGLDWRU )URP WKH SHUWLQHQW OLWHUDWXUH HJ >@ WKH WKUHH UHTXLUHPHQWV DUH IXO ILOOHGE\ S UM

YQ

Y UM

YQ

M U  F  c

+Q

N D

FRV QM + Q  N U 

FRV QM  + Q c N U  + Q  c N D

G

+HUHLQ + Q  ]  LV WKH +DQNHO IXQFWLRQ RI VHFRQG NLQG DQGQWKRUGHUDQG  + Q c ] LVWKHILUVWGHULYDWLYHZLWKUHVSHFWWRWKHDUJXPHQW,QYHU\ PDQ\ DSSOLFDWLRQVLWLVVXIILFLHQWWRZRUNZLWKWKHWZROLPLWLQJIRUPV>@

6RXQG5DGLDWLRQIURP6WUXFWXUHV  §J ] · ­ ½   M OQ ¨ ¸  Q  ° ° S ©  ¹ ° °  +Q ] | ® ¾  N U  Q    Q °§ ] ·  M Q    §  · °   Q t °¯¨©  ¸¹ °¿ S ¨© ] ¸¹

H 

DQG S§



 M  ¨© Q   ¸¹  M] H H S]

+ Q  ] |

I 

 N U !! Q   

ZKHUH(XOHU¶VFRQVWDQWLVJ « 7KH UDGLDWHG SRZHU FDQ EH GHWHUPLQHG IRU H[DPSOH IURP WKH VRXQG SUHVVXUHLQWKHIDUILHOGLHIURP(T G XVLQJWKHDV\PSWRWLFH[SUHV VLRQ I  :UDG

   U  F

S

³ S UM



UGM



U  F YQ



H Q + Q  c N D





D 

ZKHUH ­   Q   ® ¯   Q t

HQ

7KXVWKHUDGLDWLRQHIILFLHQF\GHILQHGLQ D EHFRPHV VQ

Z    S F +  c N D  N D  Q

E 

IRUDF\OLQGULFDOYLEUDWLRQZLWKQQRGHVDURXQGWKHFLUFXPIHUHQFH $ FRPSOHWHO\ DQDORJRXV DQDO\VLV FDQ EH PDGH IRU WKH FDVH WKDW WKH YLEUDWLRQVH[KLELWDSHULRGLFLW\ZLWKDZDYHOHQJWKO]DOVRLQWKHD[LDOGLUHF WLRQFIWKHOHIWSDUWRI)LJ,QVXFKDFDVHWKHF\OLQGHUYLEUDWLRQVDUH JLYHQE\ Y M  ]

YQ FRV QM H  MN] ] 

D 

7KH DVVRFLDWHG UDGLDWLRQ HIILFLHQF\ LV JLYHQ E\ E  ZKHQ WKH ZDYHQXPEHUNLVUHSODFHGE\WKHUDGLDOZDYHQXPEHUFRPSRQHQW NU

N  N ] 

E 

:KHQN]!NNUEHFRPHVLPDJLQDU\DQGWKHUDGLDWLRQHIILFLHQF\RIFRXUVH YDQLVKHVVQ 

(OHPHQWDU\5DGLDWRUV

,Q)LJDUHVKRZQWKHUDGLDWLRQHIILFLHQFLHVDVIXQFWLRQRI+HOPKROW] QXPEHUIRUWKHQLQHILUVWYLEUDWLRQPRGHVRIDQLQILQLWHF\OLQGHU)URPWKLV JUDSK DQG IXUWKHU VWXGLHV RI WKH JRYHUQLQJ HTXDWLRQV WKH IROORZLQJ UH PDUNVFDQEHPDGH x :KHQ WKH D[LDO FRPSRQHQW RI WKH VWUXFWXUDO ZDYHOHQJWK LV VPDOOHU WKDQ WKH IOXLG ZDYHOHQJWK LH O]  O WKH UDGLDWLRQ HIILFLHQF\ YDQ LVKHVV $VIRUWKHSODWHWKHUHRFFXUVDK\GURG\QDPLFVKRUWFLU FXLWFI)LJVDQG x :KHQ O  SDQ DQG O  O] RU HTXLYDOHQWO\ NUD ! Q LH WKH D[LDO ZDYHOHQJWK O] DQG DOVR WKH FLUFXPIHUHQWLDO ZDYHOHQJWK SDQ DUH ORQJHUWKDQWKDWRIWKHIOXLGDFRPSDUDWLYHO\VWURQJUDGLDWLRQRFFXUV ZLWK V   7KH PD[LPXP LV UHDFKHG DW NUD | Q DQG DPRXQWV WR V PD[  Q  x :KHQ NUD  Q RU O ! SDQ EXW O  O] VR WKDW WKH FLUFXPIHUHQWLDO ZDYHOHQJWKLVVPDOOHUWKDQWKDWRIWKHIOXLGWKHUHLVDK\GURG\QDPLF VKRUWFLUFXLWLQWKHFLUFXPIHUHQWLDOGLUHFWLRQZKLFKKRZHYHUZLOOQRW EH FRPSOHWH GXH WR WKH F\OLQGHU JHRPHWU\ +HQFH D VPDOO UDGLDWLRQ HIILFLHQF\LVIRXQGIRUWKLVUDQJHZKLFKFDQEHDSSUR[LPDWHGE\ S ­ NU D  Q   NU D   °  ° VQ | ®  Q  ° Z S  ¨§ NU D ¸·  Q t   NU D  Q   °¯ F N U Q  ©  ¹

F 

7KHODUJHUWKHRUGHUQWKHVPDOOHULVWKHUDGLDWLRQHIILFLHQF\ DWVPDOO +HOPKROW]QXPEHUV

)LJ5DGLDWLRQHIILFLHQF\IRUDF\OLQGULFDOUDGLDWRUIRUVRPHGLIIHUHQWRUGHUV

6RXQG5DGLDWLRQIURP6WUXFWXUHV

)LJXUHYLVXDOL]HVWKHVRXQGILHOGLQWKHYLFLQLW\RIDF\OLQGULFDOUD GLDWRU 7KH H[SUHVVLRQV   WR   ZKLFK DUH YDOLG IRU Q QRGHV FLUFXPIHUHQWLDOO\DQGDQD[LDOZDYHQXPEHUN]FDQEHH[WHQGHGWRDFFRP PRGDWH DUELWUDU\ YHORFLW\ GLVWULEXWLRQV DW WKH F\OLQGHU VXUIDFH VLQFH WKH VSDWLDO IXQFWLRQV FRV QM DQG H  MN ]  GHVFULEH ERWK WKH VXUIDFH YHORFLW\ DQG WKHVRXQGSUHVVXUHDWDQDUELWUDU\UHFHLYHUSRVLWLRQ7KLVPHDQVWKDWWKHUD GLDWHG PRGDO SRZHU FDQ EH VXSHULPSRVHG ,I WKHUHIRUH WKHYHORFLW\ ILHOG RIWKHF\OLQGULFDOVXUIDFHLVLQWKHIRUP ]

Y M  ]

¦ ¬ªY N FRV QM  Y N VLQ QM ¼º H FQ

]P

VQ

 MN ]P ]

]P



Q P

D 

WKHUDGLDWHGSRZHUEHFRPHV :UDG

  U  F 6 ¦ ª YFQ N ]P  YVQ N ]P º V N ]P  ¼ ¬  Q P

E 

,QWKHODWWHUH[SUHVVLRQ6PHDQVWKH YHU\ODUJH F\OLQGHUVXUIDFH

)LJ  6RXQG ILHOG LQWKHYLFLQLW\RIDF\OLQGULFDOUDGLDWRUDW ND IRUVRPH GLIIHUHQWRUGHUVQ «)RUQ H[LVWVRQO\DQHDUILHOG

(OHPHQWDU\5DGLDWRUV

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f

T Z

f

³ T W H

 MZ W



GW  T W

f

 MZ W ³ T Z H GZ  S f

D 

)RU D FRPSDFW VRXUFH RI W\SLFDO GLPHQVLRQ D IRU ZKLFK ND   LQ WKH UDQJHRILQWHUHVWWKHUDGLDWHGVRXQGLVJLYHQE\  LH S Z

U S U

MZ T Z H  MN U 

E 

7RSHUIRUP WKHLQYHUVHWUDQVIRUPLWPXVWEHREVHUYHGWKDWE\GLIIHUHQWLDW LQJ D ZLWKUHVSHFWWRWLPHWKHH[SUHVVLRQ GT W GW

 S

f

³

MZ T Z H MZ W GZ

f

UHVXOWV7KLVPHDQVWKDWWKHWLPHKLVWRU\RIWKHSUHVVXUHDWVRPHGLVWDQFHU IURPWKHVRXUFHLVREWDLQHGDV S U W

 S

f

³

S Z H MZ W GZ

f

U  GT W  U  F  GW S U

U  S U S

f

³

f

MZ T Z H

MZ W  U  F

GZ

F 

ZKHUHWKHVXEVWLWXWLRQZW±NU Z W±UF LVPDGH )XOO\DQDORJRXVO\ F \LHOGVWKHUDGLDWHGSUHVVXUHVSHFWUXPIURPD VPDOOIRUFHVRXUFHDV

6RXQG5DGLDWLRQIURP6WUXFWXUHV S U -  Z

) Z

FRV - §  MZ ·  MN U  ¨  ¸H S U © U F ¹

D 

)URPDQLQYHUVHWUDQVIRUPLVREWDLQHG S U -  W

º FRV - ª   G ) W  U  F »  « ) W  U  F  S U ¬ U F GW ¼

E 

VLQFHDPXOWLSOLFDWLRQE\ MZ FRUUHVSRQGVWRDGLIIHUHQWLDWLRQZLWKUHVSHFWWR WLPH8SRQLQWHJUDWLQJ F DQG E RYHUDODUJHVSKHULFDOVXUIDFH HQFORVLQJWKHVRXUFHVWKHSRZHUVUDGLDWHGDUHJLYHQE\ 

:T

 U  ª GT W º « »  S F ¬ GW ¼

:)

  ª G) W º « »  S U  F ¬ GW ¼



 

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

S U -  Z

ZKHUH

 U  F

N D D FRV - H  MN U  D YD  U    MN D  N D

D 

(OHPHQWDU\5DGLDWRUV 8  VLQ Z'W    Z Z'W  

YD

,QWKLVFRQWH[WIDUILHOGPHDQVWKDWUNU$VSRZHULVRIOLWWOHPHDQ LQJIRUDVLQJOHSXOVHWKHVRXQGHQHUJ\ZLOOEHGHWHUPLQHGIRUWKHGHFHO HUDWLRQ7KLVLVVLPLODUWRWKHGHWHUPLQDWLRQRIWKHYLEUDWLRQHQHUJ\GXHWR DQLPSDFWLQ(T  7KHRQO\PRGLILFDWLRQUHTXLUHGLVWRUHSODFHWKH IRUFH E\ D VXUIDFH LQWHJUDO RI WKH SUHVVXUH 7KXV WKH GHFHOHUDWLRQ VRXQG HQHUJ\LVREWDLQHGDV ( $FF

³³ S U-  W Y U-  W GWG6  S

³³ S U-  Z



 G Z G6  U F

E 

:LWK D  LQWURGXFHG WKH LQWHJUDWLRQ RYHU WKH VXUIDFH HOHPHQWV G6 SUVLQ- U G-JLYHVWKHHQHUJ\GHQVLW\VSHFWUXPDV 

G( $FF

 S  VLQ Z'W   Z D  F D U  D 8  GZ    S Z'W     Z D  F F

F 

+HUHLQLVWDNHQLQWRDFFRXQWWKHIDFWWKDWSRVLWLYHDQGQHJDWLYHIUHTXHQFLHV UHQGHU HTXDOFRQWULEXWLRQVWRWKHVSHFWUXPDQGKHQFHGZLQ F RQO\ UHIHUV WR SRVLWLYH IUHTXHQFLHV :KHQ WKH GHFHOHUDWLRQ WLPH 'W LV VR VKRUW WKDWF 'WD LHWKHVRXQGSURSDJDWLRQLVVPDOOHUWKDQWKHEDOOGLDPHWHU WKH PHWKRGV RI UHVLGXHV FDQ EH HPSOR\HG IRU WKH IUHTXHQF\ LQWHJUDWLRQ 7KHSROHVORFDWHGDWZDF rM\LHOG ( $FF

S U  D 8  

 S D   U 8  

 PD8   

G 

7KHPDVVPDLVWKHDGGHGPDVVGHILQHGLQ E  7KHVXUSULVLQJO\VLPSOHUHVXOWPHDQVWKDWWKHUDGLDWHGHQHUJ\IRUDYHU\ VKRUWDFFHOHUDWLRQHTXDOVWKHNLQHWLFHQHUJ\RIWKHDGGHGPDVV>@)RU NQRZQDGGHGPDVVHV G DSSOLHVIRUQRQVSKHULFDOERGLHV)RUH[DP SOHWKHDGGHGPDVVIRUDWKLQGLVFLVDSSUR[LPDWHO\JLYHQE\ F  8SRQ VXEVWLWXWLQJ WKH GDWD IRU D IRUJH KDPPHU LQWR G  LH 9 |˜P U |NJPPD |NJDQG8 PVWKHDFFHO HUDWLRQQRLVHHQHUJ\DPRXQWVWR($FF |:VSHULPSDFW)RUDUDWHRIWZR LPSDFWVSHUVHFRQGWKLVFRUUHVSRQGVWRDQDYHUDJHSRZHUOHYHORIG% $OWKRXJKWKDWLVVXEVWDQWLDOO\OHVVWKDQWKHSRZHUUDGLDWHGIURPWKHLPSDFW H[FLWHGIRUJHKDPPHUFDVLQJLWLVQRWWKDWOLPLWHGDVLQWXLWLYHO\FRXOGEH H[SHFWHG5HODWLYHO\KLJKDFFHOHUDWLRQQRLVHLVGHOLYHUHGDOVRE\ VODPPLQJ

6RXQG5DGLDWLRQIURP6WUXFWXUHV

GRRUV$JDLQLWFDQEHH[SHFWHGWREHIDUEHORZWKHQRLVHUDGLDWHGIURP WKH YLEUDWLQJGRRUEODGHDQGIUDPH )RU QRW VR VKRUW GHFHOHUDWLRQV RU DFFHOHUDWLRQV G  LV LQYDOLG ,Q VWHDG E DQG F \LHOGFXUYHVWKDWDUHH[HPSOLILHGLQ)LJ

)LJ$FFHOHUDWLRQQRLVHIURPDEDOORIUDGLXVDDQGDFFHOHUDWLRQWLPH 'W D  (QHUJ\ GHQVLW\ VSHFWUXP  ORJ ª¬ G( $FF  GZ F  D(PD[ º¼  DQG E  WRWDO HQHUJ\ ORJ ($FF(PD[ ZKHUH (PD[ SU  D 8   

3ODQH%DIIOHG5DGLDWRUV :KLOVWWKHVROXWLRQRIWKHUDGLDWLRQSUREOHP FDQEHUDWKHUFRPSOLFDWHGIRU DQDUELWUDULO\VKDSHGVWUXFWXUHDVROXWLRQRQFORVHGIRUPLVUHODWLYHO\VLP SO\IRXQGIRUWKHSODQHUDGLDWRULQDULJLGEDIIOH$SUHUHTXLVLWHLVWKDWWKH UDGLDWRULVORFDWHGLQDODUJHEDIIOHVRWKDWWKHYHORFLW\ ILHOGLVNQRZQRYHU

3ODQH%DIIOHG5DGLDWRUV

DQLQILQLWHSODQH)RUWKHUDGLDWRUWKHYHORFLW\LVY Y [] ZKHUHDVLWLV ]HURHOVHZKHUH7KLVPDNHVLWSRVVLEOHWRDSSO\ x HLWKHUWKHVXPPDWLRQRIPDQ\VPDOOYROXPHYHORFLW\VRXUFHVRU x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

MZU  T

H  MN U  S U

D 

$JDLQLWLVLUUHOHYDQWLIWKHSRLQWVRXUFHLVVHPLVSKHULFDORUKDVDQ\ RWKHU VKDSH,PSRUWDQWLVWKHYROXPHYHORFLW\ T Y˜6)RUDVHWRISRLQWVRXUFHV VXFK DVLQORXGVSHDNHUDUUD\VDQGLQVRQDU¶VWKHFRQWULEXWLRQVIURPHDFK VRXUFHDUHVLPSO\VXPPHGWRHVWDEOLVKWKHQHWSUHVVXUH:LWKWKHQRWDWLRQ RI)LJWKHSUHVVXUHFDQEHZULWWHQDV S

º MZU  ª T  MN U T  MN U  H  » « H S ¬ U U ¼

MZU  S

TQ

¦U

H MN UQ 

E 

Q

7KHVWHSWRDFRQWLQXRXVYHORFLW\ILHOGLVYHU\VPDOO(YHU\VXUIDFHHOHPHQW G6KDYLQJWKHYHORFLW\ Y 6 FDQEHVHHQDVDSRLQWVRXUFHZLWKWKHYROXPH YHORFLW\ GT 6  Y V G6:LWKWKHVXPPDWLRQRYHUVRXUFHFRQWULEXWLRQVUH SODFHG E\ DQ LQWHJUDWLRQ RYHU WKH DUHD WKH SUHVVXUH UHVXOWLQJ IURP WKH FRPSOHWHUDGLDWRULVJLYHQE\ S

MZU  Y V  MN U G6  H S ³ U

7KLVH[SUHVVLRQGDWHVEDFNWR/RUG5D\OHLJK>@

 

6RXQG5DGLDWLRQIURP6WUXFWXUHV

)LJ6XPPDWLRQRISRLQWVRXUFHV

,IWKHGLUHFWLYLW\LQWKHIDUILHOGLVVRXJKWLWLVVXLWDEOHWRORFDWHWKHRUL JLQIRUWKH VSKHULFDO FRRUGLQDWHLQ  VRPHZKHUHDWWKHFHQWUHRIWKH UDGLDWRUVXFKWKDWWKHUHFHLYHUSRLQWKDVWKHFRRUGLQDWHV [

5 VLQ - VLQ M



\

5 FRV -  ]

5 VLQ - FRV M 

D 

DVVKRZQLQ)LJ+HUH-LVWKHDQJOHEHWZHHQWKH\D[LVDQGWKHYHF WRUWRWKHUHFHLYHUSRLQWZKHUHDV MLVWKHDQJOHLQWKHSODQH\ )RUDQ DUELWUDU\VRXUFHSRLQW[T]TWKHGLVWDQFHWRWKHUHFHLYHUSRLQWEHFRPHV  

U

ª [  [   \   ]  ]  º T T ¬« ¼»

 

ª [T ]T [T  ]T º 5 «   VLQ - VLQ M   VLQ - FRV M  » 5 5 5  ¼» ¬«

E 

| 5  [T VLQ - VLQ M  ]T VLQ - FRV M  [T  ]T   5  

)RUWKHDSSUR[LPDWLRQLVXVHGWKHELQRPLDOIRUPXOD   H |   H ZKHQ HZKLFKLVYDOLGIRU[TDQG]TPXFKOHVVWKDQ5:LWK E LQVHUWHG LQWR  DQG5DVVXPHGODUJHRQHREWDLQV S 5 -  M

MZU   MN 5 MN VLQ - [T VLQ M  ]T FRV M H (5 G[T G]T  ³ Y [T  ]T H S 5

F 

3ODQH%DIIOHG5DGLDWRUV

ZKHUH (5 H |   7KLV HVWDEOLVKHV D UHODWLYHO\ PDQDJHDEOH H[ SUHVVLRQIRUWKHGLUHFWLYLW\RIWKHUDGLDWHGVRXQG7KHIDUILHOGDSSUR[LPD WLRQGHYHORSHGLVDSSOLFDEOHLI  MN [T  ]T   5

N  [T  ]T 5

[T  ]T 5O

S  

G 

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

)LJ  6XEVWLWXWLRQ RI D PRGDO YLEUDWLRQ ILHOG E\ D SHULRGLF DUUDQJHPHQW RI SRLQWVRXUFHVRSHUDWLQJLQDQWLSKDVH

6RPH ILQGLQJV IURP WKH DSSOLFDWLRQ RI   DQG F  ZLWK (5   DUH D  )RUDUUD\VRIDQWLSKDVHGSRLQWVRXUFHVZKLFKFDQEHFRQVLGHUHGDV DSSUR[LPDWLRQVIRUPRGDOYHORFLW\ILHOGVDVLOOXVWUDWHGLQ)LJ WKHVXPPDWLRQOHDGVWR S [ \  ] S [ \  ]

Q  Q  Q  Q MZU  T H  MN 5 ¦ ª¬ H MN D[  5 H MS º¼ ¦ ª¬ H MNE]  5 H MS º¼  S 5 Q  Q 

ZU VLQ QD VLQQ  E  T S 5 VLQ D VLQ E

D 

6RXQG5DGLDWLRQIURP6WUXFWXUHV

ZKHUH D  S  ND[5 E  S  NE]5 7KH VHFRQG IRUP LV RE WDLQHGE\VXPPLQJWKHJHRPHWULFVHULHVLQWKHILUVW5LVWKHGLVWDQFH EHWZHHQWKHSRLQWVRXUFHQ  Q DQGWKHUHFHLYHUSRLQWLQWKHIDU ILHOG )RU FORVHO\ VSDFHG SRLQW VRXUFHV VXFK WKDW ND   DQG NE WKHVLQHVLQWKHGHQRPLQDWRUVWHQGWRXQLW\ VLQFH[5DQG ]57KHUHPDLQLQJH[SUHVVLRQVKRZVWKDWVXFKDQDUUD\IXUQLVKHV WKHVRXQGSUHVVXUH S [ \  ]

ZU  T ZU  T VLQ QD VLQ Q  E d  S 5 S 5

E 

ZKLFKLVJHQHUDOO\ORZHUWKDQWKDWRIDVLQJOHSRLQWVRXUFHGXHWRWKH K\GURG\QDPLFVKRUWFLUFXLW,IRQWKHRWKHUKDQGWKHVHSDUDWLRQRI QHLJKERXULQJ SRLQW VRXUFHV LV ODUJHU WKDQ KDOI D ZDYHOHQJWK WKH SUHVVXUH FRQWULEXWLRQV VXP XS WR SRVVLEO\ VXEVWDQWLDO DPSOLWXGHV 7KHPD[LPXPRFFXUVZKHQND[5 SNE]5 SDQGDPRXQWVWR S PD[

ZU  T QQ   S 5

F 

ZKLFK FRUUHVSRQGV WR WKH VXP RI DOO SRLQW VRXUFHV RSHUDWLQJ LQ SKDVH7KHDVVRFLDWHGUDGLDWLRQDQJOHLVVLPSO\IRXQGLQWKDWDOOLQ FLGHQWZDYHFRQWULEXWLRQVH[KLELWSKDVHGLIIHUHQFHVRIƒDQGLQ WHUIHUHFRQVWUXFWLYHO\ 6RPH H[DPSOHVRIFRPSXWHGGLUHFWLYLWLHVDUH VKRZQLQ)LJ E  )RUDUHFWDQJXODUSLVWRQZLWKVLGHVDDQGEDQGWKHYHORFLW\JLYHQE\ Y []  YWKHLQWHJUDOLQ F \LHOGVWKHIDUILHOGSUHVVXUH S 5 -  M

ZU  Y VLQ $   VLQ %    S 5 $  %

G 

ZKHUH$ NDVLQ-VLQMDQG% NEVLQ-FRVM7KHUDGLDWLRQHVVHQ WLDOO\RFFXUVLQWKHGLUHFWLRQRIWKHQRUPDO7KHODUJHUNDDQGNE WKHPRUHSURQRXQFHGLVWKHFHQWUHOREH,IDWDOOVLGHOREHVDSSHDU WKHLU]HURHVDUHORFDWHGDW$  S S«DQG%  S S«7R VXSSUHVVWKHVLGHOREHVWKHYHORFLW\PXVWGHFD\WRZDUGVWKHHGJHV F  )RUDFLUFXODUPHPEUDQHRIUDGLXVDDURWDWLRQDOO\V\PPHWULFYHORF LW\GLVWULEXWLRQFDQEHH[SUHVVHGDVDVXPRIIXQFWLRQVLQWKHIRUP Y U0

Y -  N 0 U0 

H 

7KHUHE\ YLVWKHDPSOLWXGHDWWKHFHQWUH- « WKH]HURRUGHU%HV VHOIXQFWLRQU0WKHGLVWDQFHIURP WKHFHQWUHWRDVXUIDFHHOHPHQWDQG N0  SO0 LV WKH PHPEUDQH ZDYHQXPEHU ZKLFK LV GHSHQGHQW RQ

3ODQH%DIIOHG5DGLDWRUV

WKHPHPEUDQHSURSHUWLHVDQGWKHW\SHRIH[FLWDWLRQO0LVWKHDVVRFL DWHGZDYHOHQJWK%\LQWURGXFLQJ H LQ F WKHVXUIDFHHOH PHQWLVVXLWDEOHFKRVHQDVG[T G]T S U0 GU06LPLODUO\FDQEHHP SOR\HG [T VLQ M  ]T FRV M  U0 VLQFH LW LV VXIILFLHQW WR FRQVLGHU D VLQJOHGLUHFWLRQLQYLHZRIWKHURWDWLRQDOV\PPHWU\:LWKWKHVHVXE VWLWXWLRQVWKHLQWHJUDOLQ  JLYHV S 5 -

MZU  D  MN 5 N0 -  J  - J 0  N VLQ - -  J 0 - J   Y H  5 N0  N VLQ -

I 

LQ ZKLFK J NDVLQ -DQG J0 N0D7RLOOXVWUDWHWKLVUHVXOW)LJ  VKRZV WKH GLUHFWLYLW\ RI IRXU FDVHV 2ZLQJ WR V\PPHWU\ RQO\ KDOIGLDJUDPVDUHSUHVHQWHG

)LJ'LUHFWLYLW\RIOLQHDUDUUD\VRISRLQWVRXUFHV

6RXQG5DGLDWLRQIURP6WUXFWXUHV

)LJ  'LUHFWLYLW\ RI UDGLDWLRQ IURP D FLUFXODU PHPEUDQH ZLWK WKH YHORFLW\ GLVWULEXWLRQ RI H

$V FDQ EH VHHQ WKH UDGLDWLRQ LV FRPSDUDWLYHO\ KLJK DQG GLUHFWHG IRU N0N  JJ07KLVLVLQDJUHHPHQWZLWKWKHILQGLQJVIRULQILQLWHSODWHV DQGSODQDUDUUD\VRIVRXUFHVZKHQWKHIOXLGZDYHOHQJWKLVVKRUWHUWKDQWKDW RI WKH VWUXFWXUH 7KH GLUHFWLRQ RI WKH PDLQ UDGLDWLRQ LV FORVH WR WKDW IRU ZKLFKWKHGHQRPLQDWRULQ I YDQLVKHVLHVLQ- | N0N$OVRRILQWHU HVWLVWKDWWKHUDGLDWHGVRXQGSUHVVXUHLVUHODWLYHO\LQGHSHQGHQWRIZKHWKHU WKHUH LVDQRGHRUDQDQWLQRGHDWWKHSHULSKHU\IRU N0N7KHGLIIHU HQFH LV VHHQ HVVHQWLDOO\ LQ WKH GLUHFWLYLW\ ZKHUH DGMDFHQW PD[LPD DUH

3ODQH%DIIOHG5DGLDWRUV

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

YÖ N [  N ] H  MN[ [ H MN ] ] 

D 

E  6RXQGSUHVVXUHIURPWKHERXQGDU\FRQGLWLRQV U  F N YÖ N [  N ]

S % [ \  ]

N  N [  N ]

H MN[ [ H MN] ] H

 M N  N [  N ] \

 \ !  

E 

F  3DUWLFOHYHORFLW\LQWKH\GLUHFWLRQ Y \%

M YÖ N [  N ] H MN[ [ H  MN ] ] H

N  N [  N ] \



F 

,QRUGHUWRHVWDEOLVKDGHVFULSWLRQRIWKHUDGLDWLRQIURPDSODQHUDGLDWRU ZLWK DQ DUELWUDU\ YHORFLW\ ILHOG Y [ ]  (TV   DJDLQ DUH FRQVLGHUHG EXLOGLQJEORFNVIURPZKLFKWKHWRWDOVROXWLRQFDQEHFRQVWUXFWHG7KHSUR

6RXQG5DGLDWLRQIURP6WUXFWXUHV

FHGXUHLVLGHQWLFDOWRWKDWHPSOR\HGLQ6HFW7KHYHORFLW\ ILHOGRIWKH UDGLDWRULVGHVFULEHGE\WKH)RXULHULQWHJUDO f



Y% [ ]

S 

³ YÖ N  N H [

 MN [ [  MN ] ]

H

]

GN [ GN ] 

D 

f

ZKLFK LV DSSOLFDEOH IRU DOO SK\VLFDOO\ UHDOL]DEOH GLVWULEXWLRQV 7KH ZDYHQXPEHU VSHFWUXP YÖ N [  N ]  LV REWDLQHG IURP D WUDQVIRUPDWLRQ RI WKH VWDWLRQDU\YHORFLW\GLVWULEXWLRQ f

YÖ N [  N ]

³ Y [ ] H

 MN [ [  MN ] ]

H

%

G[G]

E 

f

7KHLQWHJUDWLRQLVH[WHQGHGWRLQILQLW\EXWVLQFHYV RXWVLGHWKHUDGLDWRU VXUIDFHLWVXIILFHVWRHQFRPSDVVWKHUDGLDWRUVXUIDFH $V E LVYDOLGIRUHYHU\ ZDYHQXPEHUSDLUN[N]DQGWKHVXSHUSRVL WLRQSULQFLSOHDSSOLHVWKHVRXQGSUHVVXUHLVREWDLQHGDVWKHLQWHJUDO S [ \  ]

U  F N S 

YÖ N [  N ]

f

³

 

 [

N N N

f

 ]

H MN[ [ H MN] ] H

 M N  N [  N ]

GN [ GN ] 

F 

7KLVH[SUHVVLRQFDQEHXVHGWRGHWHUPLQHWKHUDGLDWLRQIURPDUELWUDU\SODQH UDGLDWRUV LQ D ULJLG EDIIOH ,W WKXV UHDOL]HV DQ DOWHUQDWLYH WR (TV   RU F  7KH LGHQWLW\ RI WKH H[SUHVVLRQV FDQ EH SURYHQ ZLWK VRPH PDWKH PDWLFDO H[HUFLVH ,Q WKLV FRQWH[W KRZHYHU LW LV VXIILFLHQW WR GHPRQVWUDWH WKDWWKH\HVWDEOLVKWKHVDPHLQWHQVLW\SDWWHUQLQWKHIDUILHOG7RGRWKLVWKH SRZHUWUDQVPLWWHGWKURXJKDVXUIDFHDW\ \SLVFDOFXODWHGDV :

º  ªf 5H « ³ S [ \ S  ] Y [ [ \ S  ] G[G] »  ¬ f ¼ ª§ YÖ N  N ( · U F N [ ] GN [ GN ] ¸ ³ YÖ N [  N ] ( GN [ GN ] 5H ³ «¨ ³     ¸ S «¨ N   N [  N ] ¹ ¬©



º

»» G[G]

D 

¼

+HUHLQ ( LV DQ DEEUHYLDWLRQ IRU WKH H[SRQHQWLDO IXQFWLRQV LQ F  $OO LQWHJUDOV H[WHQG IURP f WR f $V VKRZQ LQ 6HFW  WKH LQWHJUDO SURGXFWUHGXFHVWRDVLQJOHGRXEOHLQWHJUDO :

ª f YÖ N  N YÖ N  N º U  F N [ ] [ ] «³ GN [ GN ] » 5H  S «¬ f »¼ N  N [  N ]

U  F N S 

NF

³

 NF

YÖ N [  N ]



N  N [  N ]

GN [ GN ] 

E 

3ODQH%DIIOHG5DGLDWRUV

7KHLQWHJUDOOLPLWVDUHJLYHQE\WKHFRQGLWLRQ NF N [  N ] N VLQFHRQO\ LQ WKDW UDQJH LV WKH UDGLFDO UHDO YDOXHG %\ PHDQV RI WKH VXEVWLWXWLRQV N[ N VLQ - FRV M DQG N]  N VLQ - FRV M WKH LQWHJUDQG LV VLPSOLILHG WKURXJK GN [ GN ] N  N [  N ]

N VLQ - FRV - G- GM N FRV -

N VLQ - G- GM 

7KLVPHDQVWKDWWKHUDGLDWHGSRZHULVREWDLQHGDV :

 U  F N S   S YÖ N VLQ - VLQ M  N VLQ - FRV M VLQ - G- GM   ³ ³ S  

F 

7KHUDGLDWHGSRZHULQ F FDQQRZEHFRPSDUHGZLWKWKDWWUDQVPLWWHG WRWKHIDUILHOGJLYHQE\ F $IWHUFKDQJLQJWRSRODUFRRUGLQDWHVWKH IDUILHOGH[SUHVVLRQUHDGV :S

  U  F

S   S

³ ³ S 5 -  M 



5  VLQ - G- GM



G 

 Z  U  S   S Y, -  M VLQ - G- GM   ³ ³ S F  

,QWKHVHH[SUHVVLRQVWKHIDFWRU(5LVVHWHTXDOWRXQLW\VXFKWKDW Y, -  M

³Y [  ] H T

T

MN VLQ - VLQ M [T

H

MN VLQ - FRV M ]T

G[T G]T 

H 

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x 7R UHDOL]H D UDGLDWRU ZLWK D SUHVFULEHG GLUHFWLYLW\ D ZDYHQXPEHU VSHFWUXPIRUWKHUDGLDWRUYHORFLW\PXVWEHIRUPHGWKDWLVSUHVFULEHG LQ WKH UDQJH N [  N ]  N  EXW FDQ EH DUELWUDU\ LQ WKH UDQJH RI ODUJHU ZDYHQXPEHUV 7KH YHORFLW\ GLVWULEXWLRQ DVVRFLDWHG ZLWK WKH SUH VFULEHGGLUHFWLYLW\FDQEHDFKLHYHGLQPDQ\ZD\V,QSULQFLSOHLWLV SRVVLEOH WR UHDOL]H D YHU\ VKDUS OREH ZLWK IRU H[DPSOH D UDGLDWRU

6RXQG5DGLDWLRQIURP6WUXFWXUHV

x

x

WKDWLVYHU\VPDOOLQFRPSDULVRQZLWKWKHZDYHOHQJWKRIWKHPHGLXP 7KHHIILFLHQF\RIVXFKDUDGLDWRUKRZHYHULVUDWKHUVPDOO$OVRWKH UDGLDWRU JHQHUDWHV DQ H[WUHPHO\ VWURQJ QHDUILHOG ZKLFK LQ FDVH RI VRPHVPDOOLQKRPRJHQHLW\WRWDOO\DOWHUVWKHGLUHFWLYLW\7KHGHVLJQ RIUDGLDWRUVZLWKSUHVFULEHGGLUHFWLYLW\DQGORZHQHUJ\ FRQVXPSWLRQ LVWUHDWHGLQ>@ &RQYHUVHO\WRWKHDERYHFRQFOXVLRQLWIROORZVWKDWWKHYHORFLW\ GLV WULEXWLRQ RI D UDGLDWRU FDQQRW EH IXOO\ UHFRQVWUXFWHG IURP PHDVXUH PHQWVLQWKHIDUILHOG7KLVLVVRVLQFHDQDUELWUDU\QHDUILHOGFDQEH VXSHULPSRVHGRQWKHPHDVXUHGORQJZDYHVLQWKHIDUILHOG %\DSSO\LQJ3DUVHYDO¶VWKHRUHP  S 





Y 6

S 

³

f



Y [ ] G[G]

³ YÖ N  N [



]

GN [ GN ] 

f

6

WKHUDGLDWLRQHIILFLHQF\GHILQHGLQ D FDQEHZULWWHQDV NF

³ YÖ N  N [

V

GN [ GN ]

 

N  N [  N ]

 NF

f

³

N



]

I 





YÖ [ ] GN [ GN ]

f

7KH  N

x x

QXPHUDWRU

LQWHJUDWHG

RYHU

WKH

UDQJH

N  7KH LPDJLQDU\ SDUW RI WKH UDGLDWLRQ

LPSHGDQFHFRUUHVSRQGLQJWRWKHDSSDUHQWPDVVFDQEHDVVHVVHGIURP D VLPLODU H[SUHVVLRQ ZKHUH WKH QXPHUDWRU LV LQWHJUDWHG RYHU NF d N [  N ] d f  ,QWKHGLUHFWLRQSHUSHQGLFXODUWRWKHUDGLDWRUVXUIDFHWKHVRXQGSUHV VXUHLVJLYHQE\WKHPHDQVTXDUHYHORFLW\RIWKHUDGLDWRU )RU VPDOO VRXUFHV FRPSDUHG ZLWK WKH ZDYHOHQJWK WKH H[SRQHQWLDO IXQFWLRQVLQ E FDQEHH[SDQGHGVXFKWKDW YÖ N [  N ]

³ Y [ ] G[G]  MN ' V



x

LV

 NF d N [  N ] d NF

[

[

 MN ] ']

N [ N '[[  N [ N ] '[]  ] ']]  

J 

ZKHUH '[ ³ [YV [ ] G[G]  '[[ ³ [  YV [ ] G[G] HWFFRUUHVSRQGVWR WKHGLSROHDQGTXDGUXSROHPRPHQWVIRUWKHUDGLDWRU :KHQ YV [ ]  LV FRPSRVHG RI VWDQGLQJ ZDYHV WKHQ YÖ N [ N ]  LV D V\PPHWULFIXQFWLRQLQN[DQGN]$FFRUGLQJO\DOVRWKHGLUHFWLYLW\ LV V\PPHWULF 7R REWDLQ D QRQV\PPHWULF GLUHFWLYLW\ YV [T ]T  PXVW FRQWDLQVDSRUWLRQRISURSDJDWLQJZDYHV

5DGLDWLRQIURP%HQGLQJ:DYHV

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

­° $ ª H  MN% [  UH MN% [  UM H N% [ º  [ d  ¬ ¼  ®  [t °¯ 

D 

LQDFFRUGDQFHZLWK6HFW$LVWKHZDYHDPSOLWXGHN%LVWKHEHQGLQJ ZDYHQXPEHUUDQGUM DUHWKHUHIOHFWLRQFRHIILFLHQWVRIWKHUHIOHFWHGSURSD JDWLQJDQGHYDQHVFHQWZDYHVUHVSHFWLYHO\

)LJ  %HQGLQJ ZDYH UDGLDWLRQ IURP D VHPLLQILQLWH SODWH D  YLEUDWLRQ ILHOG E UDGLDWLRQIURPORQJDQG F VKRUWEHQGLQJZDYHV

6RXQG5DGLDWLRQIURP6WUXFWXUHV

8SRQLQVHUWLQJ D LQ E WKHZDYHQXPEHUVSHFWUXP LVREWDLQHG DV UM ª M º U M  $« » N %  N [ N %  MN [ ¼ ¬ N%  N [

YÖ N [

E 

:LWKWKHZDYHQXPEHUVSHFWUXPVXEVWLWXWHGLQ E WKHUDGLDWHGSRZHU EHFRPHV N

:c

U  F N  7  GN [  $ ³ S N  N [ N

F 



ZKHUH 7

N % M  MU  UM  N % N [ M    U  MU  N [   U  UM

N

 %

 N [ N %  MN [



2ZLQJWRWKHIDFWWKDWKHUHDRQHGLPHQVLRQDOWUDQVIRUPLVFRQVLGHUHGWKH GHQRPLQDWRU LQ F  LV DOWHUHG WR S $FFRUGLQJO\ : c  LV WKH UDGLDWHG SRZHUSHUXQLWZLGWK 7KHLQWHJUDOLVFULWLFDOO\GHSHQGHQWRQZKHWKHUWKHSROHVDWN[ r N%DUH ORFDWHG LQVLGH RU RXWVLGH WKH LQWHJUDWLRQ UDQJH )RU ORQJ ZDYHOHQJWKVIRU ZKLFKN% NWKHLQWHJUDOLVSUDFWLFDOO\VROHO\GHWHUPLQHGE\WKHEH KDYLRXURIWKHLQWHJUDQGDURXQGN[ r N$FDOFXODWLRQZRXOGIXUQLVKDQ LQILQLWH SRZHU KRZHYHU VLQFH WKH UDGLDWLQJ VXUIDFH LV DVVXPHG VHPL LQILQLWH EXW WKH UDGLDWLRQ HIILFLHQF\ FDQ EH PHDQLQJIXOO\ GHWHUPLQHG E\ PHDQV RI I  7KHUHE\ WKH UDGLFDO N  N [ LV DSSUR[LPDWHG E\ N  N %  DQG EURXJKW RXWVLGH WKH LQWHJUDO VXFK WKDW WKH OLPLWV FDQ EH H[ WHQGHGWRLQILQLWH7KLVOHDGVWR V

N  

N  N %

 N %  N

G 

)RUVKRUWZDYHOHQJWKVZKHUHN%!N F FRQWDLQVQRSROHVDQGWKH LQWHJUDO \LHOGV D ILQLWH UHVXOW DOVR IRU LQILQLWH VXUIDFHV $ IHZ H[DPSOHV FRPSXWHG E\ PHDQV RI F  DUH SUHVHQWHG LQ )LJ  IRU ZKLFK WKH IROORZLQJUHPDUNVFDQEHPDGH x %HORZ WKH FULWLFDO IUHTXHQF\ WKH UDGLDWLRQ GHSHQGV VWURQJO\ RQ WKH ERXQGDU\FRQGLWLRQDW[  x :KHQWKHIDFWRUWR N % FDQEHPDGHWRYDQLVKWKURXJKVXLWDEOHFKRLFHV RIUDQGUMWKHUDGLDWLRQLVRIGLSROHW\SHRUZKHQDOVRWKHIDFWRUWR N%N[LVPDGHWRYDQLVKTXDGUXSROHW\SH FXUYHVDQG 

5DGLDWLRQIURP%HQGLQJ:DYHV

x )RUDYDQLVKLQJYHORFLW\DW[ YV   DGLSROHW\SHRIQHDUILHOG UDGLDWLRQFDQEHHVWDEOLVKHGWKURXJKDVXLWDEOHFKRLFHRIWKHUHIOHFWLRQ FRHIILFLHQWUM FXUYH 7KLVOHDGVWROHVVUDGLDWLRQWKDQWKDWIRUDVLP SO\VXSSRUWHGRUFODPSHGERXQGDU\ x 6LQFH E\ RWKHUZLVH XQDOWHUHG FRQGLWLRQV D FKDQJH RI WKH QHDUILHOG LHDFKDQJHRIUMDIIHFWVWKHUDGLDWHGSRZHULWFDQEHFRQFOXGHGWKDW WKHRULJLQRIWKHUDGLDWLRQOLHVFORVHWRWKHGLVFRQWLQXLW\7KLVLVDOVR FRQILUPHGE\LQWHQVLW\PHDVXUHPHQWV x 7KH IDFW WKDW WKH VRXQG UDGLDWLRQ FDQ EH UHGXFHG E\ WKH UHIOHFWHG SURSDJDWLQJDQGHYDQHVFHQWZDYHVIRUIUHTXHQFLHVEHORZWKHFULWLFDO IUHTXHQF\ GHPRQVWUDWHVWKHSRVVLELOLW\WR DFWLYHO\ UHGXFHWKHUDGLD WLRQE\PHDQVRIIRUFHVDQGPRPHQWVDSSOLHGDWWKHERXQGDULHV x 7KHVRXQGUDGLDWLRQPHFKDQLVPIRUWKHVHPLLQILQLWHSODWHEHORZWKH FULWLFDOIUHTXHQF\FDQEHLQWHUSUHWHGDVDQRQUDGLDWLQJSODWHILHOGGXH WRWKHK\GURG\QDPLFVKRUWFLUFXLWEXWDUDGLDWLQJHGJHVLQFHWKHVKRUW FLUFXLWWKHUHLVLQFRPSOHWH7KHVWUHQJWKRIWKHHGJHUDGLDWLRQGHSHQGV RQWKHÄFRPSOHWHQHVV³RIWKHVKRUWFLUFXLWDQGWKXVRQWKHGHWDLOVRI WKHERXQGDU\FRQGLWLRQV

)LJ  ,QIOXHQFH RI WKH ERXQGDU\ FRQGLWLRQV RQ WKH UDGLDWLRQ IURP IOH[XUDO ZDYHVEHORZWKHFULWLFDOIUHTXHQF\

6RXQG5DGLDWLRQIURP6WUXFWXUHV

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c  Z  Pcc 

O%

WKHFULWLFDOIUHTXHQF\LVREWDLQHGIURP N% NRUO% O FIFLH IF

F S

Pcc  %c

D 

,ILQVWHDGRIWKHUDWLR Pcc  %c WKHORQJLWXGLQDOZDYHVSHHGIRUWKHPDWHULDO F/DQGWKHSODWHWKLFNQHVVKDUHLQWURGXFHGWKHH[SUHVVLRQIRUWKHFULWLFDO IUHTXHQF\FDQEHUHZULWWHQDV IF

F  F/ K

E 

$VLPSOHPHWKRGIRUPHDVXULQJWKHFULWLFDOIUHTXHQF\FRQVLVWVRIDVVHVVLQJ WKHUDWLRRIPDVVWREHQGLQJVWLIIQHVVIURPDVWDWLFDOH[SHULPHQW7KHFHQ WUHGLVSODFHPHQW [ RIDVLPSO\VXSSRUWHGEHDPRIOHQJWKOGXHWRLWVPDVV SHUXQLWOHQJWK Pc LV[  PcO  J   % ZKHUHJLVWKHJUDYLWDWLRQDODFFHO HUDWLRQ7KLVPHDQVWKDWWKHFULWLFDOIUHTXHQF\FDQEHHVWLPDWHGIURP IF

F S O 

[  J

F 

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

5DGLDWLRQIURP%HQGLQJ:DYHV

GLDWLRQLQWRZDWHUFDQQRWEHPDGHZLWK F VLQFHLQSDUWLFXODUIRUWKLQ SODWHVWKHIOXLGORDGLQJVKLIWVWKHFULWLFDOIUHTXHQF\\HWKLJKHU7KHIOXLG ORDGLQJDFWVDVDIUHTXHQF\GHSHQGHQWPDVVOD\HUDQGWKHFULWLFDOIUHTXHQF\ LV PRVWO\ VR KLJK WKDW WKH RUGLQDU\ EHQGLQJ WKHRU\ LV LQYDOLG +HQFH WKH ZDYHQXPEHUPXVWEHGHWHUPLQHGE\PHDQVRI D 

)LJ&ULWLFDOIUHTXHQFLHVIRUUDGLDWLRQLQWRDLU

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

6RXQG5DGLDWLRQIURP6WUXFWXUHV

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

7KH VLWXDWLRQ LV VOLJKWO\ PRUH LQYROYHG IRU SODWHV WKDW DUH FRUUXJDWHG RQO\LQRQHGLUHFWLRQ$VZDVVKRZQLQ6HFWRQRUWKRWURSLFSODWHV WKHEHQGLQJZDYHQXPEHUDQGKHQFHWKHZDYHOHQJWKYDULHVZLWKWKHGLUHF WLRQRISURSDJDWLRQ,QWKHVWLIIGLUHFWLRQWKHZDYHOHQJWKLVREWDLQHGDV S  %[c  Pcc  Z 

O%[

7KHDVVRFLDWHGFULWLFDOIUHTXHQF\OLHVDW I F[

F S

Pcc  %[c

G 

:LWK %[c UHSUHVHQWLQJWKHPD[LPXP VWLIIQHVVIF[FRQVWLWXWHVWKHORZHUOLPLW DQGWKHFULWLFDOIUHTXHQFLHVDUHKLJKHUIRUZDYHVLQRWKHUGLUHFWLRQV ,Q WKH RUWKRJRQDO GLUHFWLRQ LH DFURVV WKH FRUUXJDWLRQ WKH FULWLFDO IUH TXHQF\LVJLYHQE\ I F]

F S

Pcc  %]c

H 

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

5DGLDWLRQIURP%HQGLQJ:DYHV

0RGDO5DGLDWLRQ :LWKWKHQRUPDOFRPSRQHQWRIWKHUDGLDWRUYHORFLW\ H[SUHVVHGDVDVXP RI HLJHQIXQFWLRQVRUPRGHVFI6HFW Y [ ]

¦ Y M [ ]  Q

 

Q

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

QS [ Q S[ ­   d [ d O[  Y  VLQ  °Y VLQ O[ O[  ® °    f  [   O[  [  f ¯

D 

8SRQIRUPLQJWKHZDYHQXPEHUVSHFWUXPDFFRUGLQJWR E DQGGHYHO RSLQJWKHSRZHUUDGLDWHGSHUXQLWZLGWKLQDQDORJ\ZLWK F RQHILQGV WKDW :c

N

U  F S

³

 N

N  

N N



 [

Y $ (  Y  $ ( GN [ 

E 

ZKHUH $Q  MQQ S O[ VLQ ¬ª N [ O[  QQ S  ¼º  N [ O[  QQS  DQG(Q  H  M N O  QQ S   IRU Q  > @ 7KLVFDQEHGHYHORSHGLQWR [ [



N N $ U  F  UF Y ³   GN [    Y    S S N  N [ N 

:c





N

³

 N

N $



N  N [ 

GN [

N UF Q Q · N $ $ §    Y Y  FRV ¨ M  M     S ¸ ³    GN [  S  © ¹  N N  N [ 

F 



ZKLFKFDQEHDEEUHYLDWHG : c :c  :c  :c  G  c c ,QWKLVH[SUHVVLRQ : DQG : DUHWKHUDGLDWHGSRZHUVIURPWKHLQGLYLGXDO PRGHV :c LVDPRGDOLQWHUDFWLRQWHUP,WVPDJQLWXGHDQGVLJQYDU\ZLWK

WKHSKDVHVKLIWEHWZHHQWKHPRGHV

6RXQG5DGLDWLRQIURP6WUXFWXUHV

7KHSULQFLSDOEHKDYLRXURI_$Q_LVSORWWHGLQ)LJ(VVHQWLDOO\WKHVH DUHIXQFWLRQVLQWKHIRUP VLQ[ [ZLWKPD[LPD DW N[ O[ r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c |

J 

U  F O[ 

¦ YQ



 V Q |  

D 

:KHQN O[ | QQ SWKHLQWHJUDWLRQEHFRPHVPRUHLQWULFDWH8VHIXO DSSUR[LPDWLRQVDUH :Qc |

U  F O[ 

N O [

S

YQ



 V|

 N O [  S

 QQ  

E 

)LJ3ULQFLSDOEHKDYLRXURI_$Q_

7KHGHYHORSPHQWRIDSSUR[LPDWLRQVIRUWKHLQWHUDFWLRQWHUP LVGLIILFXOW VLQFH WKH LQWHJUDQG FRQWDLQV WZR VLQXVRLGV ZLWK GLIIHULQJ DUJXPHQWV 7KHUHIRUHDUHLQFOXGHGLQ)LJVRPHH[DPSOHVRIWKHLQIOXHQFHRIWKLV WHUP 7KHUHE\ WKH QHW UDGLDWHG SRZHU : c  LV FRPSDUHG ZLWK WKH VXP RI

5DGLDWLRQIURP%HQGLQJ:DYHV

SRZHUVRIWKHWZRPRGHV :c  :c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ÄDQWLPRGHV³ZLOORI IHUVRPHUDGLDWLRQUHGXFWLRQ

)LJ 5DGLDWLRQIURP DSODWHZLWK WKHYLEUDWLRQJLYHQE\ D D DQGE UD  GLDWLRQIURPDVLQJOHPRGH :%c U  F O YQ   F ,QIOXHQFHRIWKHSKDVHRIDVHF RQGPRGHRIHTXDODPSOLWXGHDVWKHUHIHUHQFHPRGHQ DQGQ G 5DGLD WLRQUHGXFWLRQGXHWRDVHFRQGPRGHLQDQWLSKDVH ' /  ORJ ª¬: c  :c  :c º¼ 

6RXQG5DGLDWLRQIURP6WUXFWXUHV

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Ö N [  N ]

MZ SÖ N [  N ] % N [  N ]  N %

SÖ N [  N ] @)DK\)-6RXQGLQWHQVLW\ QGHG ( )16SRQ/RQGRQ >@&ULJKWRQ '*'RZOLQJ $3)IRZFV:LOOLDPV-(+HFNO0DQG/HSSLQJ WRQ)*0RGHUQPHWKRGVLQDQDO\WLFDODFRXVWLFV†6SULQJHU 9HUODJ/RQGRQ >@ *|VHOH .  6FKDOODEVWUDKOXQJ YRQ 3ODWWHQGLH]X%LHJHVFKZLQJXQJHQ DQJHUHJWVLQG$FXVWLFDS >@+HPSHO:DQG6LHGHO76WDWLVWLVFKH(UKHEXQJEHU'LHVHOPRWRUHQ JHUlXVFKH0RWRUWHFKQLVFKH=HLWVFKULIWS >@0RUVH309LEUDWLRQDQGVRXQG&K9,,0F*UDZ+LOO1HZ@ .XUW]H * DQG %ROW 5+  2Q WKH LQWHUDFWLRQ EHWZHHQ SODWH EHQGLQJ ZDYHVDQGWKHLUUDGLDWLRQORDG$FXVWLFDS

6RXQG5DGLDWLRQIURP6WUXFWXUHV >@ $EUDPRZLW] 0 DQG 6WHJXQ ,$  +DQGERRN RI PDWKHPDWLFDO IXQF WLRQV'RYHU3XEOLFDWLRQV1HZ@ 5D\OHLJK-:67KHWKHRU\RIVRXQG9RO†'RYHU3XEOLFD WLRQV1HZ@ +HFNO 0  6FKDOODEVWUDKOXQJ YRQ 3ODWWHQ EHL SXQNWI|UPLJHU $QUH JXQJ$FXVWLFDS >@ 0DLGDQLN *  5HVSRQVH RI ULEEHG SDQHOV WR UHYHUEHUDQW DFRXVWLF ILHOGV-RXUQDORIWKH$FRXVWLFDO6RFLHW\RI$PHULFDS >@ .OHPHQ]0Y(VWRUII2DQG+HFNO06FKDOODEVWUDKOXQJHLQHV.D WDO\VDWRUVPLWGHU%(0XQGGHPÄ5D\OHLJK9HUIDKUHQ³3URF'$*$6DDU EUFNHQYRO,,S >@&ROWRQ&DQG.UHVV5,QWHJUDO HTXDWLRQPHWKRGVLQ VFDWWHULQJWKH RU\-RKQ:LOH\ 6RQV1HZ@6H\EHUW$)6RHQRUNR%5L]]R)-DQG6KLSS\'-$QDGYDQFHG FRPSXWDWLRQDOPHWKRGIRUUDGLDWLRQDQGVFDWWHULQJRIDFRXVWLFZDYHVLQWKUHH GLPHQVLRQV-RXUQDORIWKH$FRXVWLFDO6RFLHW\RI$PHULFDS >@&UHPHU/DQG:DQJ0'LH6\QWKHVHHLQHVYRQHLQHP EHOLHELJHQ .|USHULQ/XIWHU]HXJWHQ)HOGHVDXV.XJHOVFKDOOIHOGHUQXQGGHUHQ5HDOLVLH UXQJLQ'XUFKUHFKQXQJXQG([SHULPHQW$FXVWLFDS >@%REURYQLWVNLL@ +HFNO 0  %HPHUNXQJ ]XU %HUHFKQXQJ GHU 6FKDOODEVWUDKOXQJ QDFK GHU0HWKRGHGHU.XJHOIHOGV\QWKHVH$FXVWLFDS >@2FKPDQQ0'LH0XOWLSROVWUDKOHUV\QWKHVH±HLQHIIHNWLYHV9HUIDK UHQ]XU%HUHFKQXQJGHU6FKDOODEVWUDKOXQJYRQVFKZLQJHQGHQ6WUXNWXUHQEH OLHELJHU2EHUIOlFKHQJHVWDOW$FXVWLFDS >@&UHPHU/7KHRULHGHU6FKDOOGlPPXQJGQQHU:lQGHEHL VFKUlJHP 6FKDOOHLQIDOO$NXVWLVFKH=HLWVFKULIWS >@ +HFNO 0  8QWHUVXFKXQJHQ EHU GLH /XIWVFKDOOGlPPXQJ YRQ 'RS SHOZlQGHQPLW6FKDOOEUFNHQ3URFUG,QWHUQDWLRQDO&RQJUHVVRQ$FRXVWLFV 6WXWWJDUWS >@5D\OHLJK-:67KHWKHRU\RIVRXQG9RO †'RYHU3XE OLFDWLRQV1HZ@,62',6$NXVWLN ±0HVVXQJGHU6FKDOOGlPPXQJLQ*HElXGHQXQG YRQ%DXWHLOHQ%HXWK9HUWULHE%HUOLQ >@ *|VHOH./XIWVFKDOOGlPPXQJ3URFUG,QWHUQDWLRQDO&RQJUHVVRQ $FRXVWLFV 6WXWWJDUW  S  (OVHYLHU 3XEOLVKLQJ &RPSDQ\ $PVWHU GDP >@ +HFNO0DQG'RQQHU86FKDOOGlPPXQJGLFNHU:lQGH5XQGIXQN WHFKQLVFKH0LWWHLOXQJHQS >@ /\RQ5+DQGGH-RQJ5*7KHRU\DQGDSSOLFDWLRQRIVWDWLVWLFDOHQ HUJ\DQDO\VLV QGHG %XWWHUZRUWK+HLQHPDQQ%RVWRQ0$ >@&URFNHU0-DQG3ULFH$-6RXQGWUDQVPLVVLRQXVLQJVWDWLVWLFDO HQ HUJ\DQDO\VLV-RXUQDORI6RXQGDQG9LEUDWLRQS >@:|KOH:6WDWLVWLVFKH(QHUJLHDQDO\VHGHU6FKDOOWUDQVPLVVLRQ† 7DVFKHQEXFK$NXVWLN )DVROG.UDDN6FKLUPHUHGV 9(%9HUODJ%HUOLQ >@*|VHOH.DQG6FKOH:6FKDOO:lUPH)HXFKWH$XIODJH† %DXYHUODJ:LHVEDGHQ >@ *|VHOH .=XU%HUHFKQXQJGHU/XIWVFKDOOGlPPXQJYRQGRSSHOVFKD OLJHQ%DXWHLOHQ RKQH9HUELQGXQJGHU6FKDOHQ $FXVWLFDS

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH %RUQH6RXQG

 0HFKDQLFDO 0HDVXUHPHQW 0HWKRGV  5HJLVWUDWLRQ RI 0RWLRQ 0RVW³VWUXFWXUHERUQHVRXQGVHQVRUV´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

)LJ$UUDQJHPHQWIRUGLUHFWREVHUYDWLRQRIWHVWREMHFWGLVSODFHPHQW

0HFKDQLFDO0HDVXUHPHQW0HWKRGV

HGJHRIDPLUURURIOHQJWK O LVGLVSODFHGE\DQDPRXQW[ZKLOHWKHRWKHU HGJH SLYRWV DERXW D IL[ D[LV DV GHSLFWHG LQ )LJ  WKH PLUURU URWDWHV DQ DQJOH[O$QLPSLQJLQJOLJKWEHDPDFFRUGLQJO\LVGHIOHFWHGWZLFHWKDWDQ JOH DQG D VSRW RI WKH OLJKW UHIOHFWHG DW D VFUHHQ DW D GLVWDQFH /  LV GLV SODFHGE\ '

[ / O 

 

UHODWLYH WKH HTXLOLEULXP UHIOHFWLRQ SRLQW 6LQFH DQ DPSOLILFDWLRQ RI /O FDQEHUHDGLO\DFKLHYHGDWHVWREMHFWGLVSODFHPHQWRIPP QRWRQO\FDQEHPHDVXUHGEXWDOVRWKHWLPHKLVWRU\FDQEHUHJLVWHUHG

)LJ2EVHUYDWLRQRIGLVSODFHPHQWE\PHDQVRIDSLYRWHGPLUURUD PLUURUDW WDFKHGWRDOHYHUDQGE PDVVOHYHUVXEVWLWXWLRQ

7KH URWDWLQJ PLUURU VHQVRU GLIIHUV LQ RQHLPSRUWDQWDVSHFWIURP WKHGL UHFWREVHUYDWLRQ:KHUHDVDGLUHFWREVHUYDWLRQOHDGVWRQRUHDFWLRQIRUFHRQ WKHWHVWREMHFWWKHPLUURULQHUWLDUHVXOWVLQDIRUFH )

 § Z P O ·  ¨ ¸[ O ©  ¹



Z P [  

 

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

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

SRUWLRQDOWRWKHSDUWLFOHYHORFLWLHVDQGWKHSURGXFWVRIIRUFHVDQGYHORFLWLHV FRQVWLWXWHWKHFRUUHVSRQGLQJSRZHUTXDQWLWLHV 7KHPRVWDSSURSULDWHZD\WRDVVHVVWKHUHDFWLRQRIDYLEUDWLRQVHQVRURQ DPHDVXUHPHQWREMHFWLVWRFRPSDUHWKHPRELOLW\RIWKHVHQVRU@WKDW 4Z

 . [[[  4Z

 . [\K 

 

ZKHUH4ZLVWKHWUDQVGXFHUFKDUJHHVWDEOLVKHG[DQGKDUHWKHFRPSUHVVLRQ DQGFURVVVHFWLRQDOFRQWUDFWLRQUHVSHFWLYHO\ZLWKWKHDVVRFLDWHGWUDQVGXFHU FRQVWDQWV .[[ DQG .[\ GHSHQGHQW RQ WKH PDWHULDO SDWFK VL]H SDWFK IRUP DQGDOLJQPHQWZLWKWKHFU\VWDOD[LV

)LJ%HKDYLRXURISLH]RHOHFWULFPDWHULDOV>@D XQGHIRUPHGVWDWHE GH IRUPDWLRQLQWKHGLUHFWLRQRI D[LDO DQGF SHUSHQGLFXODUWR WKHFXUUHQW ODWHUDO 

,Q WKH IROORZLQJ RQO\ RQH GLUHFWLRQ RI PRWLRQ ZLOO EH FRQVLGHUHG DQG WKHUHIRUHDVLQJOHWUDQVGXFHUFRQVWDQW. LVVXIILFLHQW)RUWKHDQDO\VLVPXVW EH FRQVLGHUHG PRUHRYHU WKDW WKH WHPSRUDO FKDQJH LQ FKDUJH LV SURSRU WLRQDOWRWKHFXUUHQWL G4GWDQGWKHWHPSRUDOFKDQJHLQGLVSODFHPHQWLV SURSRUWLRQDO WR WKH YHORFLW\ Y  G[GW8SRQLQWURGXFLQJWKHSKDVRUQRWD WLRQWKHILUVWIXQGDPHQWDOUHODWLRQ LZ

 . YZ

D 

LVREWDLQHGIRUSLH]RHOHFWULFWUDQVGXFHUV7KHUHFLSURFDOSLH]RHOHFWULFHI IHFWLPSOLHVWKDWDQDSSOLHGYROWDJHJLYHVULVHWRDSURSRUWLRQDOH[SDQVLRQ RIWKHPDWHULDOOHDGLQJWRDPHFKDQLFDOWUDQVGXFHUIRUFH )Z

.8 Z 

E 

7KH IDFW WKDW WKH VDPH FRQVWDQW . DSSHDUV LQ ERWK D  DQG E  LV DJDLQDFRQVHTXHQFHRIHQHUJ\FRQVHUYDWLRQ7KHSURRISURFHHGVH[DFWO\DV IRUHOHFWURG\QDPLFWUDQVGXFHUVLQ F WR H 

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

%\ FRPELQLQJ WKH ODVW WZR HTXDWLRQV WKH IRXUSROH GHVFULSWLRQ LV RE WDLQHGDV ­) Z ½ ® ¾ ¯ YZ ¿

ª  «  . ¬

. º ­ LZ ½ ® ¾  »¼ ¯8 Z ¿

 

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­ LZ ½ ® ¾ ¯8 Z ¿

ª  = (L º ­ L ½ ® ¾  «  »¼ ¯8 ¿ ¬

 

+HUHLQ=(LHVVHQWLDOO\UHSUHVHQWVWKHUHODWLYHO\ODUJHLPSHGDQFHRIWKHFD SDFLWDQFHUHDOL]HGE\WKHWUDQVGXFHU = (L

 MZ & 

D 

7KHUHE\WKHPDWULFHVLQWKHIRXUSROHFKDLQLQ)LJDUHNQRZQVLQFH WKH PHFKDQLFDO DQG HOHFWULFDO ORDGV DJDLQ FDQ EH WDNHQ IURP G  DQG D DQG E UHVSHFWLYHO\%\FDUU\LQJRXWWKHPDWUL[PXOWLSOLFDWLRQ WKHPHFKDQLFDOSDUWLVIRXQGWREHGHVFULEHGE\ ­) . ½ ® ¾ ¯ Y. ¿

ª = 06  = 00 « = 00 « « «  = 00 ¬

V = 06  = 00 º  = 06 » ) MZ = 00 » ­® Z ½¾  » ¯ YZ ¿ V  » MZ = 00 ¼

E 

DQGWKHHOHFWULFDOSDUWE\ ­) Z ½ ® ¾ ¯ YZ ¿

ª  .= (. « «   §  = (. · ¸ « . ©¨ = (L ¹ ¬

. º » ­ L. ½  » ® ¾  ¯8 . ¿ .= (L »¼

F 

,QWKHPHFKDQLFDOSDUW=06 =0L=0. LVWKHVXPRIWKHPHFKDQLFDOVHQ VRU LPSHGDQFH HVVHQWLDOO\JLYHQE\WKHPDVVRIWKHKRXVLQJDQGWKHLP

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

SHGDQFH RI WKH WHVW REMHFW ,Q FRQWUDVW WR WKH HOHFWURG\QDPLF WUDQVGXFHU =00KHUHUHSUHVHQWVWKHLPSHGDQFHRIWKHVHLVPLFPDVV

)LJ(OHFWULFDODQGPHFKDQLFDOLPSHGDQFHV IRUDSLH]RHOHFWULFWUDQVGXFHUD  &RQILJXUDWLRQRXWOLQHDQGE JHQHUDOL]DWLRQDVFDVFDGHGIRXUSROHV

8SRQFRPSDULQJ)LJVDQGWKHIROORZLQJFRUUHVSRQGHQFHVFDQ EHPDGHFI6HFW (OHFWURG\QDPLFWUDQVGXFHU 9ROWDJHVRXUFH 6HULHVFRQQHFWHGLQQHULPSHGDQFH 0DJQHWLPSHGDQFH=00

3LH]RHOHFWULFWUDQVGXFHU &XUUHQWVRXUFH 3DUDOOHOFRQQHFWHGLQQHULPSHGDQFH 6HLVPLFPDVVLPSHGDQFH=00

= 00 | MZ P00 6RIWVXVSHQVLRQV 0RYLQJFRLOLPSHGDQFH=0L

G 

6WLIISLH]RFU\VWDOV 7UDQVGXFHUKRXVLQJLPSHGDQFH=0L

= 0L | MZ P0L

H 

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

$ PXOWLSOLFDWLRQ RI E  DQG F  \LHOGV WKH FRPSOHWH PDWUL[ DJDLQLQWKHIRUPRI E $OVRLQWKLVFDVHWKHLQGLYLGXDOGHWHUPLQDQWV HTXDO“,QDQDORJ\ZLWK I WKHVHQVRUVHQVLWLYLW\LVREWDLQHGDV 8 Y.

8. 

= (. $

. §  · .  ·§ V  ¨ ¸ ¨  ¸ MZ = 00 ¹ = 00 © = (. = (L ¹ ©



D 

)RU WKH VXEVHTXHQW GLVFXVVLRQ LW LV VXLWDEOH WR OHW =00  MZP00 =(L MZ&=(. 5ZKLFKFRUUHVSRQGWRWKHXVXDOH[SHULHQFHZLWKSLH]R HOHFWULF WUDQVGXFHUV DQGWRFRQYHUW IURPYHORFLW\WRDFFHOHUDWLRQ $FFRUG LQJO\ D EHFRPHV 8 D.

8. 

P00 .   V& §  · § Z  P00 · .  ¸ ¨  ¸ ¨  MZ &5 ¹© V ¹ V& ©

E 

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

8. 

   |  . = 00  = 06  §   ·§ V = 00  = 06 · .  ¨  ¸¨ = 06  ¸ = 00 . © = (. = (L ¹ © MZ = 00 ¹

F 

7KHUDWKHUVLPSOHDSSUR[LPDWLRQLQWKHODVWSDUWRI F LVDGPLVVLEOHIRU WKHIROORZLQJFRQGLWLRQV x )RUIRUFHWUDQVGXFHUVRQHLVFRQFHUQHGZLWKPDNLQJWKHPHFKDQLFDO HOHPHQWEHWZHHQWKHIRUFHWREHUHJLVWHUHGDQGWKHSLH]RHOHFWULFHOH PHQW DV VPDOO EXW VWLII DV SRVVLEOH 7KHUHIRUH =06  =0.  =0L=00ZKHUH=00LVWKHLPSHGDQFHRIWKHH[FLWHGVWUXFWXUH x 7KH WUDQVGXFHU FRQVWDQW .  DV ZHOO DV WKH HOHFWULFDO LPSHGDQFHV =(L DQG=(.DUHQXPHULFDOO\YHU\ODUJH

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

)LJ)UHTXHQF\UHVSRQVHRISLH]RHOHFWULFWUDQVGXFHUVD DFFHOHURPHWHUDVLQ )LJE IRUFHWUDQVGXFHUDVLQ )LJDQGF SLH]RHOHFWULFH[FLWHU

)LJ3ULQFLSOHGHVLJQRISLH]RHOHFWULFIRUFHWUDQVGXFHU

$V SRLQWHG RXW WKH SLH]RHOHFWULF PDWHULDOV FDQ DOVR EH XVHG IRU VWUXF WXUHERUQH VRXQG H[FLWHUV ,Q VXFK D FDVH WKH WUDQVIHU IXQFWLRQ LV GHWHU PLQHGIURP E DQG F DV

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG ) 8.

). 

= 0. .  = 06 § = (. · § V · .  = (. ¨  ¸ ¨  ¸ = (L ¹ © MZ =7 ¹ =7 ©

G 

ZKHUH =7 = 0.  = 0L  = 00 = 00 = 0L  = 0.  DQG = 06 = 0L  = 0. |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

)LJ3ULQFLSOHFRQILJXUDWLRQRIDQHOHFWURVWDWLFWUDQVGXFHU

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

7KH FRQGHQVHU PLFURSKRQH IRU DLUERUQH VRXQG LV DOVR DQ HOHFWURVWDWLF WUDQVGXFHU,WFRQVLVWVRIDWKLQILUPO\SUHVWUHVVHGPHPEUDQHFRQVWLWXWLQJ RQH RI WKH HOHFWURGHV DQG WKH ULJLG KRXVLQJ IRUPLQJ WKH RWKHU 7KH PHP EUDQH LV IRUFHG WR YLEUDWLRQV E\ LQFLGHQW VRXQG ZKHUHE\ LWV DPSOLWXGH LV FRQYHUWHGWRDQHOHFWULFDOVLJQDOE\WKHWUDQVGXFHU>@ 7KHHOHFWURPHFKDQLFDOUHODWLRQXQGHUO\LQJWKLVWUDQVGXFHULV&RXORPE¶V ODZRIHOHFWURVWDWLFDWWUDFWLRQ$SSOLHGRQWZRSODWHVDWDGLVWDQFHGZLWK WKHHTXLOLEULXPFDSDFLWDQFH & WKHDWWUDFWLRQIRUFHLVJLYHQE\ )

48 G

&8  G

& &  &  8  8  | 8  88  G G G

D 

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

&8 8 Z G

E 

7KLVIRUFHLVH[HUWHGRQWKHWHVWREMHFWIRUDFRQVWDQWFDSDFLWDQFH & ,QWKH RSSRVLWH VLWXDWLRQ ZLWK WKH WUDQVGXFHU DV VHQVRU ZKHUH WKH PHPEUDQH LV EURXJKWWRYLEUDWHZLWKWKHYHORFLW\ YZWKHWUDQVGXFHUFXUUHQWDPRXQWVWR LZ



&8 YZ  G

F 

,QWKLVGHYHORSPHQWDJDLQWKHHTXDOLW\RIPDJQLWXGHVRIWUDQVGXFHUFRQ VWDQWVSURYHQLQ D WR H LV XVHG,QWKHIRXUSROHUHSUHVHQWDWLRQ WKHHTXDWLRQIRULGHDOHOHFWURVWDWLFWUDQVGXFHUVUHDGV ­) Z ½ ® ¾ ¯ YZ ¿

ª  « «¬  G &8

&8 G º ­ L Z ½ » ® ¾  »¼ ¯8 Z ¿

G 

'LVUHJDUGLQJ WKH V\PEROV G  LV LGHQWLFDO WR WKH UHSUHVHQWDWLRQ IRU SLH]RHOHFWULFWUDQVGXFHUV7KHFORVHUHODWLRQVKLSEHWZHHQWKHWZRW\SHVRI WUDQVGXFHUVLVDOVRYDOLGIRUWKHLQIOXHQFHRIWKHLQQHUHOHFWULFDOLPSHGDQFH

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

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

G) G[

 G48  G G[



4 G8 8 G4   G G[  G G[



 G) 8 & G   [ G   G[ G G[

4 G G) 8 G&  G &8 G[ G G[ 



 G) 8 &  œ V(  G[  G 



8 & G



§8& ·   ¨¨ ¸¸ © G ¹ &

H 

DV GHYHORSHG IURP D  +HUHLQ [ LV WKH FKDQJH LQ HOHFWURGH GLVWDQFH DQGWKHYDULDEOHFDSDFLWDQFHLVDSSUR[LPDWHGE\ &

& | &   [ G  [ G

I 

$VFDQEHH[SHFWHGV(LVQHJDWLYHVLQFHFRPSUHVVLRQLPSOLHVDQHQODUJHG DWWUDFWLRQ LQ FRQWUDVW WR DQ RUGLQDU\ PHFKDQLFDO VSULQJ $FFRUGLQJO\ WKH DGGLWLRQDOPHFKDQLFDOLPSHGDQFHUHVXOWLQJLQWKLVZD\EHFRPHV = 0(

V( MZ





 §8& · ¨ ¸  MZ & ¨© G ¸¹

J 

,QWKLVH[SUHVVLRQWKHWUDQVGXFHUFRQVWDQWVTXDUHGDSSHDUV7KLVLVW\SLFDO IRUWKHPHFKDQLFDOHIIHFWRIDQHOHFWULFDOHOHPHQWDVREVHUYHGLQ D  )RUWKH UHPDLQLQJDQDO\VLVDOOWKHUHODWLRQVSUHYLRXVO\GHULYHGIRUWKH SLH]RHOHFWULFWUDQVGXFHUVFDQEHDGRSWHG2QO\WKHLQQHUPHFKDQLFDOLP SHGDQFHPXVWEHUHSODFHGE\ 

c = 0L

= 0L

 §8& ·  ¨ ¸  MZ & ¨© G ¸¹

K 

,QWKLVZD\WKHFKDLQRIPDWULFHVEHFRPHV ­) ½ ® ¾ ¯Y¿

c ºª  D º ª  = (L º ­ L ½ ª = 0L ® ¾ «  » «  D  » «  »¼ ¯8 ¿ ¬ ¼¬ ¼¬ c D D  = 0L c D = (L º ­ L ½ ª = 0L «  D » ®8 ¾   D = (L ¬ ¼¯ ¿

D 

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

LQFDVHRIDQLPPRELOHEDFNLQJHOHFWURGH:LWK=00 Æ fWKHWUDQVIHUIXQF WLRQRIWKHWUDQVGXFHURSHUDWHGDVDVHQVRULVREWDLQHGE\PHDQVRI D  DV 8 Y.

8. 

&8= (L 8 o  G   = (L = (. MZ G    MZ &5





E 

,QWKLVGHYHORSPHQWDUHXVHGWKHLPSHGDQFHV = (L  MZ & DQG = (. 5 LQ DFFRUGDQFH ZLWK DSSOLFDWLRQV LQ SUDFWLFH 8SRQ PXOWLSO\LQJERWKVLGHVE\ MZ  WKH OHIWKDQG VLGH EHFRPHV WKH UDWLR RI YROWDJH WR GLVSODFH PHQW [ Y MZ 7KLVPHDQVWKDWIRUKLJKLPSHGDQFHVLJQDODQDO\VLVHTXLS PHQW VXFK WKDW Z !  5&  WKH HOHFWURVWDWLF WUDQVGXFHU FRQVWLWXWHV D GLV SODFHPHQWVHQVRU 7KHWUDQVGXFHUUHODWLRQVGHULYHGLQWKHVHVHFWLRQVFDQDOVREHHVWDEOLVKHG E\ PHDQV RI +DPLOWRQ¶V SULQFLSOH RQFH DJDLQ SURYLQJ LWV JHQHUDOLW\7KH SURFHGXUH ZLOO EH H[HPSOLILHG LQ WKLV VHFWLRQ RI HOHFWURVWDWLF WUDQVGXFHUV ZKHUHE\DUHUHTXLUHGWKHHQHUJ\RIDFKDUJHGFRQGHQVHU (&

 4   &

D 

DQGWKDWRIDQLQGXFWDQFH (/

  /L 

  /4  

E 

ERWKRIZKLFKFDQEHREWDLQHGIURPWH[WERRNV )RUWKHPRGHOVNHWFKHGLQ)LJZKHUHWKHGLVSODFHPHQW [. LVDV VXPHG SUHVFULEHG WKH NLQHWLF HQHUJLHV WR EH FRQVLGHUHG DUH P. [    DQG /4    ZKLOVW WKH SRWHQWLDO HQHUJLHV UHDG V[    V. [  [ .   DQG 4  4   &  & 7KLVPHDQVWKDWWKH+DPLOWRQLDQEHFRPHV ª 4  4  «    P. [  /4   V[   V. [  [ .  « & «¬







º  [ G  »»  »¼

F 

ZKHUH ERWK WKH FKDUJH DQG WKH FDSDFLWDQFH DUH VSOLW LQ GF DQG DF FRPSR QHQWVDQGWKHYDQLVKLQJWLPHGHULYDWLYHRI 4 LVRPLWWHG,QWKHODVWWHUP PRUHRYHUWKHGLVSODFHPHQWGHSHQGHQWDFFRPSRQHQWRIWKHFDSDFLWDQFHLV DSSUR[LPDWHGDFFRUGLQJWR G 

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

)LJ  0RGHO RI HOHFWURVWDWLF WUDQVGXFHU VKRZLQJ WKH QRWDWLRQ XVHG LQ (TV  WR  7KHEDFNLQJHOHFWURGH  DQGWKHPDVV  IRUPDFDSDFLWRU

8SRQ FDUU\LQJ RXW WKH YDULDWLRQ ZLWK UHVSHFW WR WKH XQNQRZQ FR RUGLQDWHV[DQG 4 RQHREWDLQV P. [  V[  V. [  [ .

4  4 

  /4  4  4   [ G &





&G





D 



7KHVXEVHTXHQWPDQLSXODWLRQVDUH x 2PLVVLRQRIWHUPVRIVHFRQGRUGHUHJ [ 4  x &KDQJH WR YHORFLW\ Y [  DQG FXUUHQW L 4  LPSO\LQJ WKDW DOO GF FRPSRQHQWVYDQLVK x ,QWURGXFWLRQRISKDVRUQRWDWLRQLH [ MZ Y  4 MZ L  x 6XEVWLWXWLRQRIWKHLQGXFWLYHLPSHGDQFHMZ/E\ =(.DQGWKHPDVV DQGVSULQJLPSHGDQFHE\ =0LVXFKWKDWVLPXOWDQHRXVO\WKHUHVXOW EHFRPHVPRUHJHQHUDODQGORVVHVFDQEHWDNHQLQWRDFFRXQW 7KHRXWFRPHRIWKHVHRSHUDWLRQVLV = 0L Y 

V. Y  EL MZ

V. Y.  MZ

E 

§  · E Y  ¨ = (.  ¸ L  MZ & ¹ © ZKHUHWKHDEEUHYLDWLRQ E 4 MZ G& 8 MZ G KDVEHHQLQWURGXFHG&OHDUO\

WKHILUVWH[SUHVVLRQKDVWKHGLPHQVLRQRIIRUFHZKHUHDVWKHVHFRQGWKDWRI YROWDJH 7KHUHIRUH WKLV VHW RI HTXDWLRQV FDQ EH FRPSDUHG ZLWK WKDW LQ D %\REVHUYLQJWKDWWKHIRUFHDQGWKHYROWDJHDUH ) V. Y .  Y MZ DQG 8 = (. L UHVSHFWLYHO\(T E FDQEHUHZULWWHQDV

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG )

= 0L Y  E L

8

E Y  L MZ & 

F 

,WLVVRPHZKDWODERULRXVWREULQJ F LQWKHIRUP RI D VLQFHWKH c QHHGEHFRQVLGHUHG,ILQVSLWHRIWKDWWKH GLIIHUHQFHEHWZHHQ=0LDQG = 0L DOJHEUDLVXQGHUWDNHQLWLVIRXQGWKDWERWKPHWKRGV\LHOGWKHVDPHUHVXOW ZLWK WKH H[FHSWLRQ RI WKH VLJQ RI WKH FXUUHQW L  DVVRFLDWHG ZLWK WKH VLJQ FRQYHQWLRQ $FRPSDULVRQRIWKHWZRSURFHGXUHVVKRZVWKDWWKHRQHEDVHGRQ+DPLO WRQ¶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



) v )  v )  )





D 

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­) Z ½ ® ¾ ¯ YZ ¿

 º ­ LZ ½ ªJ «   J » ®8 ¾  ¬ ¼¯ Z¿

E 

ZKHUH J LV WKH WUDQVGXFHU FRQVWDQW GHSHQGHQW RQ WKH GLVWDQFH G WKH VWUHQJWKRIWKHPDJQHWLFILHOGDQGWKHJHRPHWU\

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

)LJ  3ULQFLSOH FRQILJXUDWLRQ RI DQ HOHFWURPDJQHWLF WUDQVGXFHU )HUUR PDJQHWLFDUPDWXUH  DQGSHUPDQHQWPDJQHW  

$ FRPSDULVRQ ZLWK (T F  VKRZV WKDW VDYH WKH QXPHULFDO YDOXHV WKH WUDQVGXFHU UHODWLRQV HTXDO WKRVH RI LGHDO HOHFWURG\QDPLF WUDQVGXFHUV $OVRWKHUROHRIWKHLQQHUHOHFWULFDOLPSHGDQFHLVWKHVDPHVLQFHLWUHVXOWV LQDYROWDJHGURS,QERWKFDVHVWKHLQQHULPSHGDQFHFRQVLVWVRIDQLQGXF WDQFH/L DQGDUHVLVWDQFH5HJDUGLQJWKHLQQHUPHFKDQLFDOLPSHGDQFHWKHUH LVDVPDOOGLIIHUHQFH6LPLODUWRWKHHOHFWURVWDWLFWUDQVGXFHUWKHUHLVDQDW WUDFWLRQIRUFHZKLFKGHSHQGVRQWKHGLVWDQFHEHWZHHQWKHPDJQHWDQGWKH DUPDWXUHDQGWKXVRQWKHLUUHODWLYHPRWLRQ$VEHIRUHWKLVOHDGVWRDQHJD WLYH VWLIIQHVV FI H  7KLV PHDQV WKDW (T G  DJDLQ FDQ EH DGRSWHGDOEHLWZLWK=0LUHSODFHGE\ c = 0L

= 0L 

V( MZ

= 0L 

J  MZ /L

F 

$V VHHQ WKH SLYRWDO TXDQWLW\ ZLWK UHVSHFW WR WKH UHDFWLRQ RI DQ HOHFWULFDO HOHPHQW RQ WKH PHFKDQLFDO LPSHGDQFH LV DV DOZD\V WKH VTXDUH RI WKH WUDQVGXFHUFRQVWDQW %\ PXOWLSO\LQJWKHPDWULFHV G  E DQG D WKHUHODWLRQIRU WKH SK\VLFDO HOHFWURPDJQHWLF WUDQVGXFHU ZLWK DQ LPPRELOH PDJQHW LV RE WDLQHGDV ­) ½ ® ¾ ¯Y ¿

ª J   = 0L c = (L J « = (L J ¬«

c J º­ L ½  = 0L » ® ¾  J ¼» ¯8 ¿

G 

)URP WKLV UHODWLRQ DOO WKH TXDQWLWLHV RI LQWHUHVW FDQ EH GHYHORSHG ,I WKH PRWLRQ RI WKH PDJQHW PXVW EH WDNHQ LQWR DFFRXQW WKH SURFHGXUH GHPRQ VWUDWHGLQ6HFWDSSOLHV

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

(OHFWURPDJQHWLF DV HOHFWURVWDWLF WUDQVGXFHUV HVVHQWLDOO\ DUH HPSOR\HG LQWKHODERUDWRU\ 7KHLUDGYDQWDJHLVWKDWWKH\DUHRQO\ZHDNO\FRXSOHGWR WKHWHVWREMHFWDQGUHDOL]HKDUGO\ DQ\ ORDGLQJDOVRDOORZLQJIRULQYHVWLJD WLRQVRIYHU\OLJKWZHLJKWV\VWHPV 0DJQHWRVWULFWLYH7UDQVGXFHUV $ PDJQHWRVWULFWLYH WUDQVGXFHU PRVW RIWHQ FRQVLVWV RI D URG RI PDJQHWRVWULFWLYHPDWHULDOLQVHUWHGLQDFRLODVLOOXVWUDWHGLQ)LJ7KH SULPDU\ PDJQHWRVWULFWLYH PDWHULDOV DUH QLFNHO VRPH IHUULWHV DQG PRUH UHFHQWO\VSHFLDODOOR\VWHUPHG³JLDQWPDJQHWRVWULFWLYHDOOR\V´ >@

)LJ 3ULQFLSOHFRQILJXUDWLRQRIDPDJQHWRVWULFWLYHWUDQVGXFHU0DJQHWRVWULF WLYHURG  DQGFRLO  

%\SDVVLQJDFXUUHQWL WKURXJKWKHFRLOWKHPDJQHWLFIRUFHVLQWKHPDJ QHWRVWULFWLYH PDWHULDO HLWKHU VKRUWHQ RU OHQJWKHQ WKH URG DQG FRQVWUDLQLQJ WKHURGDWLWVHQGVUHVXOWVLQDUHODWLYHO\ODUJHIRUFH$VVWDWHGSUHYLRXVO\ WKHIRUFHLVSURSRUWLRQDOWRWKHVTXDUHRIWKHPDJQHWLFIOX[7RDFKLHYHDQ DOWHUQDWLQJ IRUFH SURSRUWLRQDO WR VRPH VXSSOLHG DOWHUQDWLQJ FXUUHQW D GF FRPSRQHQWLVDJDLQUHTXLUHGVXFKWKDW WKHWRWDODQGWKHWUDQVGXFHUIRUFHV FDQEHZULWWHQDV   J 6L  J 6 L LZ 

) v) )Z

   J 6 L  L  | J 6 L  J 6 L L   

D 

6LQFH WKH WUDQVGXFHU LV UHFLSURFDO IRU L !! L  DQ DOWHUQDWLQJ YROWDJH DULVHV ZKHQ WKH PDJQHWRVWULFWLYH WUDQVGXFHU LV FRPSUHVVHG ZLWK WKH YHORFLW\ YZ ZKLFKLVJLYHQE\ 8Z

J 6 L Y Z 

E 

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

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­) ½ ® ¾ ¯Y ¿ ­) ½ ® ¾ ¯Y ¿

  ª %O/  = 0L = (L  = 0L º ­ L ½ « » ® ¾ %O/ ¬ = (L  ¼ ¯8 ¿ c = (L  = 0L c º­ L ½  ªJ   = 0L « » ® ¾ J ¬ = (L  ¼ ¯8 ¿

D 

IRU WKH HOHFWURG\QDPLF DQG HOHFWURPDJQHWLF DV ZHOO DV PDJQHWRVWULFWLYH WUDQVGXFHUVUHVSHFWLYHO\:KHQDOOGLVSODFHPHQWVDUHEORFNHGWKHIRUFHLV SURSRUWLRQDOWRWKHDOWHUQDWLQJFXUUHQWIRUWKLVFODVVRIWUDQVGXFHUV )RU WKH WUDQVGXFHUV GHSHQGHQW RQ HOHFWURVWDWLF IRUFHV     D DQG D DSSO\2QHILQGV ­) ½ ® ¾ ¯Y ¿ ­) ½ ® ¾ ¯Y ¿

  ª  = 0L .  = 0L = (L º ­ L ½ « » ® ¾ . ¬   = (L ¼ ¯8 ¿ c D   = 0L = (L º ­ L ½  ª = 0L « » ® ¾ D ¬   = (L ¼ ¯8 ¿

E 

IRUWKHSLH]RHOHFWULFDQGHOHFWURVWDWLFWUDQVGXFHUV)RUWKLVFODVVRIWUDQV GXFHUV WKH IRUFHV DUH SURSRUWLRQDO WR WKH DOWHUQDWLQJ YROWDJH DV WKH GLV SODFHPHQWVYDQLVK

([FLWDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

7ZR RI WKH UHODWLRQV LQ D  DQG E  VWLOO FRQWDLQV WKH IODZ WKDW WKHHOHFWULFDOO\LQGXFHGQHJDWLYHVWLIIQHVVPXVWEHFRQVLGHUHGLQWKHLQQHU c  %\ VXEVWLWXWLQJ I  DQG F  LQ E  PHFKDQLFDO LPSHGDQFH = 0L DQG D UHVSHFWLYHO\WKHHOHFWURVWDWLFGHVFULSWLRQEHFRPHV ­) ½ ® ¾ ¯Y¿

 ª D = (L   = 0L = (L « D = (L ¬  = 0L

= 0L º ­ L ½ » ® ¾  ¼ ¯8 ¿

F 

8SRQOHWWLQJ %O/ o D = (L  F EXWWKHVLJQEHFRPHVLGHQWLFDOWRWKHIRU PXOD IRU WKH HOHFWURG\QDPLF WUDQVGXFHU 7KH HOHFWURVWDWLF DQG HOHFWUR G\QDPLFWUDQVGXFHUVDUHFDWHJRUL]HGDV0WUDQVGXFHUV HTXLYDOHQWYROWDJH VRXUFH 7KHGHVFULSWLRQLQ F PRUHRYHUZRXOGKDYHEHHQHVWDEOLVKHG GLUHFWO\IURP(T F GHYHORSHGIURP+DPLOWRQ¶VSULQFLSOH ,QDVLPLODUZD\RQHREWDLQV ­) ½ ® ¾ ¯Y¿

 ª = 0L « J ¬ 

J   = 0L = (L º ­ L ½ » ® ¾   = (L ¼ ¯8 ¿

G 

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

D: Y  8 5

D 5 Y 

D 

)RUWKHVHFRQGVWHSWKHDX[LOLDU\H[FLWHULVWXUQHGRIIDQGWKHPDVVGULYHQ E\ WKH UHFLSURFDO WUDQVGXFHU 7KH IRUFH ) DQG YHORFLW\ Y UHVXOWLQJ IRU D FXUUHQWL5DUHUHODWHGE\ Y

) =

D5 L5 = 

E  ZKHUH=LVWKHNQRZQLPSHGDQFH,QWKLVUHODWLRQLVXVHGWKDWWKHUHFLSURF LW\SULQFLSOHRUWKHODZRIPXWXDOHQHUJLHVDUHYDOLGVXFKWKDW

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG 8 5 Y

)  8 5 L 5  L5

)  Y 

F 

UHVSHFWLYHO\ 6LQFH WKH WUDQVGXFHU WR EH FDOLEUDWHG VKRXOG EH OLQHDU WKH YROWDJHRIIHUHGLV 8:

D: Y  

G 

%\VXEVWLWXWLRQRI D DQG E YD5DQGYFDQEHHOLPLQDWHGDQG D:



8 : 8 :  = 8 5 L 5 

H 

7KXVWKHGHWHUPLQDWLRQRIWKHFDOLEUDWLRQIDFWRUD:VRXJKWLVWUDQVIHUUHGWR PHDVXUHPHQWVRIHOHFWULFDOYROWDJHVDQGFXUUHQWVEHVLGHVWKHNQRZOHGJHRI DZHOOGHILQHGLPSHGDQFH

)LJ0HDVXUHPHQWRIFRPELQHGTXDQWLWLHVDVVRFLDWHGZLWKEHQGLQJPRWLRQ

&RPELQHG4XDQWLWLHV

3UHFLVH PHDVXUHPHQWV RI FXUUHQWV DQG YROWDJHV DUH XVXDOO\ VWUDLJKWIRU ZDUGDVORQJDVWKHUHFLSURFDOWUDQVGXFHULVVXIILFLHQWO\VWURQJWRDFKLHYHD PHDVXUDEOHVLJQDORYHUWKHHQWLUHUDQJHRILQWHUHVW$ZHOOGHILQHGLPSHG DQFH FDQ EH PRUH FXPEHUVRPH KRZHYHU VLQFH DOVR IRU FRPSDFW PDVVHV WKHVXVSHQVLRQPD\OHDGWRGHYLDWLRQVIURPWKHEDVLF= MZP$GGLWLRQ DOO\ LW PXVW EH WDNHQ LQWR DFFRXQW WKDW IRU VPDOO PDVVHV DOO FRQQHFWHG HOHFWULFDOHTXLSPHQWFDQQRWLFHDEOHDIIHFWWKHLPSHGDQFH

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

*HQHUDWLRQDQG0HDVXUHPHQWRI6WUXFWXUH%RUQH6RXQG

TXDQWLWLHVRQWZRVLGHVRIDWKLQVWUXFWXUH)RUWXQDWHO\VXFKH[WHQVLYHDQG HUURUSURQHPHDVXUHPHQWVDUHRQO\QHFHVVDU\RQUDUHRFFDVLRQVVLQFHIUH TXHQWO\WKHPRVW GRPLQDQW FRPSRQHQWRIPRWLRQFDQEHLGHQWLILHGDQGD SURSHUWUDQVGXFHUFRQILJXUDWLRQGHYLVHG

)LJ  (VWLPDWLRQ RI GLIIHUHQW FRPSRQHQWV RI PRWLRQ IURP VXUIDFH PHDVXUH PHQWV

)LJ  'HWHUPLQDWLRQ RI YDULRXV PRWLRQ FRPSRQHQWV IURP PHDVXUHPHQWV RQ WKHVXUIDFHVRIDSODWHRUVKHOOVWUXFWXUH

5HIHUHQFHV

,Q FRQMXQFWLRQ ZLWK PHDVXUHPHQWV RI VWUXFWXUHERUQH VRXQG LQWHQVLW\ UHJLVWUDWLRQRIVWUHVVDQGYHORFLW\LVUHTXLUHGDVLQGLFDWHGE\WKHH[SUHVVLRQ LQ 6HFW  $JDLQ WKH YHORFLW\ UHJLVWUDWLRQ LV FRPSDUDWLYHO\ VWUDLJKWIRU ZDUGDWOHDVWDIDUDVWKHQHFHVVDU\TXDQWLWLHVUHSUHVHQWLQJWKHLQWHULRUPR WLRQFDQEHGHGXFHGIURP PHDVXUHPHQWVDWWKHVXUIDFHV,QFRQWUDVWWKHGH WHUPLQDWLRQRIVWUHVVHVLVPRUHLQWULFDWHIURPWZRDVSHFWV>@ x 7KHVWUHVVVWUDLQUHODWLRQVFRPSULVHVHYHUDOVWUDLQFRPSRQHQWVFI   DQG D  VXFK WKDW D VHW RI ILQLWH GLIIHUHQFHV ZRXOG KDYH WR EH IRUPHG 'XH WRWKHFRPSOLFDWHGDQGHUURUSURQHILQLWH GLIIHUHQFHRSHUDWRUVVLPSOHVWDWHVRIGHIRUPDWLRQDUHDVVXPHGIRU ZKLFK VLPSOH VWUHVVVWUDLQ UHODWLRQV DUH DYDLODEOH        DQG I  x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

/LVW RI 6\PEROV DQG 1RWDWLRQ 

L L: M N

Nc N cc N[\] N N% N/ NU N5 N7 LMN O OP P Pc Pcc Q S T U U//U/7U77 UUO UM V Vcc V W

WM X Y

Y [ \ ] YZ Z ’

'

(OHFWULFDOFXUUHQW>$@ 7UDQVGXFHUFXUUHQW ,PDJLQDU\XQLW ,QWHJHUQXPEHUZDYHQXPEHU>P@ 5HDOSDUWRIFRPSOH[ZDYHQXPEHU N ,PDJLQDU\SDUWRIFRPSOH[ZDYHQXPEHU N :DYHQXPEHUFRPSRQHQWVLQ[\DQG]GLUHFWLRQV :DYHQXPEHULQDPELHQWIOXLG %HQGLQJZDYHQXPEHU /RQJLWXGLQDOZDYHQXPEHU :DYHQXPEHULQUDGLDOGLUHFWLRQ 5D\OHLJKZDYHQXPEHU 6KHDUZDYHQXPEHU 8QLWYHFWRUVLQ[\DQG]GLUHFWLRQV /HQJWK>P@ 0HDQIUHHSDWK>P@ 0DVV>NJ@ 0DVVSHUXQLWOHQJWK>NJP@ 0DVVSHUXQLWDUHD>NJP@ 5HIUDFWLRQLQGH[LQWHJHUQXPEHU 3UHVVXUHIRUFHSHUXQLWDUHDVRXQGSUHVVXUH>1P@ 9ROXPHYHORFLW\>PV@ )ULFWLRQDO UHVLVWDQFH >1VP@ UDGLXV >P@ UHIOHFWLRQ IDFWRU 5HIOHFWLRQ IDFWRUV IRU ORQJLWXGLQDO DQG WUDQVYHUVDO ZDYHV 5HIOHFWLRQIDFWRUVDWWKHHQGVRIDURG 5HIOHFWLRQIDFWRUIRUHYDQHVFHQWZDYH 6WLIIQHVV>1P@DUFOHQJWK>P@ 6WLIIQHVVSHUXQLWDUHD>1P@ 'LVSODFHPHQWYHFWRU 7LPH>V@WUDQPLVVLRQFRHIILFLHQW 1HDUILHOGWUDQVPLVVLRQFRHIILFLHQW *HQHUDOILHOGYDULDEOH 3DUWLFOHYHORFLW\>PV@ 3DUWLFOHYHORFLW\FRPSRQHQWV 7UDQVGXFHUYHORFLW\ 5RWDWLRQDORUDQJXODUSDUWLFOHYHORFLW\>V@ /DSODFHRSHUDWRU 'LIIHUHQFHWLPHVWHSWKLFNQHVVUDWLR

/LVWRI6\PEROVDQG1RWDWLRQ

'1 4 4c

/ 3 !

) < D E J G

G ! ! H H[ \ ] ] K

KPQ

N O

O% O/ OF OQ

P Q [ U U U//!!U/7! U77

1XPEHU RI PRGHV LQ D IUHTXHQF\ EDQG 0DVV PRPHQW RI LQHUWLD >NJP@ 0DVV PRPHQW RI LQHUWLD SHU XQLW OHQJWK >NJP@ /RJDULWKPLF!GHFUHPHQW!QRUP!RI!HLJHQIXQFWLRQ!>NJ@ WKHUPDO FRQGXFWLYLW\ >: PÂ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

/LVW RI 6\PEROV DQG 1RWDWLRQ 

V V[ \ ] W W[\! W\]! W][ M M ! I F \ Z

ZQ

[ [ [Ö   [! [ c cc [![ [ [ 5H>@ ,P>@ [ \ ]

U M ] 5!- M

 1RUPDO VWUHVV >1P! @ UHOD[DWLRQ WLPH WKLFNQHVV UDWLR  DQG!GLUHFWLRQV 6WUHVV FRPSRQHQWV LQ [ \! ]  6KHDU VWUHVV >1P! @ UDGLDWLRQ HIILFLHQF\ WUDQVPLVVLRQ HIILFLHQF\ 6KHDU VWUHVV FRPSRQHQWV 3KDVH DQJOH (LJHQIXQFWLRQ 3KDVH DQJOH $QJOH RI URWDWLRQ $QJOH DX[LOLDU\ TXDQWLW\ $QJXODU IUHTXHQF\ 5HVRQDQFH IUHTXHQF\

$YHUDJHG TXDQWLW\ &RPSOH[ TXDQWLW\ 3HDN YDOXH )RXULHU WUDQVIRUPHG YDULDEOH )LUVW DQG VHFRQG WLPH GHULYDWLYH )LUVW DQG VHFRQG VSDWLDO GHULYDWLYH $EVROXWH YDOXH &RPSOH[ FRQMXJDWHG TXDQWLW\ 5HDO SDUW ,PDJLQDU\ SDUW &DUWHVLDQ FRRUGLQDWHV &\OLQGULFDO FRRUGLQDWHV 6SKHULFDO FRRUGLQDWHV

,QGH[

DEVRUSWLRQFRHIILFLHQW DFFHOHUDWLRQQRLVH DFFHOHURPHWHU DFRXVWLFERXQGDU\OD\HU DFRXVWLFIDUILHOG DFWLYHSDUWRIWKHSRZHU DGGHGV\VWHP DGLDEDWLF DIWHUHIIHFWIXQFWLRQV DLUERUQHVRXQGUDGLDWLRQ DQDORJ\IRUFHFXUUHQW DQJOHRILQFLGHQFH DQJXODUYHORFLW\ DQWLQRGH DQWLUHVRQDQFH DSSDUHQWPDVV DUELWUDULO\VKDSHGVWUXFWXUHV DWWDFKHGOD\HUV DWWHQXDWLRQ DWWHQXDWLRQFRQVWDQW DWWHQXDWLRQUHJLRQ DXGLEOHIUHTXHQF\UDQJH DYHUDJHDEVRUSWLRQHIILFLHQF\ DYHUDJHPRGDOHQHUJ\ DYHUDJHWUDQVPLVVLRQHIILFLHQF\  EDQGZLGWK EHDPH[FLWHGQRUPDOO\E\DQ LPSDFWLQJPDVV EHDPV YLEUDWLRQV EHQGLQJPRPHQW EHQGLQJVWLIIQHVV PHDVXUHPHQW PHDVXUHPHQWRI EHQGLQJZDYHHTXDWLRQ

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

HQHUJ\YDULDWLRQV PHDVXUHPHQWV PHDVXUHPHQWVEHDP PHDVXUHV PHFKDQLVPV RIDWWDFKHGOD\HUV RIEXLOGLQJPDWHULDOV RIIOH[XUDOZDYHV RIJUDQXODUPDWHULDOV RILQWHUIDFHV RIORQJLWXGLQDOZDYHV RIPHWDOV RISODVWLFV SKDVHDQJOH UDWLR YLVFRXV ZLWKVSDFHU GDPSLQJGXHWRUHODWLYHPRWLRQ QRUPDO GDPSLQJIRUEXLOWXSV\VWHPV GDPSLQJOD\HU GDPSLQJOD\HUV GDPSLQJPDWHULDO GDPSLQJPHFKDQLVPV GDPSLQJTXDQWLWLHV GDVKSRW GHFD\WLPH GHFUHPHQW ORJDULWKPLF GHJHQHUDWH GHJHQHUDWHPRGHV GLHOHFWULFWUDQVGXFHU GLODWDWLRQ GLODWDWLRQDOZDYHV GLSROH GLSROHVRXUFHV GLSROHW\SH 'LUDF¶VGHOWDIXQFWLRQ GLUHFWLYLW\ D[LDOO\V\PPHWULFPHPEUDQH GLUHFWLYLW\RIWKHUDGLDWLRQ GLVFRQWLQXLW\ GLVORFDWLRQSURFHVVHV GLVSHUVLRQ IRUZDYHVRQVKHOOV IRUZDYHVRQVWULSV

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

HOHFWURPHFKDQLFDOWUDQVIHUIXQFWLRQ  HOHFWURVWDWLFWUDQVGXFHU HQHUJ\ LQEHDPVZLWKDWWDFKHGVSULQJV DQGZDYHV LQSODWHVZLWKDWWDFKHGOD\HUV  HQHUJ\FRQVHUYDWLRQ HQHUJ\GHQVLW\ LQEHQGLQJZDYHV HQHUJ\GLVWULEXWLRQ LQEHQGLQJZDYHV HQHUJ\GLVWULEXWLRQV LQORQJLWXGLQDOZDYHV HTXDWLRQRIPRWLRQ RUWKRWURSLFSODWHV WKLQSODWHV WKUHHOD\HUV\VWHP WZROD\HUEHDP HTXLSDUWLWLRQ HTXLYDOHQWDEVRUSWLRQOHQJWK HUURUIXQFWLRQ (XOHU¶VFRQVWDQW H[FLWLQJLPSXOVH H[WHQVLRQDOVWUDLQ H[WHQVLRQV IDUILHOG ILOOHUV ILQLWHGLIIHUHQFH ILQLWHHOHPHQWPHWKRG IODQNLQJSDWKV IODQNLQJWUDQVPLVVLRQ IOH[XUDOVWLIIQHVV PHDVXUHPHQW IOH[XUDOZDYHV IORDWLQJIORRU IORZUHVLVWDQFH IOXLGKDOIVSDFHWKHZDYHLPSHGDQFH  IOXLGORDGHGSODWH IOXLGORDGLQJ IOXLGUHDFWLRQ IRRWIDOOQRLVH IRUFHFXUUHQWDQDORJ\

,QGH[  IRUFHJDXJH IRUFHWUDQVGXFHU IRUFHGYLEUDWLRQV IRXUSROH IRXUSROHHTXDWLRQV IRXUSROHPDWUL[ IRXUSROHUHSUHVHQWDWLRQ IUHHEHQGLQJZDYHV IUHHILHOG IUHHSODWHZDYHV IUHHVXUIDFH IUHHYLEUDWLRQV EHQGLQJ ORQJLWXGLQDO IUHHZDYHV IUHTXHQF\PRGXODWLRQ IUHTXHQF\UHVSRQVHIXQFWLRQ IUHTXHQF\UHVSRQVHRISLH]RHOHFWULF WUDQVGXFHU IULFWLRQGU\ JDVSXPSLQJ *DXVVLDQHUURULQWHJUDO JHQHUDWLRQ JHRPHWULFIDUILHOG JHRPHWULFQRQOLQHDULWLHV JHRSKRQH JODVVWUDQVLWLRQWHPSHUDWXUH JUD]LQJLQFLGHQFH *UHHQ¶VIXQFWLRQ *UHHQ¶VIXQFWLRQIRUWKLQSODWH JURXSVSHHG J\URVFRSLFHOHPHQW KDOISRZHUEDQGZLGWK KDOIYDOXHEDQGZLGWK   +DPLOWRQ¶VSULQFLSOH  +DQNHOIXQFWLRQ KHDWFRQGXFWLRQSKHQRPHQD KLJKSRO\PHUPDWHULDOV +RRNH¶VODZ K\GURDFRXVWLFUDGLDWRUV K\GURG\QDPLFPDVV

K\GURG\QDPLFVKRUWFLUFXLW  LGHDOERXQGDU\FRQGLWLRQV LGHDOUHIOHFWLRQ LPDJLQDU\SDUWRIPRPHQWPRELOLW\  LPPRELOHUHIHUHQFH LPSDFWGXUDWLRQRUFRQWDFWWLPH LPSDFWH[FLWDWLRQ LPSHGDQFH GULYLQJSRLQW KDOIVSDFH KHDG PDVV PDWUL[ PHDVXUHPHQW UDGLDWLRQ VLPSO\VXSSRUWHGUHFWDQJXODU SODWH VSULQJ WUDQVGXFHU WXEH LPSHGDQFHIRUPXODH LPSXOVHRILPSDFWLQJPDVV LPSXOVHUHVSRQVH LPSXOVLYHVRXQG LQGH[RIUHIUDFWLRQ LQGXFWDQFHEULGJH LQGXFWLYHVHQVRUV LQILQLWHO\ORQJEHDP LQSXWPRELOLW\RIDQLQILQLWHO\ORQJ URG LQSXWSRLQWPRELOLWLHVRIEHDPVLQ IOH[XUH LQSXWSRLQWPRELOLW\RIIRUFHH[FLWHG SODWH LQSXWSRLQWPRELOLW\RIVHPLLQILQLWH EHDPV LQWHQVLW\ LQWHQVLW\PHWKRG LQWHQVLW\VWUXFWXUHERUQH LQWHQVLW\YHFWRU LQWHUOD\HUV LQYHUVHSUREOHP LUURWDWLRQDO

,QGH[ LVRWKHUPDOFDVH .HOYLQ9RLJWPRGHO NLQHPDWLFYLVFRVLW\ NLQHWLF NLQHWLFHQHUJ\ .LUFKKRII¶VEHQGLQJWKHRU\ .LUFKKRII+HOPKROW]LQWHJUDO HTXDWLRQ /DJUDQJH¶VHTXDWLRQV /DJUDQJH¶VPXOWLSOLHU /DSODFHRSHUDWRU ODVHUGRSSOHUYLEURPHWHU ODZRIUHIUDFWLRQ OHYHO SRZHU VRXQGSUHVVXUH OLPLWLQJDQJOH OLPLWLQJDQJOHIRUWRWDOUHIOHFWLRQ  ORDGLQJWUDQVGXFHU ORFDOHODVWLFLW\ ORJDULWKPLFGHFUHPHQW ORQJLWXGLQDOSHULRGLFV\VWHP ORQJLWXGLQDOYLEUDWLRQV ORQJLWXGLQDOZDYH HIIHFWRIHODVWLFLQWHUOD\HUV HIIHFWVRIEORFNLQJPDVVHV  HIIHFWVRIGLVFRQWLQXLWLHV LQEHDPVDQGSODWHV LQSHULRGLFVWUXFWXUHV ORVVIDFWRU EHDPVZLWKDWWDFKPHQWV GHILQLWLRQ PHDVXUHPHQWV PHWDOV SODVWLFV SODWHVZLWKDWWDFKHGOD\HUV UHODWLRQVWRRWKHUGDPSLQJ PHDVXUHV ORVVIDFWRUDQGPRGXOXVRIHODVWLFLW\ IRUKLJKSRO\PHUV ORXGVSHDNHUHOHFWURG\QDPLF

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

,QGH[  ODWWLFH SRZHU PRGDOLQWHUDFWLRQWHUP PRGHVKDSHV PRGHV PRGLILHGVKHDUVWLIIQHVV PRGXOXVRIHODVWLFLW\ PHDVXUHPHQW PRPHQW H[FLWDWLRQ PRELOLWLHV PRQRSROH 0WUDQVGXFHU PXOWLOD\HUHGZDOO PXOWLSOHGHJUHHRIIUHHGRP PXOWLSROHVRXUFHV PXWXDOHQHUJLHV QDWXUDOIUHTXHQFLHV QDWXUDOYLEUDWLRQV 1DYLHU6WRNH¶VHTXDWLRQV QHDUILHOGV QRGH QRQFRQWDFWLQJGHYLFH QRQOLQHDULW\ QRUP 1WUDQVGXFHU QXPEHURIPRGHVDQGPRGDO GHQVLWLHV REOLTXHLQFLGHQFH DLUERUQHVRXQG IOH[XUDOSODWHZDYHV RQIUHHVXUIDFH RUWKRJRQDOLW\ RUWKRJRQDOLW\UHODWLRQ RUWKRWURSLFSODWHV  SDUDPHWULFH[FLWDWLRQ 3DUVHYDO¶VWKHRUHP SDUWLFXODUVROXWLRQ SDVVDQGVWRSEDQGV SHULRG SHULRGLFEHQGLQJZDYHJXLGH

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

,QGH[ UHVSRQVHWRGLVWULEXWHGH[FLWDWLRQ  UHVSRQVHWRSRLQWH[FLWDWLRQ VLPSOHOD\HU VSDFHU WKLFN ZDYHVIUHH SODWHVRQD:LQNOHUEHG SOD\EDFNRIWKHVXUIDFHURXJKQHVV  SRLQWH[FLWHGLQILQLWHEHDP SRLQWH[FLWHGWKLFNSODWH SRLQWLPSHGDQFH SRLQWPRELOLW\ 3RLQWPRELOLW\RIDKRPRJHQHRXV SODWH SRLQWPRELOLW\RIDSODWH SRLQWPRELOLW\RIDSODWHVWULS SRLQWVRXUFH 3RLVVRQ¶VUDWLR SRURVLW\ SRWHQWLDODQGNLQHWLFHQHUJ\ SRWHQWLDOHQHUJLHV SRWHQWLDOHQHUJ\ SRZHU DLUERUQHVRXQG EDODQFH EHQGLQJZDYHV IORZ OHYHO TXDVLORQJLWXGLQDOZDYHV WUDQVPLWWHG WUDQVPLWWHGWRLQILQLWHO\H[WHQGHG VWUXFWXUHV WUDQVPLWWHGWRSODWHOLNHVWUXFWXUHV  SUHVVXUHOHYHO SUHVWUHVVHGSODWHV SUHVWUHVVHGSODWHVRUPHPEUDQHV  SURSDJDWLRQ FRQVWDQW IXQFWLRQ SXUHORQJLWXGLQDOZDYHV SXUHWUDQVYHUVHZDYHV

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

,QGH[  IDFWRU UHIOHFWLRQDQGWUDQVPLVVLRQ HIILFLHQFLHV UHIOHFWLRQVDQGWUDQVPLVVLRQVIDFWRUV  UHIUDFWLRQ UHIUDFWLRQLQGH[ UHOD[DWLRQIXQFWLRQ UHOD[DWLRQSURFHVV UHOD[DWLRQWLPH UHVLOLHQWWUDQVPLVVLRQHOHPHQWV UHVRQDQFHIUHTXHQF\ UHVRQDQFHIXQFWLRQ UHVRQDQWYLEUDWLRQV EHDPV GDPSLQJPHDVXUHPHQW UHVRQDQWO\RUUHVLVWLYHO\FRQWUROOHG  UHYHUEHUDQWURRP UHYHUEHUDWLRQWLPH UHYHUVLEOHPHFKDQLFDOHQHUJ\ ULJLGWHUPLQDWLRQ ULQJIUHTXHQF\ ULQJUHVRQDQFHV UROOLQJQRLVH URWDWLQJPLUURUVHQVRU URWDWLRQDO LQHUWLD RSHUDWRU VWLIIQHVV URWDWLRQV UXEEHU VDQGZLFKSODWHV GDPSLQJ VFDWWHUHU VFDWWHULQJ VHFRQGPRPHQWRIDUHD VHFXODUVROXWLRQ VHLVPLFPDVV VHLVPLFZDYHV VHPLFRQGXFWRUV VHPLLQILQLWHEHDP VHPLLQILQLWHFRQWLQXXP VHQVRU DFFHOHUDWLRQ

FDSDFLWDWLYH GLVSODFHPHQW IRUFH LQGXFWLYH VWUDLQ YHORFLW\ VKHDU DQJOH IRUFH PRGXOXV SDUDPHWHU VWLIIQHVV VWUDLQ VKHOO HTXDWLRQ UDGLDWLRQ ZDYHVSHHG VLGHOREHV 6QHOO¶VODZ 6RPPHUIHOGUDGLDWLRQFRQGLWLRQ VRXQGEULGJHV VRXQGSRZHUOHYHO VRXQGSUHVVXUH VRXQGSUHVVXUHOHYHO VRXQGWUDQVPLVVLRQ VRXUFH VRXUFHRI]HURRUGHU VRXUFHYROXPHV VRXUFHIUHH VRXUFHIUHHILHOG VSDFHU VSDWLDODYHUDJHRIWKHVTXDUHRIWKH YHORFLW\ VSDWLDOGHFD\ VSDWLDOSHULRG VSDWLDOO\SHULRGLFPDVVHV VSHFLILFKHDW VSKHULFDOUDGLDWRU VSKHULFDOVRXQGVRXUFH VSULQJLPSHGDQFH VSULQJVWLIIQHVV VTXHH]HILOPGDPSLQJ 6W9HQDQW¶VSULQFLSOH VWDWLVWLFDOHQHUJ\DQDO\VLV VWHDG\VWDWHUHVSRQVH VWLFNDQGVOLS

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

VWLIIQHVVHVRIUHFWDQJOHV ZDYHLPSHGDQFH ZDYHV WRWDODWWHQXDWLRQ WRWDOUHIOHFWLRQ WRWDOWUDQVPLVVLRQ WUDFHVSHHG WUDFHYHORFLW\ WUDFHZDYHPDWFKLQJ WUDFHZDYHOHQJWK WUDQVGXFHU DFFHOHUDWLRQ FRQVWDQW FU\VWDO HOHFWURG\QDPLF HOHFWURPDJQHWLF HOHFWURVWDWLF IRUFH LQGXFWLYH PDJQHWRVWULFWLYH PDWUL[ SLH]RHOHFWULF VWUDLQ VWUXFWXUHERUQHVRXQG YHORFLW\ WUDQVIHUIXQFWLRQ WUDQVIHUIXQFWLRQRIHOHFWURG\QDPLF WUDQVGXFHU WUDQVIHULPSHGDQFH WUDQVIHUPRELOLWLHV WUDQVLWLRQUHJLRQ WUDQVPLVVLELOLW\ WUDQVPLVVLRQ WUDQVPLVVLRQFRHIILFLHQW WUDQVPLVVLRQHIILFLHQF\  DYHUDJH EHQGLQJZDYHV EORFNLQJPDVVHV FRUQHUV HODVWLFLQWHUOD\HUV ORQJLWXGLQDOZDYHV PXOWLOD\HUHGZDOO SODWHV VWLIIHQHUV WUDQVPLVVLRQHOHPHQWV

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

ZDYHILHOG ZDYHLPSHGDQFH  ZDYHLPSHGDQFHIRUDSUHVWUHVVHG SODWHRQD:LQNOHUEHG ZDYHLPSHGDQFHRIWKHWKLQSODWH  ZDYHPRELOLW\ RUWKRWURSLFSODWH WKLQSODWH ZDYHVSHHG EHQGLQJ ORQJLWXGLQDO TXDVLORQJLWXGLQDO WRUVLRQDO WUDQVYHUVH ZDYHWUDFHPDWFKLQJ ZDYHWUDLQFORVXUHSULQFLSOH ZDYHOHQJWK EHQGLQJZDYHV ZDYHQXPEHU EHQGLQJZDYHV FRPSOH[ ORQJLWXGLQDO VSHFWUXP ZDYHV EHQGLQJ ERG\ ORQJLWXGLQDO 5D\OHLJK VXUIDFH WRUVLRQDO WUDQVYHUVH ZDYHWUDLQFORVXUHSULQFLSOH ZHDNFRXSOLQJ :KHDWVWRQHEULGJH :LQNOHUEHG