Dynamic Mean Variance Analysis

' < 1 $ 0 ,& 0 ( $ 1 9 $ 5 ,$ 1 & ( $ 1 $ / < 6 ,6

%\ 3KLOLSSH +HQURWWH

*URXSH +(& )LQDQFH DQG (FRQRPLFV 'HSW -RX\HQ-RVDV &HGH[ )UDQFH

$XJXVW

$ E VWU D F W

:H DQDO\]H WKH FRQGLWLRQDO YHUVLRQV RI WZR FORVHO\ FRQQHFWHG PHDQYDULDQFH LQYHVW PHQW SUREOHPV WKH UHSOLFDWLRQ RI D FRQWLQJHQW FODLP RQ WKH RQH KDQG DQG WKH VHOHFWLRQ RI DQ HsFLHQW SRUWIROLR RQ WKH RWKHU KDQG LQ D JHQHUDO GLVFUHWH WLPH VHWWLQJ ZLWK LQ FRPSOHWH PDUNHWV

:H H[KLELW D SRVLWLYH SURFHVVK ZKLFK VXPPDUL]HV WZR SLHFHV RI HFRQRPLFDOO\ PHDQ LQJIXO LQIRUPDWLRQ $V D IXQFWLRQ RI WLPH LW GHVFULEHV WKH WLPH GLPHQVLRQ RI WKH LQ YHVWPHQW RSSRUWXQLW\ VHW WKURXJK LWV OLQN ZLWK WKH QRWLRQ RI G\QDPLF 6KDUSH UDWLR $V D IXQFWLRQ RI WKH VWDWHV RI WKH ZRUOG LW FDQ EH XVHG DV D FRUUHFWLRQ OHQV IRU P\RSLF LQYHVWRUV DQG LW UHYHDOV WKH JDS EHWZHHQ VWDWLF DQG G\QDPLF PHDQYDULDQFH LQYHVWPHQW VWUDWHJLHV $ VKRUW VLJKWHG LQYHVWRU ZKR FRUUHFWV WKH SUREDELOLW\ GLVWULEXWLRQ ZLWK WKH KHOS RIK DFWV RSWLPDOO\ IRU ORQJ KRUL]RQV

:H GHVFULEH WKH G\QDPLF PHDQYDULDQFH HsFLHQW IURQWLHU FRQGLWLRQHG RQ WKH LQIRU PDWLRQ DYDLODEOH DW D IXWXUH GDWH LQ WKH IRUP RI D WZR IXQG VHSDUDWLRQ WKHRUHP 7KH G\QDPLF 6KDUSH UDWLR PHDVXUHV WKH GLVWDQFH IURP RI DQ LQYHVWPHQW VWUDWHJ\ WR WKH Hs FLHQW IURQWLHU :H H[SODLQ KRZ RSWLPDO G\QDPLF 6KDUSH UDWLRV DJJUHJDWH WKURXJK WLPH DQG ZH VWXG\ WKH WLPH FRQVLVWHQF\ UXOHV ZKLFK HsFLHQW SRUWIROLRV PXVW IROORZ :H LQ YHVWLJDWH WKH HpHFW RI D FKDQJH RI LQYHVWPHQW KRUL]RQ LQ SDUWLFXODU ZH VKRZ WKDW P\RSLD LV RSWLPDO DV VRRQ DV WKH SURFHVV K LV GHWHUPLQLVWLF

.H\ZRUGV 6HOI )LQDQFLQJ 3RUWIROLR (sFLHQW )URQWLHU 6KDUSH 5DWLR 0\RSLD

_{, K D Y H J UHD WO\ E HQ Hq WHG IUR P K HOS IX O G LVFX VVLR Q V Z LWK 3 UR IHVVR UV 5 D P D & R Q W % HUQ D UG ' X P D V 1 LFR OH ( O} . D UR X L 5 R E HUW ( Q J OH 7 K LHUU\ ) R X FD X OW & K ULVWLD Q * R X ULxHUR X [ 0 R Q LT X H - HD Q E OD Q F - D FT X HV 2 OLY LHU - HD Q & K D UOHV 5 R FK HW 0 LFK D HO 5 R FN LQ J HU % UX Q R 6 R OQ LN - D Q : HUQ HU D Q G S D UWLFLS D Q WV WR WK H UHVHD UFK VHP LQ D UV D W

,Q W U R G X F W LR Q

,W K D V O R Q J E H H Q U H F R J Q L] H G LQ qQDQFH WKDW WKH DELOLW\ WR UHEDODQFH D SRUWIROLR RI VHFXULWLHV WKURXJK WLPH FDQ GUDPDWLFDOO\ LPSURYH WKH VHW RI LQYHVWPHQW RSSRUWXQLWLHV DV FRPSDUHG WR VLPSOH EX\ DQG KROG VWUDWHJLHV '\QDPLF VHOI qQDQFLQJ VWUDWHJLHV KDYH EHHQ XVHG RQ WKH RQH KDQG WR UHSOLFDWH WKH SD\RpV RI FRQWLQJHQW FODLPV DQG RQ WKH RWKHU KDQG WR LPSURYH WKH SHUIRUPDQFH RI LQYHVWPHQW SRUWIROLRV 7KH SXUSRVH RI WKLV DUWLFOH LV WR SURYLGH D GHWDLOHG DQDO\VLV RI WKHVH WZR FORVHO\ UHODWHG LQYHVWPHQW SUREOHPV LQ D PHDQYDULDQFH VHWWLQJ WKDW LV IRU D TXDGUDWLF ORVV RU RE M HFWLYH IXQFWLRQ

:H VWXG\ WKH FRQGLWLRQDO YHUVLRQ RI WKHVH SUREOHPV ZLWK LQYHVWPHQW VWUDWHJLHV VWDUWLQJ IURP D IXWXUH GDWH ZLWK VWRFKDVWLF ZHDOWK :H LPSRVH YHU\ IHZ FRQVWUDLQWV RQ WKH VHFXULW\ SULFH SURFHVV DQG WKH PDUNHW LV LQ JHQHUDO LQFRPSOHWH :KHQ H[DFW UHSOLFDWLRQ FDQQRW EH DFKLHYHG WKH PHDQYDULDQFH KHGJH LV WKH G\QDPLF VWUDWHJ\ ZKLFK EHVW UHSOLFDWHV D JLYHQ

SD\Rp IRU WKH / QRUP FRQGLWLRQHG RQ WKH DYDLODEOH LQIRUPDWLRQ 7KLV SUREOHP LV DQDO\W LFDOO\ YHU\ FORVH WR WKH VHOHFWLRQ RI D PHDQYDULDQFH HsFLHQW G\QDPLF SRUWIROLR ZKHUH WKH RE M HFWLYH LV WR PLQLPL]H WKH FRQGLWLRQDO YDULDQFH RI WKH UHWXUQ RI D SRUWIROLR IRU D JLYHQ H[SHFWHG UHWXUQ

&HQWUDO WR RXU DQDO\VLV LV WKH VHW RI G\QDPLF SRUWIROLRV ZKR VKDUH WZR EDVLF SURSHUWLHV WKH\ DUH VHOI qQDQFLQJ HYHU\ WUDGLQJ SHULRG DQG WKH YDULDQFH RI WKHLU YDOXHV DW D JLYHQ KRUL]RQ LV qQLWH :H VHOHFW DPRQJ WKLV VHW DQ RSWLPDO UHSOLFDWLRQ VWUDWHJ\ RU DQ HsFLHQW G\QDPLF SRUWIROLR :H LQYHVWLJDWH WKH SURSHUWLHV RI WKHVH G\QDPLF SRUWIROLRV LQ D YHU\ JHQHUDO GLVFUHWH WLPH PRGHO ZKHUH D qQLWH QXPEHU RI VHFXULWLHV ZKLFK PD\ GLVWULEXWH GLYLGHQGV DUH WUDGHG RYHU 7 SHULRGV LQ D IULFWLRQOHVV DQG SHUIHFWO\ FRPSHWLWLYH PDUNHW ZLWK QR OLPLW RQ VKRUW VDOHV ([FHSW SRVVLEO\ IRU WKH qQDO GDWH ZH PDNH QR UHTXLUHPHQW RQ WKH LQWHJUDELOLW\ RI WKH VHFXULW\ SULFHV :H GR QRW UXOH RXW SRVVLEOH UHGXQGDQF\ EHWZHHQ WKH VHFXULWLHV DQG ZH RQO\ UHTXLUH D ZHDN QRWLRQ RI DEVHQFH RI DUELWUDJH WKH ODZ RI RQH SULFH WZR SRUWIROLRV ZLWK LGHQWLFDO YDOXHV WRPRUURZ LQ HYHU\ VWDWH RI WKH ZRUOG PXVW DOVR VKDUH WKH VDPH YDOXH WRGD\ 7KH QRWLRQ RI 6KDUSH UDWLR LV D FRPPRQ DQG UHFXUULQJ WKHPH IRU WKH WZR PHDQYDULDQFH SUREOHPV ZKLFK ZH VWXG\ :H GHqQH WKH G\QDPLF 6KDUSH UDWLR RI D VHOI qQDQFLQJ SRUWIROLR DV WKH G\QDPLF JHQHUDOL]DWLRQ RI WKH XVXDO VWDWLF 6KDUSH UDWLR D GHqQLWLRQ ZKLFK ZH ODWHU H[WHQG WR JHQHUDO FRQWLQJHQW FODLPV /LNH LWV VWDWLF FRXQWHUSDUW WKH G\QDPLF 6KDUSH UDWLR PHDVXUHV WKH ULVNUHWXUQ WUDGHRp RI DQ LQYHVWPHQW VWUDWHJ\ ,W UHDFKHV D PD[LPXP YDOXH IRU SRUWIROLRV RQ WKH G\QDPLF HsFLHQW IURQWLHU 7KH VSUHDG EHWZHHQ WKH G\QDPLF 6KDUSH

UDWLR RI DQ LQYHVWPHQW VWUDWHJ\ DQG WKH RSWLPDO G\QDPLF 6KDUSH UDWLR PHDVXUHV WKH GLVWDQFH

IURP WKH VWUDWHJ\ WR WKH HsFLHQW IURQWLHU ZKLFK LV WKH / QRUP EHWZHHQ WKH SD\Rp RI WKH VWUDWHJ\ DQG LWV RSWLPDO DSSUR[LPDWLRQ E\ HsFLHQW SRUWIROLRV

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pHUHQW LQYHVWPHQW VSDQV

$V D IXQFWLRQ RI WKH VWDWHV RI WKH ZRUOG RQ WKH RWKHU KDQG WKH SURFHVV K UHYHDOV WKH JDS EHWZHHQ VWDWLF DQG G\QDPLF PHDQYDULDQFH LQYHVWPHQW VWUDWHJLHV ,W FDQ LQGHHG EH LQWHUSUHWHG DV D FRUUHFWLRQ OHQV IRU P\RSLF LQYHVWRUV $ VKRUW VLJKWHG LQYHVWRU ZKR FRUUHFWV WKH SUREDELOLW\ GLVWULEXWLRQ ZLWK WKH KHOS RI K DFWV RSWLPDOO\ IRU ORQJ KRUL]RQV (DFK LQYHVWPHQW KRUL]RQ UHTXLUHV D GLpHUHQW FRUUHFWLRQ OHQV SURYLGHG E\ WKH YDOXH RI WKH SURFHVVK

DW WKH FRUUHVSRQGLQJ PDWXULW\ $OWHUQDWLYHO\ WKLV FRUUHFWLRQ PD\ EH LQWHUSUHWHG DV D FKDQJH RI QXPHUDLUH LQ ZKLFK P\RSLD EHFRPHV RSWLPDO ,Q WKH VSHFLDO FDVH ZKHUH K LV GHWHUPLQLVWLF P\RSLD LV G\QDPLFDOO\ RSWLPDO 7KLV KDSSHQV LQ SDUWLFXODU ZKHQ ERWK LQWHUHVW UDWHV DQG RQH SHULRG VWDWLF RSWLPDO 6KDUSH UDWLRV DUH WKHPVHOYHV GHWHUPLQLVWLF

7KH VLJQLqFDQFH RI WKH SURFHVV K LV IXUWKHU HYLGHQFHG E\ D NH\ UHJXODUL]DWLRQ SURSHUW\ %HVLGHV WKH ODZ RI RQH SULFH ZH LPSRVH QR UHJXODULW\ FRQGLWLRQ RQ WKH SULFH SURFHVV RI WKH WUDGHG VHFXULWLHV 7KH SURFHVV K UHJXODUL]HV ERWK VHFXULW\ SULFHV DQG WKH YDOXH SURFHVVHV RI WKH VHOI qQDQFLQJ SRUWIROLRV LQ WKH VHQVH WKDW WKH\ DOO KDYH qQLWH YDULDQFH HYHU\ SHULRG RQFH ZH PXOWLSO\ WKHP E\ WKH VTXDUH URRW RI K )XUWKHUPRUH RQFH PXOWLSOLHG E\ K WKH VTXDUH RI WKH YDOXH SURFHVV RI HYHU\ VHOI qQDQFLQJ SRUWIROLR LV D VXEPDUWLQJDOH XS WR WLPH 7 DQG

K LV WKH ODUJHVW SURFHVV HQMR\LQJ WKLV SURSHUW\ 7KLV UHJXODUL]DWLRQ LV LPSOLFLW LI ZH XVH K DV D FRUUHFWLRQ OHQV ZKLFK VKRZV WKDW WKH KFRUUHFWHG SUREDELOLW\ GLVWULEXWLRQV DUH QDWXUDO LQ WKH G\QDPLF PHDQYDULDQFH VHWWLQJ

7KH G\QDPLF QDWXUH RI RXU DQDO\VLV DOORZV XV WR WDFNOH WZR LQYHVWPHQW WLPLQJ LVVXHV :H LQYHVWLJDWH RQ WKH RQH KDQG WKH WLPH FRQVLVWHQF\ UXOHV RI DQ RSWLPDO G\QDPLF VWUDWHJ\ IRU D q[HG LQYHVWPHQW KRUL]RQ 7KH ULVN UHWXUQ SURqOH RI DQ HsFLHQW SRUWIROLR YDULHV WKURXJK WLPH DFFRUGLQJ ERWK WR LWV SDVW SHUIRUPDQFH DQG WR WKH VWRFKDVWLF HYROXWLRQ RI WKH RSWLPDO G\QDPLF 6KDUSH UDWLR $ VLPSOH QXPHULFDO H[DPSOH ZLWK WZR VHFXULWLHV VKRZV WKDW HYHQ

ZKHQ P\RSLD LV RSWLPDO WLPH FRQVLVWHQF\ UXOHV DUH RI WKH HVVHQFH IRU DQ RSWLPDO G\QDPLF VWUDWHJ\

$W D q[HG GDWH RQ WKH RWKHU KDQG DQ HsFLHQW SRUWIROLR LV LQ JHQHUDO QR ORQJHU RSWLPDO DV ZH YDU\ WKH LQYHVWPHQW KRUL]RQ XQOHVV ZH FRUUHFW WKH SUREDELOLWLHV ZLWK WKH KHOS RI K :KHQ WKH FRUUHFWLRQ OHQV LV DSSOLHG WKH H[SHFWHG UHWXUQ WDUJHW RI DQ HsFLHQW SRUWIROLR GHSHQGV RQ WKH KRUL]RQ WKURXJK WKH RSWLPDO G\QDPLF 6KDUSH UDWLR DQG WKHUHIRUH WKURXJK WKH VWRFKDVWLF HYROXWLRQ RI WKH SURFHVV K

:H SURYLGH QRZ DQ LQIRUPDO GHVFULSWLRQ RI RXU PDLQ UHVXOWV DQG ZH UHODWH WKHP WR WKH OLWHUDWXUH 3UHFLVH GHqQLWLRQV DQG UHVXOWV DUH GHYHORSHG LQ WKH DUWLFOH

:H VWXG\ WKH VHW ; _W7 RI G\QDPLF VHOI qQDQFLQJ SRUWIROLRV ; VWDUWLQJ IURP SHULRG W DQG qQLVKLQJ DW KRUL]RQ 7 ZLWK D YDOXH Z₇; ZLWK qQLWH VHFRQG PRPHQW 7KHUH H[LVWV D SRVLWLYH SURFHVV K VXFK WKDW K₇ DQG HYHU\ G\QDPLF SRUWIROLR ; LQ ;_W7 VDWLVqHV

K L K L

K Z_{V V}; _„ ( _V K_{V V} Z ; _„ ( _V Z₇;

IRU HYHU\ SHULRG V EHWZHHQ W DQG 7 b ZKHUH Z _V; GHQRWHV WKH YDOXH RI SRUWIROLR ; DW WLPH V :H FRQVWUXFW WKH ODUJHVW SURFHVV K KDYLQJ WKLV UHJXODUL]DWLRQ SURSHUW\

7KH VHW RI YDOXHV DW KRUL]RQ7 RI SRUWIROLRV LQ; W7 LV FORVHG LQ / $QDORJRXV UHVXOWV DUH GHULYHG LQ D GLVFUHWH WLPH VHWWLQJ E\ 0RWRF]\QVNL ZKLFK H[WHQGV 6FKZHL]HU DQDO\VLV IURP RQH WR VHYHUDO ULVN\ DVVHWV ,Q FRQWLQXRXV WLPH ZH UHIHU WR WKH ZRUNV RI 0RQDW DQG 6WULFNHU DQG 'HOEDHQ HW DO 7KH H[LVWLQJ OLWHUDWXUH ORRNV DW VWUDWHJLHV

ZKRVH YDOXHV DUH LQ / DW DOO WLPHV DQG GHULYHV D VLPLODU UHVXOW E\ UHTXLULQJ WKDW WKH ORFDO PHDQYDULDQFH WUDGHRp EH ERXQGHG :H GR QHHG VXFK DQ DVVXPSWLRQ EHFDXVH ZH GR QRW LPSRVH LQWHJUDELOLW\ FRQVWUDLQWV RQ WKH G\QDPLF VWUDWHJLHV EHIRUH WKH qQDO KRUL]RQ

:H qUVW XVH WKH SURSHUWLHV RI ; _W7 LQ RUGHU WR VWXG\ WKH SUREOHP RI RSWLPDO UHSOLFDWLRQ

LQ LQFRPSOHWH PDUNHWV )RU HYHU\ FRQWLQJHQW FODLP )₇ LQ / DQG HYHU\ ZHDOWK OHYHO Z_W DW WLPH W ZH VROYH WKH IROORZLQJ RSWLPL]DWLRQ SUREOHP

K L

HVVLQI( _W )₇ b Z ₇;

; ;W7

V W

Z W; Z W

7KH HVVHQWLDO LQqPXP DQ LQqPXP LQ WKH IDPLO\ RI UDQGRP YDULDEOHV ZULWHVK_W)_Wb Z _W J_W ZKHUH )_W DQG J_W DUH WZR UDQGRP YDULDEOHV ZKLFK UHSUHVHQW UHVSHFWLYHO\ WKH YDOXH DQG WKH KHGJLQJ TXDOLW\ RI WKH RSWLPDO UHSOLFDWLRQ VWUDWHJ\ 7KLV VWUDWHJ\ FDQ EH REWDLQHG E\ D

VXFFHVVLRQ RI QDLYH RQH SHULRG RSWLPL]DWLRQV SURYLGHG WKDW ZH ZRUN DW WLPHWLQ WKH PRGLqHG KW

SUREDELOLW\ GLVWULEXWLRQ 3 ZKLFK LV GHULYHG IURP WKH RULJLQDO RQH 3 WKURXJK WKH 5DGRQ 1LNRG\P GHULYDWLYH K_W ( >K_W @

7KH OLWHUDWXUH KDV VWXGLHG WKH QRQ FRQGLWLRQDO YHUVLRQ RI WKLV LV SUREOHP DQG KDV LPSRVHG

WKDW WKH HQWLUH YDOXH SURFHVV RI WKH SRUWIROLRV EH LQ / 6FKZHL]HU DQG *RXULxHURX[ DQG /DXUHQW VWXG\ D VLPLODU GLVFUHWH WLPH VHWWLQJ :H HPSKDVL]H WKH UROH RI WKH SURFHVV K LQ WKH FRQVWUXFWLRQ RI WKH RSWLPDO UHSOLFDWLRQ VWUDWHJ\ ZKLFK ZH SURSRVH 7KH FRQWLQXRXV WLPH OLWHUDWXUH LV ZHOO GHYHORSHG DQG ZH UHIHU DPRQJ RWKHUV WR )†ROOPHU DQG 6RQGHUPDQ 'XsH DQG 5LFKDUGVRQ 6FKZHL]HU DQG 6FK†DO 0RQDW DQG 6WULFNHU 5KHLQO†DQGHU DQG 6FKZHL]HU *RXULxHURX[ HW DO 3KDP HW DO /DXUHQW DQG 3KDP DQG %LDJLQL HW DO 2XU PRGHO LV IUHH RI WKH WHFKQLFDO FRQVLGHUDWLRQV DQG UHJXODULW\ FRQGLWLRQV ZKLFK DUH QHHGHG LQ FRQWLQXRXV WLPH IRU WKH FRQVWUXFWLRQ RI WKH VWRFKDVWLF LQWHJUDO RU IRU WKH H[LVWHQFH RI D VROXWLRQ WR WKH UHSOLFDWLRQ SUREOHP

:H QH[W LQYHVWLJDWH WKH PHDQYDULDQFH HsFLHQF\ RI G\QDPLF SRUWIROLRV LQ; _W7 $ GHWDLOHG OLVW RI UHIHUHQFHV RQ WKH VXE MHFW FDQ EH IRXQG LQ /L DQG 1J 7KHVH DXWKRUV VWXG\ D GLVFUHWH WLPH PRGHO ZKHUH VHFXULW\ UHWXUQV DUH VHULDOO\ LQGHSHQGHQW :H OHW

I

( _W>5 _W7; @ _b 5 _W7

T

65_W7;

9DU >_W 5 _W7; @

I

EH WKH FRQGLWLRQDO G\QDPLF 6KDUSH UDWLR RI D SRUWIROLR; LQ; W7 ZKHUH5 W7 DQG5 W7;

Z ₇; Z _W; GHQRWH UHVSHFWLYHO\ WKH ULVN IUHH UDWH DQG WKH UHWXUQ IURP WWR 7 RI SRUWIROLR

; 7KH 6KDUSH UDWLR 65_W7; LV D UDQGRP YDULDEOH ZKLFK GHSHQGV RQ WKH LQIRUPDWLRQ DYDLODEOH DW GDWH W ,WV PD[LPDO YDOXH LV

Y

X I

X 5

X W7

# $

W

65_W7 b

K_W

7KH RSWLPDO 6KDUSH UDWLR REWDLQV IRU SRUWIROLRV RQ WKH FRQGLWLRQDO G\QDPLF PHDQYDULDQFH _{: H UHIHU WR 6 K D US H IR U D E UR D G UHY LHZ R I WK H Q R WLR Q R I 6 K D US H UD WLR}

HsFLHQW IURQWLHU _{( ) W7} ZKLFK DUH VROXWLRQ WR HVVLQI 9DU >_W 5 _W7; @

!

! ; ;W7

!

V W Z _W; Z_W

! ! !

( W>5 W7; @ 5 W7

IRU VRPH H[SHFWHG UHWXUQ WDUJHW_{5 W7} DQG VRPH LQLWLDO ZHDOWKZ _WDW WLPHW 7KH VHWZ_{7( ) W7} RI YDOXHV DW WLPH 7 RI HsFLHQW SRUWIROLRV LQ ( ) _W7 LV FORVHG LQ Z ₇; _W7 8QGHU VRPH UHJXODULW\ FRQGLWLRQV ZH GHULYH D WZR IXQG VHSDUDWLRQ WKHRUHP HYHU\ SRUWIROLR RQ WKH G\QDPLF HsFLHQW IURQWLHU FDQ EH GHVFULEHG DV D EX\ DQG KROG FRPELQDWLRQ RI D ULVN IUHH

ERQG DQG WKH G\QDPLF VHOI qQDQFLQJ SRUWIROLR ZLWK PLQLPXP / QRUP ZLWK ZHLJKWV WKDW PD\ GHSHQG RQ WKH LQIRUPDWLRQ DYDLODEOH DW WKH VWDUW RI WKH VWUDWHJ\ ,Q D FRQWLQXRXV WLPH VHWWLQJ ZLWK FRPSOHWH PDUNHWV %DM HX[^%HVQDLQRX DQG 3RUWDLW GHULYH D VLPLODU WZR IXQG VHSDUDWLRQ WKHRUHP LQ DQ XQFRQGLWLRQDO IRUP ZLWK D WZR VWHS SURFHGXUH 7KH\ GHULYH DQ HsFLHQW SD\Rp DW PDWXULW\ ZKLFK WKH\ WKHQ H[DFWO\ UHSOLFDWH ZLWK D G\QDPLF SRUWIROLR 7KLV VWUDWHJ\ LV QR ORQJHU SRVVLEOH ZKHQ PDUNHWV DUH LQFRPSOHWH

:H H[WHQG WKH QRWLRQ RI G\QDPLF 6KDUSH UDWLR WR DQ\ FRQWLQJHQW FODLP )7 E\ VHWWLQJ I

( _W>)₇@_b 5 )_W

G HI W7

a

S

65_W7)₇

9DU >_W )₇@

ZKHUH )_WLV WKH YDOXH DW WLPHWRI WKH RSWLPDO UHSOLFDWLRQ RI)₇ 7KLV 6KDUSH UDWLR LV DOZD\V VPDOOHU WKDQ WKH RSWLPDO G\QDPLF 6KDUSH UDWLR DQG

w x

K L 65a ₎

W7 7 HVVLQI( W )7 b Z7; 9DU W )7 b _65W7 V W ; _{( ) W7}

a

ZKLFK VKRZV WKDW 65_W7)₇ PHDVXUHV WKH GLVWDQFH IURP WKH FRQWLQJHQW FODLP WR WKH HsFLHQW IURQWLHU

:H VWXG\ WKH EHKDYLRU RI DQ HsFLHQW G\QDPLF SRUWIROLR WKURXJK WLPH ,W UHPDLQV HsFLHQW HYHU\ SHULRG ZLWK DQ H[SHFWHG UHWXUQ XS WR 7 ZKLFK WHQGV WR VPRRWK WKH SDVW SHUIRUPDQFH RI WKH VWUDWHJ\ LQ D ZD\ ZKLFK ZH FKDUDFWHUL]H 7KLV JLYHV DQ DQVZHU LQ WKH PHDQYDULDQFH VHWWLQJ WR WKH LVVXH RI WLPH GLYHUVLqFDWLRQ VWXGLHG IRU LQVWDQFH LQ 6DPXHOVRQ 7KRU OH\ 9DQ (DWRQ DQG &RQRYHU DQG +DQVVRQ DQG 3HUVVRQ /HVV DJ JUHVVLYH EHKDYLRU QHDU WKH LQYHVWPHQW KRUL]RQ LQ RQO\ MXVWLqHG LQDVPXFK DV WKH VWUDWHJ\ RXWSHUIRUPHG WKH ULVNOHVV LQYHVWPHQW LQ WKH SDVW

%HVLGHV LWV UHJXODUL]DWLRQ SURSHUWLHV DQG LWV OLQN ZLWK RSWLPDO G\QDPLF 6KDUSH UDWLRV WKH SURFHVV K KDV DQRWKHU LPSRUWDQW HFRQRPLF LQWHUSUHWDWLRQ WKH ZHLJKW K FDQ EH XVHG DV D FRUUHFWLRQ OHQV IRU P\RSLF LQYHVWRUV )RU WKH WZR PHDQYDULDQFH SURJUDPV ZKLFK ZH VWXG\ WKH UHSOLFDWLRQ DQG WKH SRUWIROLR VHOHFWLRQ D VKRUW VLJKWHG LQYHVWRU DFWV RSWLPDOO\ LQ WKH ORQJ UXQ LI VKH VROYHV WKH PHDQYDULDQFH SUREOHP ZLWK D VKRUW KRUL]RQ LQ D PRGLqHG SUREDELOLW\ GLVWULEXWLRQ ,I VKH FKRRVHV SHULRG V DV LQYHVWPHQW KRUL]RQ VKH VKRXOG XVH WKH YDOXH RI

K DW WLPH V SURSHUO\ QRUPDOL]HG DV D 5DGRQ1LNRG\P GHULYDWLYH LQ RUGHU WR PRGLI\ WKH SUREDELOLW\ GLVWULEXWLRQ 7KLV GRHV QRW PHDQ KRZHYHU WKDW DQ\ FRPELQDWLRQ RI IUDJPHQWV RI HsFLHQW SRUWIROLRV ZLOO IRUP DQ RSWLPDO VWUDWHJ\ 6RPH WLPH FRQVLVWHQF\ UXOHV PXVW EH VDWLVqHG DQG ZH VKRZ WKURXJK D QXPHULFDO H[DPSOH WKDW D VHYHUH ORVV LQ HsFLHQF\ PD\ UHVXOW RWKHUZLVH

:KHQ K LV GHWHUPLQLVWLF DQG RQO\ D IXQFWLRQ RI WLPH QR PRGLqFDWLRQ LQ WKH SUREDELOLW\ GLVWULEXWLRQ LV QHHGHG DQG P\RSLD LV RSWLPDO 7KLV UHVXOW DQVZHUV D ORQJ VWDQGLQJ LVVXH LQ WKH OLWHUDWXUH q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sFLHQW IURQWLHU ,Q WKLV VSHFLDO FDVH RSWLPDO 6KDUSH UDWLRV DJJUHJDWH LQ WLPH FRQYHQLHQWO\ DV IROORZV

Vb_{u v}

<

65 65_{WV X X}

X W

,Q PRUH JHQHUDO VHWWLQJV ZH SURYLGH WLPH DJJUHJDWLRQ UHVXOWV ZKLFK LQYROYH 6KDUSH UDWLRV FRPSXWHG LQ WKH PRGLqHG SUREDELOLW\ GLVWULEXWLRQV

7KH DUWLFOH LV RUJDQL]HG DV IROORZV 6HFWLRQ GHVFULEHV WKH PDUNHW VWUXFWXUH DQG WKH G\QDPLF SRUWIROLRV 6HFWLRQ GHDOV ZLWK WKH RSWLPDO UHSOLFDWLRQ RI FRQWLQJHQW FODLPV ZKLOH 6HFWLRQ VWXGLHV WKH PHDQYDULDQFH SRUWIROLR VHOHFWLRQ SUREOHP 6HFWLRQ LQYHVWLJDWHV WKH WLPH FRQVLVWHQF\ RI DQ RSWLPDO VWUDWHJ\ DQG VWXGLHV WKH XVH RI K DV D FRUUHFWLRQ OHQV 6HFWLRQ GHVFULEHV KRZ RSWLPDO G\QDPLF 6KDUSH UDWLRV DJJUHJDWH WKURXJK WLPH DQG FKDU

DFWHUL]HV VLWXDWLRQV ZKHUH P\RSLD LV RSWLPDO 6HFWLRQ SURYHV ZLWK D VLPSOH QXPHULFDO H[DPSOH WKDW WLPH FRQVLVWHQF\ LV RI WKH HVVHQFH IRU DQ HsFLHQW VWUDWHJ\ 6HFWLRQ FRQFOXGHV ZLWK VRPH UHPDUNV RQ WKH FKRLFH RI DQ LQYHVWPHQW KRUL]RQ 0RVW SURRIV FDQ EH IRXQG LQ WKH DSSHQGLFHV

' \ Q D P LF 3 R U W I R O LR V

0 D UN HW 6 WUX FWX UH

:H FRQVLGHU D qQLWH QXPEHUQ RI VHFXULWLHV WUDGHG LQ D IULFWLRQOHVV DQG FRPSHWLWLYH PDUNHW RYHU D VHW RI GLVFUHWH WLPHV ZLWK qQLWH KRUL]RQ :H LQGH[ WKH WUDGLQJ GDWHV E\ WKH LQWHJHUV

G HI

EHWZHHQ DQG D qQDO KRUL]RQ 7 ,QIRUPDWLRQ LV GHVFULEHG E\ D qOWUDWLRQ ) I) JW „W„7 RYHU D SUREDELOLW\ VSDFH o_{) 7}3

7KURXJKRXW WKH DUWLFOH HTXDOLWLHV DQG LQHTXDOLWLHV EHWZHHQ UDQGRP YDULDEOHV DUH XQGHU VWRRG WR KROG 3 DOPRVW VXUHO\ :H GHQRWH UHVSHFWLYHO\ ( >) @ DQG ( _W>) @ WKH H[SHFWHG YDOXH

DQG WKH FRQGLWLRQDO H[SHFWDWLRQ ZLWK UHVSHFW WR) WRI D UDQGRP YDULDEOH) LQ / 3 :H OHW

Q

/_W3 DQG/ _W3 ,5 EH UHVSHFWLYHO\ WKH VSDFH RI UDQGRP YDULDEOHV DQG UDQGRP YHFWRUV LQ

Q

5

, ZKLFK DUH ERWK PHDVXUDEOH ZLWK UHVSHFW WR _{) W} DQG LQ / 3 ,II UHVS I_W LV SRVLWLYH

S S

DQG PHDVXUDEOH ZLWK UHVSHFW WR ) ₇ UHVS ) _W ZH OHW / 3 I UHVS /_W3 I_W EH WKH

S S

VHW RI UDQGRP YDULDEOHV ) VXFK WKDW I ) UHVS I )_W EHORQJV WR / 3 UHVS /_W3

S S

Q Q Q

:H GHqQH LQ WKH VDPH ZD\ / 3 I ,5 DQG /_W3 I_W ,5 IRU UDQGRP YHFWRUV LQ ,5 S

:H FRQVLGHU RQ / 3 I WKH WRSRORJ\ GHULYHG IURP WKH VFDODU SURGXFW ( >I ) ) @ IRU WZR S

IXQFWLRQV ) DQG ) LQ / 3 I

I I

,II LV D SRVLWLYH UDQGRP YDULDEOH LQ/ 3 ZH GHQRWH3 DQG( WKH SUREDELOLW\ GLVWULEX WLRQ DQG LWV FRUUHVSRQGLQJ H[SHFWDWLRQ RSHUDWRU REWDLQHG IURP WKH RULJLQDO SUREDELOLW\ GLV WULEXWLRQ3 WKURXJK WKH SRVLWLYH 5DGRQ1LNRG\P GHULYDWLYHI ( >I@ )RU HYHU\ UDQGRP YDUL

I

I

DEOH ) VXFK WKDW I ) LV LQ / 3 ZH KDYH ( >) @ ( >I ) @ ( >I@ DQG( _W>) @ ( _W>I ) @ ( _W>I@

I I I

:H GHQRWH YDU >_W ) @ ( _W>) @ b ( _W>) @ WKH FRQGLWLRQDO YDULDQFH DW WLPH W RI D UDQGRP S

Q

YDULDEOH ) LQ / 3 I )RU WZR YHFWRUV [ DQG\ LQ ,5 ZH GHQRWH [ DQG[ \ UHVSHFWLYHO\ WKH WUDQVSRVH RI[ DQG WKH XVXDO VFDODU SURGXFW EHWZHHQ [ DQG \

$Q XQVSHFLqHG QXPHUDLUH LV q[HG HYHU\ SHULRG DQG ZH OHW S_WEH WKH YHFWRU RI SULFHV RI WKH

Q VHFXULWLHV LQ WKLV QXPHUDLUH DW WLPH W :H OHW G_W EH WKH QXPHUDLUH GLYLGHQG GLVWULEXWHG E\ WKH VHFXULWLHV DW WLPH W 7KH RZQHU RI RQH XQLW RI VHFXULW\ LDW WLPH WLV HQWLWOHG WR UHFHLYH

G HI L

WKH GLYLGHQG G_W LQ QXPHUDLUH WKH QH[W SHULRG :H OHW ƒ_W S_WG_W EH WKH FXPGLYLGHQG

SULFH YHFWRU RI WKH VHFXULWLHV DW WLPH W 7KH YHFWRU SURFHVVHV _IS_{WJ „W„7 I}G_{WJ „W„7} DQG Iƒ_{W „W„7J} DUH DGDSWHG WR WKH qOWUDWLRQ ₎

:H GR QRW UXOH RXW WKDW VRPH VHFXULW\ PLJKW EH UHGXQGDQW DW VRPH WUDGLQJ SHULRG DQG LQ VRPH VWDWH RI WKH ZRUOG EXW ZH GR LPSRVH WKDW WKH ODZ RI RQH SULFH KROGV 7KLV LV D YHU\ ZHDN IRUP RI DEVHQFH RI DUELWUDJH DV H[SODLQHG IRU LQVWDQFH LQ +DQVHQ DQG 5LFKDUG )RU WKH UHPDLQGHU RI WKH DUWLFOH ZH VKDOO DVVXPH WKDW WKH IROORZLQJ WZR DVVXPSWLRQV DUH VDWLVqHG

$ VV X P S WLR Q 3 ULFHV D Q G UHWX UQ V R I WK H VHFX ULWLHV G R Q R W YD Q LVK ) R U HYHU\ SHULRG W

EHWZ HHQ D Q G 7 D Q G IR U HYHU\ SHULRG V EHWZ HHQ D Q G 7 WK H S ULFH YHFWR UV SW D Q G ƒV D UH 3 D OP R VW VX UHO\ G Lp HUHQ W IUR P WK H Q X OO YHFWR U

$ VV X P S WLR Q / D Z R I R Q H S ULFH ) R U HYHU\ SHULRG WEHWZ HHQ D Q G 7 _b D Q G IR U HYHU\

Q

UD Q G R P YHFWR UV; _WD Q G <_WLQ ,5 P HD VX UD EOH Z LWK UHVSHFW WR ) _W WK H HTX D OLW\ ƒ_W ; _W ƒ_W <_W

LP S OLHV S ;_{W W} S <_{W W}

6 HOI ) LQ D Q FLQ J 3 R UWIR OLR V

Q

$ G\QDPLF SRUWIROLR ; VWDUWLQJ DW WLPH W LV D SURFHVV LQ ,5 DGDSWHG WR ₎ DQG LQGH[HG L

E\ WLPH V ZLWK W „ V „ 7 ZKHUH ; V UHSUHVHQWV WKH QXPEHU RI XQLWV RI VHFXULW\ L KHOG LQ SRUWIROLR ; DW WLPHV $ QHJDWLYH TXDQWLW\ FRUUHVSRQGV WR D VKRUW VDOH :H OHWZ ; EH WKH

G HI YDOXH SURFHVV RI SRUWIROLR ; QDWXUDOO\ GHqQHG E\Z _V; S ;_{V V}

:H VD\ WKDW D G\QDPLF SRUWIROLR ; VWDUWLQJ DW WLPH WLV VHOI qQDQFLQJ DW WLPH V ZKHQHYHU

Z _V; ƒ ;_{V Vb} DQG WKDW LW LV VHOI qQDQFLQJ ZKHQHYHU LW LV VHOI qQDQFLQJ IURP W WR 7 7KH VHOI qQDQFLQJ FRQGLWLRQ LV DQ DFFRXQWLQJ HTXDOLW\ ZKLFK VWDWHV WKDW YDOXH LV SUHVHUYHG ZKHQ WKH SRUWIROLR LV UHEDODQFHG

,W LV HDVLO\ FKHFNHG WKDW WKH ODZ RI RQH SULFH LPSOLHV WKDW WZR VHOI qQDQFLQJ SRUWIROLRV ZLWK LGHQWLFDO qQDO YDOXH DW WLPH 7 VKDUH WKH VDPH YDOXH SURFHVV 7KLV SURSHUW\ ZLOO ODWHU DOORZ XV WR LGHQWLI\ WZR VXFK G\QDPLF SRUWIROLRV

7KH PDLQ SXUSRVH RI WKLV VHFWLRQ LV WR FKDUDFWHUL]H WKH VHW RI VHOI qQDQFLQJ G\QDPLF SRUW S

IROLRV VWDUWLQJ DW WLPHWZLWK WKH SURSHUW\ WKDW WKHLU qQDO ZHDOWK DW WLPH7 LV LQ/ 3 K₇

ZKHUH K7 LV D SRVLWLYH UDQGRP YDULDEOH PHDVXUDEOH ZLWK UHVSHFW WR ) 7 :H GHQRWH ;W7 S

G HI

WKLV VHW DQG ZH OHW Z _{7; W7 I}Z ₇; ; _{; W7J} EH WKH VHW LQ / 3 K₇ RI WHUPLQDO YDOXHV RI SRUWIROLRV LQ ;_W7 %HVLGHV WKH VHOI qQDQFLQJ FRQGLWLRQ ZH LPSRVH QR UHVWULFWLRQ RQ WKH YDOXH SURFHVV RI WKH SRUWIROLRV DW SHULRGV SULRU WR WKH qQDO KRUL]RQ

7KH qQDO ZHLJKW K₇ FDQ EH XVHG LQ VLWXDWLRQV ZKHUH VHFXULW\ SULFHV ƒ₇ GR QRW KDYH qQLWH VHFRQG PRPHQWV DW WKH qQDO GDWH 7 7KH ZHLJKW K₇ FDQ WKHQ EH FKRVHQ VR DV WR GHFUHDVH WKH VL]H RI WKH WDLOV LQ WKH SUREDELOLW\ GLVWULEXWLRQ RI ƒ₇ ,I VHFXULW\ SULFHV DW WLPH 7 KDYH qQLWH VHFRQG PRPHQWV DQG LQ SDUWLFXODU LI ZH ZLVK WR LQYHVWLJDWH WKH PHDQYDULDQFH WUDGHRp DW KRUL]RQ 7 LQ WKH SUREDELOLW\ 3 LW ZLOO EH QDWXUDO WR VHW K₇ 7KLV LV DFWXDOO\ KRZ ZH SUHVHQWHG RXU UHVXOWV LQ WKH ,QWURGXFWLRQ $QRWKHU PRWLYDWLRQ IRU WKH XVH RI D JHQHUDO qQDO ZHLJKW K₇ ZLOO EHFRPH FOHDU ODWHU RQ ZKHQ ZH VKRZ WKDW LW DOORZV XV WR NHHS D FOHDU V\PPHWU\ EHWZHHQ WKH qQDO KRUL]RQ DQG WKH LQWHUPHGLDWH WUDGLQJ SHULRGV

7KH IROORZLQJ SURSRVLWLRQ EXLOGV D SRVLWLYH SURFHVV K E\ EDFNZDUG LQGXFWLRQ IURP WKH qQDO ZHLJKW K₇ 7KLV SURFHVV SOD\V D FHQWUDO UROH LQ WKH GHVFULSWLRQ RI WKH VWUXFWXUH RI; _W7 DQG PRUH JHQHUDOO\ LQ WKH PHDQYDULDQFH DQDO\VLV :H VKDOO VHH ODWHU WKDW LW LV FORVHO\ OLQNHG WR WKH QRWLRQ RI RSWLPDO G\QDPLF 6KDUSH UDWLR DQG WKDW LW FDQ EH LQWHUSUHWHG DV D FRUUHFWLRQ OHQV IRU P\RSLF LQYHVWRUV ,W ZLOO WKHQ EH XVHIXO WR UHO\ RQ D JHQHUDO VHWWLQJ ZKHUH WKH qQDO ZHLJKW K₇ DW WLPH 7 PD\ EH GLpHUHQW IURP RQH

Q

:H GHQRWH 1 WKH 0RRUH3HQURVH JHQHUDOL]HG LQYHUVH RI D V\PPHWULF PDWUL[ 1 LQ ,5 d

Q

5

, 7KH PDWUL[ 1 LV LWVHOI V\PPHWULF FRPPXWHV ZLWK1 DQG VDWLVqHV

1 1 1 1

1 1 1 1

,I1 LV D UDQGRP PDWUL[ PHDVXUDEOH ZLWK UHVSHFW WR) _W WKHQ 1 LV GHqQHG IRU HYHU\

LQ o DQG 1 LV DOVR PHDVXUDEOH ZLWK UHVSHFW WR _)W

3 U R S R VLWLR Q 7 K H D G D S WHG S URFHVV K G Hq Q HG E\ WK H ED FNZ D UG HTX D WLR Q

u vb

G HI

K S 1 S

_{W W W W}

e f

G HI

Z LWK 1 _W ( _W K_{W W W} ƒ ƒ IR U „ W_„ 7 _b LV Z HOO G Hq Q HG 3 D OP R VW VX UHO\ SR VLWLYH

S

Q

D Q G VD WLVq HVƒ_W /_W3 K_W ,5 IR U HYHU\ SHULRG WEHWZ HHQ D Q G 7 D V VRR Q D V WK H IR OOR Z LQ J WZ R FR Q G LWLR Q V D UH P HW

S

Q

D ƒ₇ / 3 K₇ ,5

S

Q

E G_W /_W3 K_W ,5 IR U HYHU\ SHULRG W Z LWK „ W„ 7 b

6 HH 7 K HLO IR U D J HQ HUD O G HVFULS WLR Q R I WK H 0 R R UH3 HQ UR VH LQ Y HUVH

7 K H IR OOR Z LQ J S UR SHUWLHV WK HQ K R OG

L ) R U HYHU\ G \ Q D P LF SR UWIR OLR ; _; W7 WK H S URFHVV IK ZV V; _JW„V„7 LV D VX EP D UWLQ JD OH WK D W LV IR U HYHU\ SHULRG V Z LWK W„ V „ 7 b Z H K D YH

K L K L

K Z_{V V}; „ ( _V K_{V V} Z ; „ ( _V K Z_{7 7};

LL 7 K H VHW; _W7 LV WK H VHW R I VHOI q Q D Q FLQ J G \ Q D P LF SR UWIR OLR V VWD UWLQ J D W WLP H WVX FK WK D W

S

Z V; /V3 KV IR U HYHU\ SHULRG W_„ V _„ 7

S

LLL 7 K H VHWZ ₇; _W7 LV FOR VHG LQ / 3 K₇

$OWKRXJK WKH SURRI RI 3URSRVLWLRQ FDQ EH IRXQG LQ WKH DSSHQGL[ ZH UHSRUW KHUH DQ LPSRUWDQW OHPPD ZKLFK SOD\V D FHQWUDO UROH LQ WKLV SURRI

/ H P P D / HW i EH D €D OJHEUD R Q WK H S UR ED ELOLW\ VSD FH o3 / HW + ) D Q G * EH

Q

UD Q G R P YD ULD EOHV D Q G p D Q G = EH D UD Q G R P YHFWR UV LQ ,5 Z K LFK VD WLVI\ WK H IR OOR Z LQ J WK UHH FR Q G LWLR Q V

D p LV P HD VX UD EOH Z LWK UHVSHFW WR i D Q G ER WK p D Q G = D UH 3 D OP R VW VX UHO\ G Lp HUHQ W IUR P WK H ]HUR YHFWR U

Q

E ) R U HYHU\ YHFWR U [ LQ ,5 P HD VX UD EOH Z LWK UHVSHFW WR i WK H HTX D OLW\ = [ LP S OLHV

p [

S _Q S

F + ! D Q G + = / 3 ,5 + ) / 3 * / 3

G HI

,I Z H G HQ R WH 1 ( >+ = = Mi@ WK HQ WK H IR OOR Z LQ J S UR SHUWLHV K R OG

Q

L ( YHU\ UD Q G R P YHFWR U [ LQ ,5 P HD VX UD EOH Z LWK UHVSHFW WR i VD WLVq HV = [ = 1 1 [

LL p 1 1 p D Q G p 1 p !

Q

LLL ( YHU\ UD Q G R P YHFWR U [ LQ ,5 P HD VX UD EOH Z LWK UHVSHFW WR i VD WLVq HV

K L

c d c d

K I _b p [ J _„ ( + ) _b = [ * _Mi

Z LWK

c db

G HI

K p 1 p

G HI

I p 1 ( >+ ) = Mi@

K L

e f

G HI

J ( >* Mi@ ( + ) Mi b ( + ) = Mi 1 ( >+ ) = Mi@

"

LY ( TX D OLW\ R EWD LQ V LQ ,Q HTX D OLW\ IR U [ [ Z Z LWK

G HI

"

[ Z KZ _b I1 p 1 ( >+ ) = _Mi@ Z K HUH Z LV D Q \ UD Q G R P YD ULD EOH P HD VX UD EOH Z LWK UHVSHFW WR i

S

Y J _w ( >* _Mi@J LV LQ / 3 D Q G K I LV LQ / 3

&RQGLWLRQ E RI 3URSRVLWLRQ LQYROYHV WKH YDULDEOHK_WZKLFK LV GHULYHG UHFXUVLYHO\ WKURXJK (TXDWLRQ 7KH IROORZLQJ OHPPD JLYHV D VXsFLHQW FRQGLWLRQ LQ WHUPV RI WKH qQDO ZHLJKW

K₇ RQO\

S

Q

/ H P P D ,Iƒ₇ LV D Q HOHP HQ W R I/ 3 K₇ ,5 & R Q G LWLR Q D R I 3 UR SR VLWLR Q WK HQ

S

Q

G_W EHOR Q JV WR /_W3 K_W ,5 IR U HYHU\ SHULRG WIUR P WR 7 b & R Q G LWLR Q E R I 3 UR SR

N N

VLWLR Q LI R Q H VHFX ULW\ VD \ 6 HFX ULW\ N SD \ V Q R G LYLG HQ G D Q G LV VX FK WK D WS S7 WGW LV D Q

S

Q

HOHP HQ W R I/ 3 K₇ ,5 IR U HYHU\ SHULRG W IUR P WR 7 b

)RU WKH UHPDLQGHU RI WKH DUWLFOH ZH DVVXPH WKDW &RQGLWLRQV D DQG E RI 3URSRVLWLRQ DUH VDWLVqHG VR WKDW WKH UHVXOWV RI WKLV SURSRVLWLRQ DSSO\

$ VV X P S WLR Q & R Q G LWLR Q V D D Q G E R I 3 UR SR VLWLR Q D UH VD WLVq HG

:H UHPDUN WKDW WKH SURRI RI 3URSRVLWLRQ \LHOGV WZR HTXDWLRQV WKDW ZLOO SURYH XVHIXO )RU HYHU\ SHULRG WEHWZHHQ DQG 7 b

S 1 1 S

_{W W W W}

DQG IRU HYHU\ SRUWIROLR ; LQ ; _W7 DQG HYHU\ SHULRG V EHWZHHQ WDQG 7 b

Z ; ƒ ; ƒ 1 1 ;

_{V V V V V V V}

3URSRVLWLRQ VKRZV WKDW K DFWV DV D ZHLJKW ZKLFK UHJXODUL]HV WKH SULFHV DQG WKH YDOXHV RI WKH VHOI qQDQFLQJ SRUWIROLRV LQ ;_W7 HYHU\ SHULRG RQFH ZH PXOWLSO\ WKHVH SURFHVVHV E\ WKH VTXDUH URRW RI K WKH\ DOO KDYH qQLWH VHFRQG PRPHQWV HYHU\ SHULRG 7KH XVH RI D JHQHUDO ZHLJKW K₇ DW WLPH 7 DOORZV XV WR WUHDW V\PPHWULFDOO\ WKH qQDO KRUL]RQ DQG HDUOLHU WUDGLQJ SHULRGV

6FKZHL]HU DQG 0RWRF]\QVNL VWXG\ G\QDPLF VHOI qQDQFLQJ SRUWIROLRV ZKRVH HQWLUH YDOXH SURFHVVHV KDYH qQLWH VHFRQG PRPHQWV DW HYHU\ SHULRG 6FKZHL]HU H[KLELWV

LQ WKLV VHWWLQJ DQ H[HPSOH GXH WR : 6FKDFKHUPD\HU ZKHUH FRQWUDU\ WR RXU UHVXOW WKH VHW RI

qQDO YDOXHV RI VHOI qQDQFLQJ SRUWIROLRV LV QRW FORVHG LQ / 3 ,Q RUGHU WR GHULYH WKDW WKLV VHW LV FORVHG 6FKZHL]HU LQWURGXFHV D NH\ UHJXODULW\ DVVXPSWLRQ FDOOHG WKH QRQGHJHQHUDF\ FRQGLWLRQ ZKLFK LQ RXU VHWWLQJ FRUUHVSRQGV WR UHTXLULQJ WKDW WKH SURFHVV K EH ERXQGHG 0RWRF]\QVNL JHQHUDOL]HV WKLV DSSURDFK IURP RQH WR VHYHUDO ULVN\ VHFXULWLHV :KHQ WKLV FRQGLWLRQ LV VDWLVqHG ZH UHPDUN WKDW RXU DQDO\VLV \LHOGV SULFHV DQG SRUWIROLR YDOXHV LQ

/ 3 HYHU\ SHULRG DV ZHOO DQG ZH UHFRYHU WKHLU UHVXOW :KHQ K LV QRW ERXQGHG KRZHYHU

ZH VWLOO REWDLQ WKDW Z ₇; _W7 LV FORVHG LQ / 3 E\ GURSSLQJ WKH UHTXLUHPHQW RQ WKH VHFRQG PRPHQWV SULRU WR WKH qQDO KRUL]RQ

:H DUH QRZ UHDG\ WR VWXG\ WKH SURSHUWLHV RI WKH VHOI qQDQFLQJ SRUWIROLRV LQ; _W7 :H VWDUW ZLWK WKH UHSOLFDWLRQ RI D FRQWLQJHQW FODLP LQ 6HFWLRQ DQG ZH SURFHHG ZLWK WKH VHOHFWLRQ RI DQ HsFLHQW SRUWIROLR LQ 6HFWLRQ ,Q ERWK FDVHV ZH VKDOO FRQVLGHU DV ORVV RU RE M HFWLYH

S

IXQFWLRQ DW WLPH 7 WKH QRUP RI/ 3 K₇ ZKLFK LV ZHOO GHqQHG IRU SRUWIROLRV LQ ;_W7 $V PHQWLRQHG DERYH WKH JHQHUDO SRVLWLYH ZHLJKW K₇ WUHDWV V\PPHWULFDOO\ WKH qQDO KRUL]RQ DQG LQWHUPHGLDWH WUDGLQJ SHULRGV 7KLV ZLOO SURYH XVHIXO DV ZH VWXG\ ODWHU WKH HpHFW RI D FKDQJH RI LQYHVWPHQW KRUL]RQ

2 S W LP D O 5 H S O LF D W LR Q

7KLV VHFWLRQ LQYHVWLJDWHV WKH UHSOLFDWLRQ SURSHUWLHV RI WKH VHOI qQDQFLQJ G\QDPLF SRUWIROLRV :H qUVW VKRZ KRZ WR FRQVWUXFW D G\QDPLF VWUDWHJ\ ZKLFK EHVW UHSOLFDWHV D FODLP )₇ DW

S

KRUL]RQ 7 IRU WKH QRUP RI / 3 K₇ VWDUWLQJ IURP D ZHDOWK Z _W DW WLPH W :H WKHQ VWXG\ WKH FRVW RI WKH RSWLPDO UHSOLFDWLRQ DQG ZH VKRZ WKDW WKH YDOXH SURFHVV RI WKH RSWLPDO VROXWLRQ LV XQLTXH :KHQ WKH qQDO SD\Rp )₇ LV ]HUR ZH REWDLQ DV D VSHFLDO FDVH WKH

PLQLPXP / SRUWIROLR ZKLFK LV WKH KHGJLQJ QXPHUDLUH XVHG E\ *RXULHURX[ HW DO :H LQWURGXFH LQWHUHVW UDWHV E\ PHDQV RI GHIDXOW IUHH ]HUR FRXSRQ ERQGV :H SURYH WKDW K LV WKH ODUJHVW SURFHVV ZKLFK UHJXODUL]HV WKH YDOXH SURFHVV RI G\QDPLF SRUWIROLRV LQ ;_W7 DQG ZH UHODWH RXU ZRUN ZLWK WKH FRQFHSW RI YDULDQFHRSWLPDO VLJQHG PDUWLQJDOH PHDVXUH LQWURGXFHG E\ 6FKZHL]HU 6LQFH ERWK LQWHUHVW UDWHV DQG WKH UHSOLFDWLRQ VWUDWHJ\ DUH QXPHUDLUH VHQVLWLYH LW LV QDWXUDO WR LQYHVWLJDWH WKH HpHFW RI D FKDQJH RI QXPHUDLUH

& R Q VWUX FWLR Q R I D Q 2 S WLP D O 5 HS OLFD WLR Q

7KH RSWLPDO UHSOLFDWLRQ VWUDWHJ\ RI D SD\Rp)₇ LV REWDLQHG E\ D WZR VWHS SURFHGXUH :H qUVW FRQVWUXFW D SURFHVV) E\ EDFNZDUG LQGXFWLRQ ZKLFK UHSUHVHQWV WKH RSWLPDO YDOXH RI WKH UHSOLFDWLRQ :H WKHQ XVH ) LQ RUGHU WR EXLOG DQ RSWLPDO KHGJLQJ VWUDWHJ\ LQ D IRUZDUG DQG SDWK GHSHQGHQW ZD\

3 U R S R VLWLR Q ) R U HYHU\ SHULRG WVX FK WK D W _„ W_„ 7 _b IR U HYHU\ LQ LWLD O Z HD OWK Z W LQ

S S

/_W3 K_W D Q G IR U HYHU\ SD \ R p )₇ LQ / 3 K₇

ˆ z

K L u v

WZ )W 7

HVVLQ I( _W K₇ )_{7 b} Z₇; ( _W K₇ )_{7 b} Z₇; K_W)_Wb Z_W J _W

_{; ;}

W7

V W

Z W; Z W

G HI

)_W D Q G J_W D UH G Hq Q HG E\ ED FNZ D UG LQ G X FWLR Q E\ J₇ D Q G IR U W„ V „ 7 b

G HI

)_V S 1_{V V} ( _V>K_{V V V} ) ƒ @

K L _{e f}

G HI

J_V ( _V>J_V @ ( _V K_V )_V b ( _V K_{V V} ) ƒ_V 1 _V ( _V>K_{V V V} ) ƒ @

S

) R U HYHU\ SHULRG V EHWZ HHQ WLP H W D Q G 7 b )_V /_V3 K_V J_V LV P HD VX UD EOH Z LWK

WZ )W 7

UHVSHFW WR ) VJV / 3 D Q G JV _w 7 K H G \ Q D P LF SR UWIR OLR ; LV G Hq Q HG UHFX UVLYHO\ E\

G HI

WZ )W 7

; K Z _b ) 1 S 1 ( >K ) ƒ @

_{W W W W W W W W W W W}

u v

G HI WZ )

WZ )W 7 W 7

; K ƒ ; b ) 1 S 1 ( >K ) ƒ @

_{V V V Vb V V V V V V V V}

WZ )W 7 Q

IR U W „ V „ 7 b D Q G D W WLP H 7 Z H OHW ; ₇ EH D Q \ UD Q G R P YHFWR U LQ ,5

WZ )W WZ )W

7 7

P HD VX UD EOH Z LWK UHVSHFW WR ) 7 VX FK WK D W S ;7 7 ƒ ;7 7 b 7 K H G \ Q D P LF SR UWIR OLR

WZ )W 7

; EHOR Q JV WR ; _W7 VWD UWV D W WLP H W Z LWK LQ LWLD O Z HD OWK Z _W D Q G VD WLVq HV

ˆ z

u _{WZ )} v u _{WZ )} v

W 7 W 7

( K ) _b Z ; K ) _b Z ; J

_{V 7 7 7 V V V V}

IR U HYHU\ SHULRG V EHWZ HHQ W D Q G 7

S

D E D E

,W LV HDVLO\ FKHFNHG WKDW LI )₇ )₇ DQG )₇ DUH LQ / 3 K₇ LIZ _W Z_W DQG Z _W DUH LQ

S S

/_W3 K_W DQG LIr_W LV PHDVXUDEOH ZLWK UHVSHFW WR _{) W} ZLWK r )_{W 7} LQ / 3 K₇ DQG r Z_{W W}

S

LQ / _W3 K_W WKHQ

D D E E D E D E

WZ )_{W 7} WZ )_{W 7} WZ Z ) )_{W W 7 W}

; ; ;

WZ )W 7 Wr Z r )W W W 7

r ; ;

_W

,W LV FOHDU IURP 3URSRVLWLRQ WKDW WKH RSWLPL]DWLRQ SURJUDP

K L

HVVLQI( _W K₇ )₇ b Z ₇; V W ; ;W7

W) )W 7

LV VROYHG LQ ; ZLWK J_WDV RSWLPDO YDOXH 7KH LQLWLDO YDOXH )_WFDQ EH VHHQ DV WKH LQLWLDO FRVW RI WKH EHVW UHSOLFDWLRQ VWUDWHJ\ ZKLOH J_W GHVFULEHV WKH TXDOLW\ RI WKLV UHSOLFDWLRQ

)RU D SHULRG V EHWZHHQ WDQG 7 WKH UDQGRP YDULDEOHV )_V DQG J_V DUH GHULYHG E\ EDFNZDUG LQGXFWLRQ IURP WKH SD\Rp )₇ DQG WKH\ GR QRW GHSHQG RQ WKH VWDUWLQJ WLPH W 7KH\ DUH IXUWKHUPRUH UHVSHFWLYHO\ OLQHDU DQG TXDGUDWLF LQ WKH SD\Rp)₇ 7KHVH UHPDUNV DOORZ XV WR FRQVWUXFW D OLQHDU RSHUDWRU 4 _WDQG D TXDGUDWLF RSHUDWRU* _WIRU HYHU\ SHULRGWEHWZHHQ DQG

S

7 IURP WKH VSDFH RI UDQGRP YDULDEOHV LQ/ 3 K7 WR WKH VSDFH RI UDQGRP YDULDEOHV PHD

G HI G HI

VXUDEOH ZLWK UHVSHFW WR _)W VXFK WKDW 4 _W)₇ )_WDQG * _W)₇ J_W DV GHqQHG UHFXUVLYHO\ S

LQ 3URSRVLWLRQ 7KLV SURSRVLWLRQ VKRZV WKDW 4 _W)₇ EHORQJV WR /_W3 K_W ZKLOH * ₇)₇

LV DQ HOHPHQW RI/ 3 $W WLPH 7 WKH RSHUDWRUV 4 ₇ DQG * ₇ DUH WULYLDOO\ UHVSHFWLYHO\ WKH LGHQWLW\ DQG WKH QXOO RSHUDWRU (TXDWLRQ RI 3URSRVLWLRQ ZULWHV

ˆ _{u v}z _{u v}

WZ )W 7 WZ )W 7

* _V)₇ ( _V K₇ )₇ b Z₇ ; b K_V 4 _V)₇b Z_V;

S

IRU DQ\ LQLWLDO ZHDOWK OHYHO Z_W LQ /_W3 K_W ,Q SDUWLFXODU IRU V W DQG Z_W 4 _W)₇ ZH REWDLQ

ˆ _{u v}z

W4 ) )W 7 7

* ) ( K ) b Z ;

_{W 7 W 7 7 7}

7KH IROORZLQJ OHPPD OLVWV VRPH SURSHUWLHV RI WKHVH RSHUDWRUV

/ H P P D / HW V D Q G W EH WZ R SHULRG V VX FK WK D W W„ V „ 7 OHW)₇ D Q G Z _W EH WZ R UD Q G R P

S S

YD ULD EOHV UHVSHFWLYHO\ LQ / 3 K₇ D Q G LQ /_W3 K_W

L ) R U HYHU\ G \ Q D P LF SR UWIR OLR ; LQ ;_W74 _VZ ₇; Z _V; LL * _W)₇ LI D Q G R Q O\ LI)₇ Z ₇; _W7

e f

WZ W WZ W

LLL K Z_{V V}; 4 _V)₇ ( _V K Z_{7 7}; )₇

LY ) R U HYHU\ G \ Q D P LF SR UWIR OLR ; LQ ;_V7

u v K u v L

WZ )W 7 WZ )W 7

K_V 4 _V)₇ b Z _V; Z_V; ( _V K₇ )₇ b Z ₇; Z₇;

3URSHUWLHV L DQG LL VLPSO\ VWDWH WKDW H[DFW UHSOLFDWLRQ FDQ RQO\ EH DFKLHYHG IRU D FRQWLQJHQW FODLP ZKLFK LV WKH qQDO YDOXH RI D VHOI qQDQFLQJ SRUWIROLR 3URSHUWLHV LLL DQG LY

WZ )W 7 UHVXOW IURP WKH RSWLPDOLW\ RI;

8 Q LFLW\ R I WK H 2 S WLP D O 5 HS OLFD WLR Q

7KH QH[W UHVXOW VKRZV WKDW WKH RSWLPL]DWLRQ SUREOHP RI 3URSRVLWLRQ KDV D XQLTXH VROXWLRQ DW OHDVW LQ WHUPV RI YDOXH DW WLPH 7 DQG WKHUHIRUH DOVR LQ WHUPV RI YDOXH SURFHVV :H UHFDOO WKDW ZH FDQQRW H[SHFW WR REWDLQ D XQLTXH SRUWIROLR EHFDXVH ZH GR QRW UXOH RXW UHGXQGDQF\ EHWZHHQ WKH VHFXULWLHV

S

/ H P P D : H FR Q VLG HU D SHULRG WEHWZ HHQ D Q G 7 b D Q LQ LWLD O Z HD OWK Z _WLQ /_W3 K_W

S

D Q G D FR Q WLQ JHQ W FOD LP )₇ LQ / 3 K₇ ) R U HYHU\ G \ Q D P LF SR UWIR OLR < LQ ; _W7 WK H

WZ )W 7

HTX D OLW\ Z ₇< Z ₇; K R OG V 3 D OP R VW VX UHO\ R Q WK H VHW $ _W< LQ ) _W G Hq Q HG E\

k e f

G HI

$ _W< o VX FK WK D W ( _W K₇)_{7 b} Z ₇< K_W4 _W)_7b Z_W * _W)₇

D Q G Z _W< Z _WJ

/ 0 LQ LP X P 3 R UWIR OLR

2QH VSHFLDO FKRLFH RI qQDO SD\Rp)₇ DQG LQLWLDO ZHDOWK Z _WDW WLPHWZLOO SURYH LPSRUWDQW S

LW LV REWDLQHG IRU )₇ DQG Z _W K_W 1RWLFH WKDW IRU WKLV FKRLFH RI LQLWLDO ZHDOWK S

S _WG HI _{W K W}

K Z_{W W} ZKLFK EHORQJV WR / 3 :H LQWURGXFH WKH VLPSOLqHG QRWDWLRQ ; ;

G HI

W W

DQG Z _V Z _V; IRU W „ V „ 7 (TXDWLRQV DQG RI 3URSRVLWLRQ VKRZ WKDW WKH VHOI W

qQDQFLQJ VWUDWHJ\ ; LV REWDLQHG E\ LQYHVWLQJ HYHU\ SHULRG V EHWZHHQ W DQG 7 _b WKH

W

YDOXH Z_V LQ WKH SRUWIROLR K 1_{V V} S_V ZKRVH YDOXH DW WLPH V LV K S 1_{V V V} S_V 3URSRVLWLRQ SURYHV WKDW WKLV VWUDWHJ\ \LHOGV D G\QDPLF SRUWIROLR LQ _;W7 ZKLFK VDWLVqHV

e f e W f W

HVVLQI( W K Z7 7; ( W K7Z 7 KWZ W

; ;W7

V W _S

ZW; KW

S

DQG ZKLFK PLQLPL]HV WKHUHIRUH WKH / QRUP RI/ 3 K₇ DPRQJ SRUWIROLRV LQ ;_W7 S

7KLV SRUWIROLR LV LPSRUWDQW IRU WZR UHDVRQV )RU D SD\Rp)₇ LQ/ 3 K₇ DQG D ZHDOWKZ _W

S

WZ )W 7

LQ /_W3 K_W WKH RSWLPDO UHSOLFDWLRQ VWUDWHJ\ ; LQYHVWV RU qQDQFHV DQ\ GLVFUHSDQF\

EHWZHHQ WKH LQLWLDO ZHDOWKZ _WDQG WKH RSWLPDO FRVW4 _W)₇ WKURXJK WKH/ PLQLPXP SRUWIROLR

W WZ )W 7

; 7KLV ZH VKRZ E\ GHFRPSRVLQJ WKH VWUDWHJ\ ; DFFRUGLQJ WR (TXDWLRQV DQG DV IROORZV

WZ )W 7 W4 ) )W 7 7 WZ b4 ) W W 7

; ; ;

S S

W4 ) )W 7 7 W K W

; K_WZ_Wb 4 _W)₇ ;

S

W4 ) )W 7 7 W

; K_WZ_Wb 4 _W)₇ ;

W 6HFRQG WKH RSWLPDO FRVW 4 _W)₇ PD\ EH REWDLQHG GLUHFWO\ IURP )₇ ZLWK WKH KHOS RIZ ₇ LQVWHDG RI WKH LWHUDWLYH SURFHGXUH GHVFULEHG LQ 3URSRVLWLRQ 6WDWHPHQW LLL RI /HPPD

S

ZLWK V WDQG Z _W K_WLPSOLHV LQGHHG WKDW

K L

_W

S

4 ) ( K Z )

_{W 7 W 7 7 7}

K_W

W W

7KLV SURYHV WKDW 4 _W LV D SRVLWLYH RSHUDWRU ZKHQHYHU Z₇ LV LWVHOI SRVLWLYH ,I Z_V GRHV QRW YDQLVK DW WLPH V EHWZHHQ W DQG 7 ZH DOVR KDYH

K L

_W

4 _V)_{7 W}( _V K Z_{7 7})₇ K Z_{V V}

,Q WHUHVW 5 D WHV

:H LQWURGXFH IURP QRZ RQ D PRQH\ PDUNHW )RU WKH UHVW RI WKH DUWLFOH ZH DVVXPH WKDW 6HFXULW\ LV D ULVN IUHH ]HUR FRXSRQ ERQG SD\LQJ D XQLTXH GLYLGHQG RI RQH XQLW RI QXPHUDLUH DW PDWXULW\ 7

$ VV X P S WLR Q K₇ LV LQ / 3 D Q G IR U HYHU\ SHULRG W EHWZ HHQ D Q G 7 b WK H S ULFH S_W R I WK H ]HUR FR X SR Q ER Q G LV SR VLWLYH

G HI

I

6LQFH WKH SULFH RI WKH ERQG LV DVVXPHG SRVLWLYH ZH GHqQH 5 _W7 S_W IRU _„ W_„ 7 _b

I

5 W7 LV WKH QRPLQDO ULVN IUHH UHWXUQ IURP LQYHVWLQJ LQ WKH ]HUR FRXSRQ ERQG IURP WLPH WXS WR KRUL]RQ 7 $VVXPSWLRQ PDNHV VXUH WKDW WKLV EX\ DQG KROG VWUDWHJ\ EHORQJV WR ; W7 ZH GHQRWH LW W7

,W ZLOO EH XVHIXO WR LQWURGXFH ODWHU RQ ULVN IUHH LQWHUHVW UDWHV ZLWK PDWXULW\ VKRUWHU WKDQ

7 DV ZH VWXG\ WKH PHDQYDULDQFH LQYHVWPHQW SUREOHPV ZLWK DQ KRUL]RQ V VPDOOHU WKDQ 7 :H VKDOO GR VR E\ DVVXPLQJ WKDW WKHUH H[LVWV DW WLPH WD GHIDXOW IUHH ]HUR FRXSRQ ERQG ZLWK QRQ YDQLVKLQJ SULFH ZKLFK GLVWULEXWHV RQH XQLW RI QXPHUDLUH DW PDWXULW\ V :H VKDOO GHQRWH

I

5 _WV WKH LQYHUVH RI WKH ERQG SULFH ZKLFK UHSUHVHQWV WKH ULVN IUHH UDWH IURP WLPH WWR PDWXULW\

V

I

:H UHPDUN WKDWZ_VW7 4 _V S_V 5 _V7 IRUV w WDQG ZH OHDUQ IURP 6WDWHPHQW L

e f G HI I

RI 3URSRVLWLRQ WKDW K_WS_W „ ( _W K_W S_W „ ( _W>K₇@ :H GHqQH + _W K _W 5 _W7 VR WKDW ZLWK WKLV QRUPDOL]DWLRQ WKLV ODVW LQHTXDOLW\ ZULWHV

+ _W„ ( _W>+ _W @ _„ ( _W>+ ₇@ ( _W>K₇@

DQG WKH QRUPDOL]HG SURFHVV + LV D SRVLWLYH VXEPDUWLQJDOH 1RWLFH WKDW ZH KDYH

e f

HVVLQI( _W K Z_{7 7}; + _W

; ;W7

V W _I

Z W; S_W 5 _W7

S _W

ZKLFK LV UHDFKHG LQ + ;_W :H GHULYH IURP 6WDWHPHQW LLL RI /HPPD ZLWK )₇

S

Z _W K_W DQG W_„ V _„ 7

W

K _WL K ZV V

( K Z

_{V 7 7 I}

5 _V7

S

K L K_{W S}

W

( K Z +

_{W 7 7 I W}

5 _W7

N N

:H UHPDUN WKDW WKH VXsFLHQW FRQGLWLRQ RI /HPPD ZKLFK UHTXLUHV WKDW S S_{7 W}G_W EH LQ

S _I S

Q Q

/ 3 K₇ ,5 KROGV ZLWK N DV VRRQ DV 5 _W7G_W LV LQ / 3 K₇ ,5 IRU HYHU\ SHULRG I

W EHWZHHQ DQG 7 _b 7KLV LV WKH FDVH IRU LQVWDQFH LI5 _W7 LV ERXQGHG DQG G_W EHORQJV WR S

Q

/ 3 K₇ ,5

7KH SUHVHQFH RI WKH ERQG DOORZV XV WR IXUWKHU GHVFULEH WKH RSWLPDO UHSOLFDWLRQ VWUDWHJ\

RI 3URSRVLWLRQ :H UHPDUN WKDW VLQFH K₇ LV SRVLWLYH DQG EHORQJV WR / 3 WKH PRGL K7

qHG SUREDELOLW\ GLVWULEXWLRQ 3 GHULYHG IURP 3 WKURXJK WKH 5DGRQ1LNRG\P GHULYDWLYH

K (₇ >K₇@ LV ZHOO GHqQHG

S

/ H P P D / HW )₇ EH D SD \ R p LQ / 3 K₇ D Q G W D SHULRG EHWZ HHQ D Q G 7 b : H K D YH

K u vL

K7 K7 W4 ) )W 7 7

( _W >)₇@ ( _W Z₇ ; D Q G

K u vL K u vL

* _W)_{7 K7 W4 ) )W 7 7 K7 K7 W4 ) )W 7 7}

9 D U_W )₇ b Z₇ ; 9 D U_W >)₇@ b 9 D U_W Z₇ ;

( _{W 7}>K @

7KH SURRI RI WKLV UHVXOW UHOLHV RQ 6WDWHPHQW LY RI /HPPD ZLWKV WDQGZ _W 4 _W)₇

W4 ) )W 7 7

,I ZH VHW ; VXFFHVVLYHO\ WRW7 DQG ; ZH REWDLQ

K u vL

W4 ) )W 7 7

( _W>K )_{7 7}@ ( _W K Z_{7 7} ;

ˆ z

K u _{W4 ) )} vL u _{W4 ) )} v

W 7 7 W 7 7

( _W K ) Z_{7 7 7} ; ( _W K Z_{7 7} ;

ZKLFK WRJHWKHU ZLWK (TXDWLRQ \LHOG /HPPD

K7 W4 ) )W 7 7

,Q WKH PRGLqHG SUREDELOLW\ 3 WKH VWUDWHJ\ ; PLQLPL]HV WKH YDULDQFH RI WKH UHVLGXDO )₇ b Z ₇; XQGHU WKH FRQVWUDLQWV WKDW WKH LQLWLDO ZHDOWK Z _W; EH HTXDO WR

K7

4 _W)₇ DQG WKDW WKH H[SHFWHG qQDO YDOXH ( _W >Z₇; @ EH LGHQWLFDO WR WKH H[SHFWHG YDOXH RI WKH SD\Rp)₇ 7KLV H[SODLQV ZK\ LW LV FDOOHG DQ RSWLPDO PHDQYDULDQFH UHSOLFDWLRQ VWUDWHJ\

K D V / D UJ HVW 5 HJ X OD UL]LQ J 3 UR FHVV

:H KDYH VHHQ LQ 6HFWLRQ WKDW K DFWV DV D UHJXODUL]LQJ SURFHVV IRU WKH YDOXH SURFHVVHV RI G\QDPLF SRUWIROLRV LQ _{; W7} :H SURYH KHUH WKDW K LV WKH ODUJHVW SURFHVV KDYLQJ WKLV SURSHUW\ IRU D JLYHQ WHUPLQDO YDOXH K₇ DW KRUL]RQ 7

{ {

/ H P P D / HWK EH D Q D G D S WHG S URFHVV VX FK WK D WK₇ K₇ D Q G VX FK WK D W IR U HYHU\ SR UWIR OLR

{ {

; LQ ; W7 WK H S URFHVV IK ZV V; _JW„V„7 LV D VX EP D UWLQ JD OH 7 K HQ KV _„ KV IR U HYHU\ SHULRG V EHWZ HHQ WD Q G 7

{

:H SURYH WKLV UHVXOW E\ EDFNZDUG LQGXFWLRQ /HW XV DVVXPH WKDW KV „ KV IRU D SHULRG

V EHWZHHQ WDQG 7 _b :H FRQVWUXFW D VHOI qQDQFLQJ SRUWIROLR ; VWDUWLQJ DW WLPH WLQ WKH IROORZLQJ ZD\ 7KH VWUDWHJ\ ; EX\V DQG KROGV WKH ULVN IUHH ERQG IURP WLPH W XS WR WLPH V

VS V

DW ZKLFK GDWH LW IROORZV WKH VWUDWHJ\ ; XS WR KRUL]RQ 7 ,W LV VHOI qQDQFLQJ LQ SHULRG

S

VS V

V ZLWK SRVLWLYH YDOXH Z _V; S_V LQ /_V3 K_V DQG LW EHORQJV WR_{; W7} VLQFH Z ₇; LV S

LQ / 3 K7 :H NQRZ E\ DVVXPSWLRQ WKDW

K L K L

{ {

K Z_{V V}; „ ( _V K_{V V} Z ; „ ( _V K_{V V} Z ;

DQG ZH GHULYH IURP (TXDWLRQ RI 3URSRVLWLRQ WKDW

ˆ z

u v u v K L

VS V VS V

K Z_{V V}; K_V Z _V; ( _V K_V Z _V ; ( _V K_{V V} Z ;

{ {

:H FRQFOXGH WKDW K Z_{V V}; „ K Z_{V V}; DQG K_V „ K_V VLQFH Z_V; LV SRVLWLYH ZKLFK HQGV WKH EDFNZDUG LQGXFWLRQ SURRI

1RWLFH WKDW WKH SURRI RI /HPPD UHTXLUHV WKH H[LVWHQFH RI D VWUDWHJ\ LQ; _WV ZLWK SRVLWLYH YDOXH DW WLPH V DQG ZH XVH KHUH D EX\ DQG KROG LQYHVWPHQW LQ WKH ERQG

{

:H DOVR UHPDUN WKDW LI K LV QRQ QHJDWLYH DQG UHJXODUL]HV WKH YDOXH SURFHVV RI G\QDPLF SRUWIROLRV LQ ;W7 DV GHVFULEHG LQ /HPPD WKHQ LW DOVR UHJXODUL]HV WKH FXP DQG H[ GLYLGHQG

S S SULFH SURFHVVHV ƒ DQG S ,QGHHG ZH NQRZ IURP 3URSRVLWLRQ WKDW ERWK K ƒ_{V V} DQG K S_{V V}

S S

Q { { {

DUH LQ /_V3 ,5 DQG VLQFH _„ K_{V „} K_V ZH FRQFOXGH WKDW K ƒ_{V V} DQG K S_{V V} DUH DOVR LQ

Q

/_V3 ,5 IRU HYHU\ SHULRG V EHWZHHQ WDQG 7

9 D ULD Q FH2 S WLP D O 0 D UWLQ J D OH 0 HD VX UH

,W ZDV QRWHG LQ 6HFWLRQ WKDW WKH RSHUDWRU 4 _W LV SRVLWLYH DV VRRQ DV WKH qQDO ZHDOWK

W W W

Z ₇ Z₇; RI WKH / PLQLPXP SRUWROLR ; LV LWVHOI SRVLWLYH :H VKRZ WKDW ZKHQ WKLV KDSSHQV WKH FRVW 4 _V)₇ DW WLPH V EHWZHHQ W DQG 7 b RI WKH RSWLPDO UHSOLFDWLRQ

S

RI D FRQWLQJHQW FODLP )₇ LQ / 3 K₇ FDQ EH H[SUHVVHG DV WKH GLVFRXQWHG FRQGLWLRQDO H[SHFWDWLRQ RI)₇ LQ D SUREDELOLW\ GLVWULEXWLRQ GLpHUHQW IURP WKH RULJLQDO SUREDELOLW\ 3 ,Q WKH FDVH ZKHUH K₇ WKLV QHZ SUREDELOLW\ GLVWULEXWLRQ LV FDOOHG WKH PLQLPXP YDULDQFH SUREDELOLW\ GLVWULEXWLRQ RU WKH YDULDQFHRSWLPDO PDUWLQJDOH SUREDELOLW\

W W

:H qUVW QRWLFH IURP (TXDWLRQ WKDW LIZ₇ LV SRVLWLYH WKHQ WKH YDOXH Z _V RI WKH VWUDWHJ\ W

; DW WLPH V LV DOVR SRVLWLYH 6WDWHPHQW LLL RI /HPPD WRJHWKHU ZLWK (TXDWLRQ \LHOGV

e W f

( _V K Z_{7 7})₇

4 _V)_{7 I W}

( _V>K Z₇ @

5 _{V7 7}

S S

W W

:H UHPDUN WKDW K Z_{7 7} LV LQ / 3 VLQFH ERWK K Z_{7 7} DQG K₇ DUH LQ / 3 DQG ZH ZULWH W

K Z7 7

4 _V)_{7 I} ( _V >)₇@ 5 _V7

ZKLFK VKRZV WKDW 4 _V)₇ FDQ LQGHHG EH ZULWWHQ DV D GLVFRXQWHG H[SHFWDWLRQ LQ WKH PRGLqHG W

K Z7 7 SUREDELOLW\ GLVWULEXWLRQ 3

W

:H VKDOO VKRZ ODWHU LQ 6HFWLRQ WKDW RQH FDQ XVXDOO\ QRW H[SHFW Z ₇ WR EH SRVLWLYH ZKHQ WKH FXPGLYLGHQG SULFHV DVVXPH XQERXQGHG YDOXHV 7KLV IDFW KDV EHHQ QRWHG LQ 6FKZHL]HU :KHQ WKLV KDSSHQV WKH PLQLPXP YDULDQFH SUREDELOLW\ EHFRPHV WKH YDULDQFHRSWLPDO VLJQHG PDUWLQJDOH PHDVXUH DQG WKH RSHUDWRU 4 _W DOWKRXJK VWLOO ZHOO GHqQHG LV QRW SRVLWLYH

W

,Q D FRQWLQXRXV WLPH VHWWLQJ *RXULHURX[ HW DO VKRZ WKDW Z 7 LV DOZD\V SRVLWLYH DV ORQJ DV SULFHV IROORZ FRQWLQXRXV VHPLPDUWLQJDOHV ZLWK QR GLYLGHQG GLVWULEXWLRQ 7KH\ DVVXPH WKH H[LVWHQFH RI D ULVN QHXWUDO SUREDELOLW\ GLVWULEXWLRQ D QR DUELWUDJH FRQGLWLRQ ZKLFK LV VWURQJHU WKDQ WKH ODZ RI RQH SULFH

& K D Q J H R I 1 X P HUD LUH

:H KDYH FRQVLGHUHG VR IDU DQ DUELWUDU\ QXPHUDLUH HYHU\ SHULRG DQG ZH ZLVK WR VWXG\ QRZ KRZ RXU UHVXOWV GHSHQG RQ WKLV FKRLFH :H LQWURGXFH D QHZ QXPHUDLUH HYHU\ SHULRG DQG ZH OHW s_W EH LWV YDOXH DW WLPH W LQ WKH ROG QXPHUDLUH :H DVVXPH WKDW WKH SURFHVV s

G HI G HI

s s

LV DGDSWHG DQG SRVLWLYH :H OHW S_W S s_{W W} DQG ƒ_W ƒ s_{W W} EH UHVSHFWLYHO\ WKH H[ DQG FXP GLYLGHQG SULFHV RI WKH WUDGHG VHFXULWLHV DW WLPH WLQ WKH QHZ QXPHUDLUH DQG ZH GHQRWH

G HI

s s

Z _W; S ;_{W W} Z_W; s_W WKH YDOXH RI D SRUWIROLR ; DW WLPH WLQ WKH QHZ QXPHUDLUH

T G HI

s s s

:H FKRRVH WKH QRUPDOL]DWLRQK₇ s K_{7 7} VR WKDW HYHU\ SRUWIROLR; VDWLVqHV K Z_{7 7};

S S

K Z_{7 7}; DQG D G\QDPLF SRUWIROLR KDV D qQDO YDOXH LQ / 3 K₇ IRU WKH ROG QXPHUDLUH

T

s

LI DQG RQO\ LI LWV qQDO YDOXH LQ WKH QHZ QXPHUDLUH LV LQ / 3 K₇ 6LQFH WKH VHOI qQDQFLQJ FRQGLWLRQ GRHV QRW GHSHQG RQ WKH FKRLFH RI QXPHUDLUH ZH UHPDUN WKDW WKH VHW ; _W7 UHPDLQV LGHQWLFDO DV ZH FKDQJH ERWK WKH QXPHUDLUH DQG WKH YDOXH RI WKH YDULDEOH K DW WLPH 7 IURP

s

K₇ WR K₇

I s s s I

:H OHW 5 _W7 K_W DQG + _W EH UHVSHFWLYHO\ WKH YDULDEOHV 5 _W7 K_W DQG + _W IRU WKH QHZ I s

QXPHUDLUH 7KH LQWHUHVW UDWH 5 W7 IURPW WR 7 LQ WKH QHZ QXPHUDLUH LV JLYHQ E\ G HI

I s s I

5 _W7 S_W s 5_{W W7}

DQG ZH KDYH

G HI I s

s s

+ _W K _W 5 _W7

7KH QH[W OHPPD VKRZV WKDW WKH SURFHVV + LV QXPHUDLUH LQYDULDQW DV ORQJ DV ZH FRUUHFWO\ s

QRUPDOL]H WKH YDULDEOH K₇ DV H[SODLQHG DERYH

/ H P P D ) R U HYHU\ SHULRG W EHWZ HHQ D Q G 7 Z H K D YH

s

L K_W s K _{W W}

s

LL + _W + _W

,W LV RIWHQ FRQYHQLHQW LQ qQDQFLDO HQJLQHHULQJ WR FKRRVH WKH ULVN IUHH ]HUR FRXSRQ ERQG I

ZLWK PDWXULW\ 7 DV QXPHUDLUH WKDW LV WR VHWs_W S_W 5 _W7 )RU WKLV SDUWLFXODU FKRLFH RI I s

s

QXPHUDLUH ZH KDYHK₇ K₇ VLQFHs₇ DQG5 _W7 :H DOVR FKHFN WKDW IRU HYHU\ SHULRG s s

W ZH KDYH KW + W + W VR WKDW WKH QXPHUDLUH LQYDULDQW YDULDEOH + W FDQ EH LQWHUSUHWHG DV WKH YDULDEOH KW RQFH ZH FKRRVH WKH ERQG DV QXPHUDLUH

0 H D Q 9 D U LD Q F H 3 R U W I R O LR 6 H O H F W LR Q

:H LQYHVWLJDWH LQ WKLV VHFWLRQ WKH PHDQYDULDQFH SURSHUWLHV RI VHOI qQDQFLQJ G\QDPLF SRUWIROLRV $IWHU IRUPDOO\ GHqQLQJ WKH FRQFHSWV RI G\QDPLF 6KDUSH UDWLR DQG HsFLHQW IURQWLHU LQ 6HFWLRQ ZH GHULYH DQ LPSRUWDQW ULVNUHWXUQ LQHTXDOLW\ LQ 6HFWLRQ ZKLFK LV WKH EDVLV RI RXU DQDO\VLV 7KH XSSHU ERXQG WR WKH G\QDPLF 6KDUSH UDWLRV RI VHOI qQDQFLQJ SRUWIROLRV LV VKRZQ LQ 6HFWLRQ WR EH D IXQFWLRQ RI WKH SURFHVV + 8QGHU WKH UHJXODULW\ FRQGLWLRQ WKDW WKLV RSWLPDO G\QDPLF 6KDUSH UDWLR LV ERXQGHG RU QHYHU JHWV FORVH WR ]HUR 6HFWLRQ SURGXFHV D WZR IXQG VHSDUDWLRQ WKHRUHP LQ D FRQGLWLRQDO IRUP HYHU\ HsFLHQW SRUWIROLR

FDQ EH GHVFULEHG DV D FRPELQDWLRQ RI WZR VWUDWHJLHV WKH / PLQLPXP SRUWIROLR DQG WKH ULVNOHVV VHFXULW\ DQG WKH DPRXQW DOORFDWHG WR HDFK VWUDWHJ\ PD\ GHSHQG RQ WKH LQIRUPDWLRQ DYDLODEOH DW WKH WLPH ZKHQ WKH LQYHVWPHQW LV LQLWLDWHG 6HFWLRQ JHQHUDOL]HV WKH QRWLRQ RI G\QDPLF 6KDUSH UDWLR IURP VHOI qQDQFLQJ SRUWIROLRV WR DQ\ FRQWLQJHQW FODLP 7KH JHQHUDOL]HG G\QDPLF 6KDUSH UDWLR LV DJDLQ VPDOOHU WKDQ WKH RSWLPDO G\QDPLF 6KDUSH UDWLR DQG WKHLU VSUHDG PHDVXUHV WKH GLVWDQFH WR HsFLHQF\ RI WKH FRQWLQJHQW FODLP

' Hq Q LWLR Q V

:H FRQVLGHU D WLPH SHULRG W EHWZHHQ DQG 7 b DQG ZH LQWURGXFH WKH QRWLRQV RI K7

G\QDPLF 6KDUSH UDWLR DQG G\QDPLF HsFLHQW IURQWLHU LQ WKH SUREDELOLW\ 3 FRQGLWLRQHG K7

RQ WKH LQIRUPDWLRQ DW GDWH W :H UHFDOO WKDW 3 LV GHULYHG IURP WKH RULJLQDO SUREDELOLW\ GLVWULEXWLRQ 3 E\ PHDQV RI WKH 5DGRQ1LNRG\P GHULYDWLYH K (_{7 W}>K₇@ DQG WKDW ZH ZRUN LQ

K7

WKH PRGLqHG SUREDELOLW\ 3 VR DV WR UHPDLQ FRQVLVWHQW ZLWK RXU JHQHUDO FRQGLWLRQ WKDW WKH S

YDOXHV DW WLPH 7 RI WKH VHOI qQDQFLQJ SRUWIROLRV RI; _W7 EHORQJ WR / 3 K₇ ZKLFK LV DOVR K7

/ 3

K7

)RU HYHU\ G\QDPLF SRUWIROLR ; LQ ; W7 ZH GHQRWH 65W7; WKH 6KDUSH UDWLR LQ WKH K7

PRGLqHG SUREDELOLW\ 3 FRQGLWLRQHG RQ WKH LQIRUPDWLRQ DW WLPH W ZKLFK UHVXOWV IURP IROORZLQJ WKH VHOI qQDQFLQJ LQYHVWPHQW VWUDWHJ\ ; IURP WLPH WXS WR KRUL]RQ 7 :H OHW

I K7

( _W >Z ₇; @ _b 5 Z _W;

G HI W7

K7

T

65_W7;

K7

9DU_W >Z ₇; @

G HI

K7 K7

ZKHQ 9DU_W >Z ₇; @ LV GLpHUHQW IURP ]HUR DQG ZH VHW 65_W7; IRU WKH VWDWHV RI WKH K7

ZRUOG ZKHUH 9DU_W >Z ₇; @ G HI

:H GHQRWH 5 W7; Z7; Z W; WKH WRWDO UHWXUQ IURP SHULRG W WR KRUL]RQ 7 RI D

G\QDPLF SRUWIROLR; LQ_{; W7} ZLWK QRQ YDQLVKLQJ YDOXHZ_W; DW GDWHW ,Q SDUWLFXODU ZH KDYH I

5 _W7 5 _W7W7 ZKHQ ; _W7 LV WKH VWUDWHJ\ ZKLFK LQYHVWV ZLWKRXW UHEDODQFLQJ LQ WKH K7

GHIDXOW IUHH ]HUR FRXSRQ ERQG ZLWK PDWXULW\7 IURP WLPHWRQ ,IZ_W; DQG 9DU_W >Z₇; @ DUH 3 DOPRVW VXUHO\ GLpHUHQW IURP ]HUR ZH DOVR KDYH

I K7

( _W >5 _W7; @ _b 5 _W7

K7

T

65_W7;

K7

9DU_W >5 _W7; @ WKH XVXDO GHqQLWLRQ RI D 6KDUSH UDWLR

2XU FKRLFH RI QRQ DQQXDOL]HG VLPSOH WRWDO UHWXUQV LV QRW DQ LQQRFXRXV VSHFLqFDWLRQ ,W DOORZV XV WR OLQN WKH WZR G\QDPLF LQYHVWPHQW SUREOHPV RI RSWLPDO UHSOLFDWLRQ DQG RSWLPDO SRUWIROLR VHOHFWLRQ 7KLV QLFH FRQYHUJHQFH PD\ QRW KROG IRU RWKHU GHqQLWLRQV RI WKH UHWXUQV :H OHW WKH G\QDPLF PHDQYDULDQFH HsFLHQW IURQWLHU DW WLPH WZLWK KRUL]RQ 7 ZKLFK ZH GHQRWH ( ) _W7 EH WKH VHW RI SRUWIROLRV LQ; _W7 ZKLFK DUH VROXWLRQ WR WKH RSWLPL]DWLRQ SURJUDP

K7

HVVLQI 9DU_W >Z ₇; @

!

! ; ;W7

!

V W Z _W; Z_W

! !

! K7

( _W >Z 7; @ : W7

IRU VRPH H[SHFWHG qQDO YDOXH WDUJHW : _W7 PHDVXUDEOH ZLWK UHVSHFW WR ) _W DQG VRPH LQLWLDO S

ZHDOWK ZW LQ / W3 KW :KHQ WKH LQLWLDO ZHDOWK Z WGRHV QRW YDQLVK WKLV SURJUDP FDQ EH ZULWWHQ LQ WHUPV RI UHWXUQV DV IROORZV

K7

HVVLQI 9DU_W >5 _W7; @

!

! ; ;W7

!

V W Z _W; Z _W

! !

! K7

( _W >5 W7; @ 5 W7

ZKHUH WKH H[SHFWHG UHWXUQ WDUJHW 5 _W7 LV : _W7 Z _W

$ 5 LVN 5 HWX UQ ,Q HT X D OLW\

7KH QH[W UHVXOW SURYHV WKDW WKH ULVN SUHPLXP RI DQ\ G\QDPLF SRUWIROLR LQ_{; W7} LV ERXQGHG E\ WKH YDULDQFH RI LWV UHWXUQ :H VWDWH WKH UHVXOW LQ WHUPV RI SRUWIROLR YDOXHV VR DV WR WDFNOH WKH JHQHUDO FDVH ZKHUH WKH LQLWLDO YDOXH RI WKH VWUDWHJ\ PD\ YDQLVK DQG WKH UHWXUQ PD\ QRW EH ZHOO GHqQHG ,Q WKH SURRI RI WKLV SURSRVLWLRQ LW LV LPSRUWDQW WR LVRODWH WKH VWDWHV RI WKH ZRUOG ZKHUH + _W ( _{W 7}>K @ DQG ZH qUVW VWXG\ WKLV VSHFLDO FDVH

3 U R S R VLWLR Q L 2 Q WK H VHWI+ W ( W>K7@_J LQ ) W

I

W W

Z7; 5 _W7ZW;

S _{I K}

7

IR U HYHU\ SD \ R p )₇ LQ / 3 K₇4 _W)₇ 5 _W7( _W >)₇@

I K7

HYHU\ G \ Q D P LF SR UWIR OLR ; LQ ;_W7 VD WLVq HV ( _W >Z₇; @ 5 _W7Z _W; LL ( YHU\ G \ Q D P LF SR UWIR OLR ; LQ ;_W7 VD WLVq HV

w x

u _{K I} v ( _{W 7}>K @ _K

7 7

( >Z ; @b 5 Z ; „ b 9 D U >Z ; @

_{W 7 W7 W W 7}

+ _W

W

LLL 7 K H G \ Q D P LF VWUD WHJ\ ; VD WLVq HV ,Q HTX D OLW\ Z LWK D Q HTX D OLW\

,QHTXDOLW\ \LHOGV D qUVW LQWXLWLYH UHVXOW HYHU\ G\QDPLF SRUWIROLR ; LQ ;_W7 UHWXUQV K7

WKH ULVN IUHH UDWH IURP W WR 7 RQ WKH VHW I9DU_W >Z₇; @ J ZKHUH WKH qQDO YDOXH RI WKH VWUDWHJ\ LV NQRZQ DW WLPH WZLWK FHUWDLQW\

2 S WLP D O ' \ Q D P LF 6 K D US H 5 D WLR

,QHTXDOLW\ VWDWHV WKDW HYHU\ SRUWIROLR ; LQ ; _W7 VDWLVqHV

( _{W 7}>K @ K7

65_W7; „ b

+ _W

DQG 6WDWHPHQW LLL RI 3URSRVLWLRQ VKRZV WKDW WKH XSSHU ERXQG ( _{W 7}>K @ + _Wb LV UHDFKHG

W

E\ WKH / PLQLPXP SRUWIROLR ; LQ _{; W7} :H OHW

V

w_{( >K @} x G HI W 7

K7

65_{W7 b}

+ _W

EH WKH RSWLPDO G\QDPLF 6KDUSH UDWLR IURP WLPH W WR KRUL]RQ 7 FRQGLWLRQHG RQ WKH LQIRU PDWLRQ DYDLODEOH DW WLPH W DQG ZH KDYH

u v

K7 K7 W K7 HVVVXS 65W7; 65W7; 65W7

V W ; ;W7

:H KDYH VHHQ LQ 6HFWLRQ WKDW + _WVDWLVqHV WKH LQHTXDOLWLHV + _W„ ( _{W 7}>K @ 7KH OLPLW K7

YDOXHV RI+ _WFRUUHVSRQG WR DQ RSWLPDO 6KDUSH UDWLR 65_W7 ZKLFK YDQLVKHV ZKHQ+ _W ( _{W 7}>K @ DQG ZKLFK LQFUHDVHV WR LQqQLW\ DV + _W QHDUV ]HUR

S S _I 1RWLFH DOVR WKDW LW UHVXOWV IURP (TXDWLRQ DQG WKH HTXDOLW\ K_W + 5_{W W7} WKDW

S

e f S I

K L ( W K Z7 W + W KW + 5W W7 KW

K7 W 7

( 5 ;

_{W W7 W I}

Z (W W>K7@ ( W>K7@ ( W>K7@ 5 W7( W 7>K @

W

7KLV DOORZV XV WR ZULWH WKH H[SHFWHG UHWXUQ RI WKH / PLQLPXP SRUWIROLR ; DV D IXQFWLRQ K7

RI WKH RSWLPDO 6KDUSH UDWLR 65W7

I

K L 5 _W7

K7 W

( _W 5 _W7; _K

7

65_W7

RU HTXLYDOHQWO\ LQ WHUPV RI ULVN SUHPLXP

K7

K L _{I I} 65W7

K7 W

( 5 ; _b 5 _b 5

_{W W7 W7 W7 K}

7

65_W7

u v

K7 W K7

7KH QH[W OHPPD JHQHUDOL]HV WKH HTXDOLW\ 65_W7; 65_W7 WR D IDPLO\ RI G\QDPLF W

SRUWIROLRV LQ _;W7 ZKLFK FRPELQH WKH G\QDPLF SRUWIROLR ; ZLWK WKH VWDWLF VWUDWHJ\ _W7 ZKLFK LQYHVWV LQ WKH ERQG ZLWK PDWXULW\ 7 IURP WLPH WRQ

/ H P P D / HW rW D Q G )W EH WZ R UD Q G R P YD ULD EOHV P HD VX UD EOH Z LWK UHVSHFW WR )W D Q G OHW

W

; r ;_W )_{W W7} EH WK H VHOI q Q D Q FLQ J G \ Q D P LF SR UWIR OLR Z K LFK FR P ELQ HV IUR P WLP H WR Q r_W

W

X Q LWV R I WK H VWUD WHJ\ ; Z LWK )WX Q LWV R I WK H ]HUR FR X SR Q ER Q G Z LWK P D WX ULW\ 7 7 K H G \ Q D P LF

S S S

SR UWIR OLR ; LV LQ ;_W7 LI D Q G R Q O\ LI ER WK ( _{W 7}>K @b + _W)_W D Q G r_W + )_{W W} K Z_{W W};

D UH LQ /_W3 : K HQ WK LV K D S SHQ V

K L

( K Z ; K Z ; ( >K @ b + )

_{W 7 7 W W W 7 W W}

D Q G ; K D V WK H IR OOR Z LQ J S UR SHUWLHV

L ; VD WLVq HV ,Q HTX D OLW\ Z LWK D Q HTX D OLW\ D Q G

S

+ _W

I

K7 K7

( _W >Z₇; @ b 5 _W7Z _W; b 6 5 _W7 r _W ( _{W 7}>K @

+ _W

K7 K7

9 D U_W >Z₇; @ 6 5_W7 r _W

( _{W 7}>K @

u v _I

K7 K7 K7

6 5 _W7 )_W 6 5 _W7 ( _W >Z ₇; @_b 5 _W7Z _W; 7 K H OD VW HTX D OLW\ D OVR Z ULWHV

I

( >K Z ; @ + 5 Z ; ( >K @b + )

_{W 7 7 W W7 W W 7 W W}

I

LL 7 K H VWUD WHJ\ ; LV R Q WK H Hs FLHQ W IUR Q WLHU( ) _W7 7 K H YD ULD EOHVb r_W)_Wb 5 _W7Z _W;

I K7

D Q G ( _W >Z ₇; @ b 5 _W7Z _W; K D YH WK H VD P H VLJQ R Q Z K LFK G HSHQ G V WK H G \ Q D P LF 6 K D USH UD WLR R I; IUR P WWR 7 LQ WK H IR OOR Z LQ J Z D \

I

K7 K7

6 5 W7; R Q WK H VHWI( W >Z 7; @ 5 W7Z W; _J

I

K7 K7 K7

6 5 _W7; 6 5 _W7 R Q WK H VHWI( _W >Z ₇; @ ! 5 _W7Z _W; J

I

K7 K7 K7

6 5 _W7; b 6 5 _W7 R Q WK H VHW I( _W >Z ₇; @ 5 _W7Z_W; J

LLL ,IZ _W; LV 3 D OP R VW VX UHO\ G Lp HUHQ W IUR P ]HUR WK HQ WK H VWUD WHJ\ ; LQ YHVWV IUR P WLP H

W

W R Q D IUD FWLR Q p_W R I LWV LQ LWLD O Z HD OWK LQ WK H G \ Q D P LF SR UWIR OLR ; Z K LOH WK H UHVW LV

LQ YHVWHG LQ WK H ER Q G Z LWK

)_W r_W

S

p_{W b I}

K Z ;

5 _W7Z _W; W W

7 K H UHWX UQ 5 _W7; IUR P W WR 7 VD WLVq HV

I W

5 _W7; p 5_{W W7}; b p_W5 _W7

K7 6 5 _W7

I I

K7

( _W >5 _W7; @ _b 5 _{W7 b K} 5 _W7p _W

7 6 5 _W7

K7

6 5 _{W7 I}

K7

9 D U_W >5 _W7; @ _{u v} 5 _W7 p _W

K7 6 5 _W7

D Q G

u v

I I

K7 K7 K7

6 5 _W7 5 _W7b p_W 6 5 _W7 ( _W >5 _W7; @ b 5 _W7

' \ Q D P LF ( s FLHQ W ) UR Q WLHU

/HPPD KDV GLVSOD\HG D IDPLO\ RI G\QDPLF SRUWIROLRV RQ WKH HsFLHQW IURQWLHU :H VWXG\ QRZ WKH UHFLSURFDO SUREOHP :H VHHN WR FRQVWUXFW D VROXWLRQ WR WKH PHDQYDULDQFH SRUWIROLR VHOHFWLRQ 3UREOHP IRU D JLYHQ LQLWLDO ZHDOWK Z _W DQG D JLYHQ H[SHFWHG qQDO YDOXH : _W7 %RWK Z _W DQG : _W7 DUH NQRZQ DW WLPH W EXW PD\ EH UDQGRP YDULDEOHV LQ )_W 8QGHU VRPH

K7

UHJXODULW\ FRQGLWLRQV RQ 65_W7 WKLV ZLOO WKHQ DOORZ XV WR GHVFULEH DOO WKH SRUWIROLRV RQ WKH HsFLHQW IURQWLHU

S

3 U R S R VLWLR Q : H FR Q VLG HU D Q LQ LWLD O Z HD OWK Z W D W WLP H W LQ /W3 KW D Q G D Q H[SHFWHG q Q D O YD OX H : _W7 P HD VX UD EOH Z LWK UHVSHFW WR ) _W ,I: _W7 D Q G Z _W D UH VX FK WK D W WK HUH H[LVWV D UD Q G R P YD ULD EOH )_W P HD VX UD EOH Z LWK UHVSHFW WR ) _W Z K LFK VD WLVq HV

u v _I

K7 K7

6 5 ) 6 5 : b 5 Z

_{W7 W W7 W7 W7 W}

S W

D Q G VX FK WK D W ( _W>K₇@b + _W)_WLV LQ /_W3 WK HQ WK H G \ Q D P LF SR UWIR OLR ; r ;_W )_WW7

S S

Z LWK r_W K Z_{W W}b + )_{W W} EHOR Q JV WR WK H Hs FLHQ W IUR Q WLHU ( ) _W7 D Q G VR OYHV

K7

HVVLQ I 9 D U_W >Z ₇; @

!

! ; ;W7

!

V W Z_W; Z _W

! !

! K7

( _W >Z7; @ : W7

K7

:H UHPDUN WKDW WKH YDULDEOH )_W LV QRW GHqQHG E\ (TXDWLRQ RQ WKH VHW I65_W7 J W 2Q WKLV VHW ZH OHDUQ IURP 6WDWHPHQW L RI 3URSRVLWLRQ WKDW WKH WZR VWUDWHJLHV ; DQG _W7 UHWXUQ WKH ULVN IUHH UDWH VR WKDW ; FDQ EH LGHQWLqHG ZLWK DQ LQYHVWPHQW LQ WKH ERQG

7KH QH[W ORJLFDO VWHS LV WR LQYHVWLJDWH LI WKH FRQGLWLRQ RQ _{: W7} DQGZ _W VWDWHG LQ 3URSRVL WLRQ LV HDV\ WR PHHW 7KH QH[W UHVXOW VKRZV WKDW XQGHU VRPH UHJXODULW\ FRQGLWLRQ RQ WKH RSWLPDO G\QDPLF 6KDUSH UDWLR DQ H[SHFWHG qQDO YDOXH WDUJHW : _W7 DQG DQ LQWLDO ZHDOWK Z _W

DW WLPH W VDWLVI\ WKH GHVLUHG FRQGLWLRQ LI WKH\ DUH WKH H[SHFWHG qQDO YDOXH DQG WKH LQLWLDO ZHDOWK DW WLPH WRI DW OHDVW RQH G\QDPLF SRUWIROLR LQ ; _W7 7KLV LV D QDWXUDO UHTXLUHPHQW IRU WKH PHDQYDULDQFH SUREOHP WR EH ZHOO SRVHG :H VWDUW E\ VWDWLQJ WZR W\SHV RI UHJXODULW\

K7 FRQGLWLRQ RQ WKH 6KDUSH UDWLR 65W7

K7

& R Q G LWLR Q 7 K H R S WLP D O G \ Q D P LF 6 K D USH UD WLR 6 5_W7 LV ER X Q G HG WK HUH H[LVWV D FR Q VWD Q W

K7

. VX FK WK D W 6 5 _W7 „ .

K7

& R Q G LWLR Q 7 K H R S WLP D O G \ Q D P LF 6 K D USH UD WLR 6 5 _W7 LV X Q LIR UP O\ ER X Q G HG D Z D \ IUR P

K7

]HUR WK HUH H[LVWV D FR Q VWD Q W t VX FK WK D W t 6 5 _W7

:H NQRZ WKDW + _W„ ( _{W 7}>K @ ,W LV HDVLO\ FKHFNHG WKDW &RQGLWLRQV DQG KROG DV VRRQ DV WKH YDULDEOH + _WUHPDLQV XQLIRUPO\ ERXQGHG DZD\ IURP UHVSHFWLYHO\ DQG ( _{W 7}>K @ / H P P D : H FR Q VLG HU D SR VLWLYH LQ LWLD O Z HD OWK Z_W D W WLP H W D Q G D Q H[SHFWHG q Q D O YD OX H

WD UJHW : W7 VX FK WK D W WK HUH H[LVWV D W OHD VW R Q H SR UWIR OLR ; LQ ;W7 Z K LFK VD WLVq HV ER WK

K7

Z _W; Z_W D Q G ( _W >Z₇; @ : _W7 ,I & R Q G LWLR Q R U K R OG V WK HQ WK HUH H[LVWV D UD Q G R P YD ULD EOH )_W P HD VX UD EOH Z LWK UHVSHFW WR )_W VR OX WLR Q WR ( TX D WLR Q R I 3 UR SR VLWLR Q WK D W LV VR OX WLR Q WR

u v _I

K7 K7

6 5 _W7 )_W 6 5 _{W7 : W7 b} 5 _W7Z _W

S

D Q G VX FK WK D W ( W>K7@_b + W)W LV LQ / W3

K7

1RWLFH WKDW &RQGLWLRQ LPSOLHV WKDW 65_W7 DQG )_W LV WKHQ XQLTXHO\ GHqQHG XS WR D ]HUR SUREDELOLW\ VHW WKURXJK (TXDWLRQ &RQGLWLRQ GRHV QRW H[FOXGH KRZHYHU

K7

WKH SRVVLELOLW\ WKDW WKH VHW I65W7 J KDYH QRQ ]HUR SUREDELOLW\ 2Q WKLV VHW KRZHYHU I

6WDWHPHQW L RI 3URSRVLWLRQ SURYHV WKDW: _W7 5 _W7Z _W WKH ULJKW KDQG VLGH RI (TXDWLRQ LV HTXDO WR ]HUR DQG ZH DUH IUHH WR SLFN DQ\ YDOXH IRU WKH YDULDEOH )_W

7KH QH[W UHVXOW \LHOGV D IXOO GHVFULSWLRQ RI ( ) _W7 ,W LV D WZR IXQG VHSDUDWLRQ WKHRUHP LQ D FRQGLWLRQDO IRUP 8QGHU VRPH UHJXODULW\ FRQGLWLRQ HYHU\ HsFLHQW SRUWIROLR RQ _{( ) W7}

W

FDQ EH LGHQWLqHG ZLWK D FRPELQDWLRQ RI WKH VWUDWHJ\ ; DQG WKH ERQG %HFDXVH RI SRVVLEOH UHGXQGDQF\ EHWZHHQ WKH VHFXULWLHV WKH LGHQWLqFDWLRQ LV RQO\ PHDQLQJIXO LQ WHUPV RI WKH YDOXH SURFHVVHV RI WKH VWUDWHJLHV

3 U R S R VLWLR Q 7 Z R IX Q G VHSD UD WLR Q ,I & R Q G LWLR Q R U LV VD WLVq HG WK HQ IR U HYHU\ SR UW

IR OLR ; R Q WK H Hs FLHQ W IUR Q WLHU ( ) _W7 WK HUH H[LVWV D UD Q G R P YD ULD EOH )_W LQ )_W VX FK WK D W WK H G \ Q D P LF VWUD WHJ\ ; K D V D YD OX H S URFHVV EHWZ HHQ W D Q G 7 LG HQ WLFD O WR WK H YD OX H S URFHVV R I

S S

W

WK H SR UWIR OLR < r ;W )W W7 LQ ;W7 Z LWK ZW Z W; D Q G rW K ZW W_b + )W W 7 K H

S

UD Q G R P YD ULD EOH )_W VD WLVq HV ( _{W 7}>K @b + _W)_W /_W3 D Q G LV VR OX WLR Q WR ( TX D WLR Q R I 3 UR SR VLWLR Q WK D W LV

u v _I

K7 K7 K7

6 5 _W7 )_W 6 5 _W7 ( _W >Z ₇; @ b 5 _W7Z_W;

3URSRVLWLRQ SURYHV LQ SDUWLFXODU WKDW ( ) W7 LV D YHFWRU VXEVSDFH RI ;W7 ,W LV D JHQ HUDOL]DWLRQ RI WKH ZHOO NQRZQ WZR IXQG VHSDUDWLRQ WKHRUHP ZKLFK REWDLQV LQ D RQH SHULRG VHWWLQJ DV H[SRVHG IRU LQVWDQFH LQ +XDQJ DQG /LW]HQEHUJHU %DM HX[%HVQDLQRX DQG 3RUWDLW GHULYH D VLPLODU UHVXOW LQ D FRQWLQXRXV WLPH DQG FRPSOHWH PDUNHW VHWWLQJ

:H FRQFOXGH WKH VWXG\ RI WKH VWUXFWXUH RI WKH HsFLHQW IURQWLHU E\ VKRZLQJ WKDW WKH VHW G HI

Z ₇( ) _W7 IZ ₇; ; ( ) _W7J RI YDOXHV DW WLPH 7 RI SRUWIROLRV RQ WKH HsFLHQW S

IURQWLHU _{( ) W7} LV FORVHG LQ/ 3 K₇

3