:DYH0RWLRQ
:DYHW\SHV
0HFKDQLFDO:DYHV'HVFULEHGZLWK1HZWRQLDQ0HFKDQLFV
(OHFWURPDJQHWLF:DYHV$UULYHGDW8VLQJ0D[ZHOO(TXDWLRQV
0DWWHU:DYHV7UHDWHG:LWKLQWKH)UDPHZRUNRI4XDQWXP0HFKDQLFV
0HFKDQLFDOZDYHVOLNHVRXQGZDYHVZDYHVDORQJDURSHRURQDVXUIDFHDQGWKHZDYHV
LQDODNHFRQVLVWRIGLVWXUEDQFHVWKDWGLVSODFHWKHPHGLXPWKURXJKZKLFKWKH\WUDYHO
6XFKZDYHVWUDQVIHUHQHUJ\IURPRQHORFDWLRQWRDQRWKHUXWLOL]LQJWKHPHGLXP
0HFKDQLFDOZDYHVDUHFDWHJRUL]HGE\WKHGLUHFWLRQLQZKLFKWKHPROHFXOHVRIWKHPHGLXP
DUHGLVSODFHGUHODWLYHWRWKHSURSDJDWLRQGLUHFWLRQDVDSXOVHPRYHVWKURXJKWKHPHGLXP
7UDQVYHUVH:DYHV(OHPHQWVRIWKHPHGLXPDUHGLVSODFHGSHUSHQGLFXODUO\WR
WKHZDYHSURSDJDWLRQGLUHFWLRQ
/RQJLWXGLQDO:DYHV0HGLXPHOHPHQWVDUHGLVSODFHGSDUDOOHOWRWKHZDYH
SURSDJDWLRQGLUHFWLRQ
6XUIDFH:DYHV'LVSODFHPHQWVDUHERWKORQJLWXGLQDODQGWUDQVYHUVH
7KH6FRUSLRQ
:LWKLQWKHVROLGHDUWKORQJLWXGLQDOZDYHVSURSDJDWHIURPSRLQWWRSRLQWIDVWHUWKDQ
WUDQVYHUVHZDYHV$VDKDSOHVV%HHWOHEXJPRYHVERWKW\SHVRIZDYHVDUHJHQHUDWHGLQ
WKHJURXQG$QHDUE\6FRUSLRQZLOOILUVWGHWHFWWKHORQJLWXGLQDOZDYHLQGLFDWLQJWKH
%HHWOHVGLUHFWLRQDQGWKHWLPHLQWHUYDOEHWZHHQWKLVHYHQWDQGWKHWUDQVYHUVHZDYHDUULYDO
LVXVHGE\WKHFRPSXWDWLRQDOO\DGHSW6FRUSLRQWRIL[DGLVWDQFHWRWKLV%HHWOHVQDFN
6SHHGVQHDUWKHVXUIDFHDUH7UDQVYHUVH9aPV/RQJLWXGLQDO9aPV
'W
G
G
9 9 G
9 9
Î
G
P V 'W
7UDYHOLQJ:DYHV
&RQVLGHUDQRQUHIOHFWHGWUDQVYHUVHZDYHJHQHUDWHGE\DVRXUFHRVFLOODWLQJLQVLPSOH
KDUPRQLFPRWLRQ/HWWKHZDYHSURSDJDWHLQWKH;GLUHFWLRQZLWKRXWHQHUJ\ORVV
(OHPHQWVRIWKHPHGLXPDUHGLVSODFHGYHUWLFDOO\DFFRUGLQJWRWKHZDYHIXQFWLRQ
\ [ W $6LQN[ ZW $ ,VWKHPDJQLWXGHRIPD[LPXPGLVSODFHPHQWZUWHTXLOLEULXPWKH$PSOLWXGH
N ,VWKH$QJXODU:DYH1XPEHULQXQLWVRIUDGLDQPHWHU N S O
Z ,VWKH$QJXODU)UHTXHQF\ Z SI
O ,VWKH:DYHOHQJWKRIWKHZDYH&UHVWWRFUHVWGLVWDQFHIRUH[DPSOH
N[ ZW ,VWKH3KDVHLQUDGLDQV
)RUQRQGLVSHUVLYHZDYHVZKHUH
LQFUHDVLQJ
Z z Z W WKHQDFRQVWDQWSKDVHLPSOLHVWKDWIRU
W [ PXVWDOVRLQFUHDVHLHWKHZDYH \ [ W UHSUHVHQWVDZDYHPRYLQJLQWKH[GLUHFWLRQ
$6LQN[ ZW 2WKHUZDYHSDUDPHWHUVDUH
&\FOH2QHFRPSOHWHPHGLXPHOHPHQWRVFLOODWLRQ
)UHTXHQF\I1XPEHURIF\FOHVRFFXUULQJSHUVHFRQG
3HULRG77KHWLPHLQWHUYDOEHWZHHQUHSHWLWLRQVRIWKHZDYHVKDSH
W
\ [
$6LQN[
$6LQN[
\ [ O $WWLPH
LVWKHDPSOLWXGHRIWKHZDYHDW
DQ\JLYHQ;FRRUGLQDWH)RUDSHULRGLFZDYHWKLVLVDOVRWKHZDYHDPSOLWXGHDWRQH
ZDYHOHQJWKVGLVWDQFH
\ [
$6LQN [ O )URPWKHVLQHRIDVXPRIWZRDQJOHVWKHFRQGLWLRQDERYHLVWUXHLI
NO
S
N
S O
)UHTXHQF\LVUHODWHGWRWKHZDYHDQJXODUIUHTXHQF\DQGWKHSHULRGRIWKHZDYHDV
I
Z
S
7
:DYH9HORFLWLHV
7KHWUDQVYHUVHYHORFLW\RIDPHGLXPPDVVHOHPHQWUHVXOWLQJIURPWKHSDVVDJHRID
WUDQVYHUVHPHFKDQLFDOZDYHLVJLYHQE\WKHSDUWLDOGHULYDWLYHRIWKHSRVLWLRQIXQFWLRQIRU
WKDWHOHPHQWZLWKUHVSHFWWRWKHYDULDEOH W $6LQN[ ZW \ [ W w
$6LQN[ ZW Z$&RVN[ ZW wW
X [ W )XUWKHUIRU
N[ ZW FRQVWDQWWKHSURSDJDWLRQYHORFLW\PD\EHIRXQGDV
G
N[ ZW GW
G[
GW
Z N
IO
Y
IO N
G[
Z
GW
*HQHUDOO\WKHZDYHYHORFLW\GHSHQGVRQWKHPHGLXPHODVWLFLW\DQGWKHPHGLXPLQHUWLD
$SSO\LQJ1HZWRQ
V /DZWRDWUDQVYHUVHZDYHRQDVWULQJZHFDQGHULYHWKH
SURSDJDWLRQYHORFLW\LQWHUPVRIWKHVWULQJWHQVLRQDQGLWVOLQHDUPDVVGHQVLW\
&RQVLGHUDVQDSVKRWRIWKHVWULQJ
Y
¦)\ W ˜ 6LQT W ˜ 6LQT PD 'OP 5 P
W6LQT
PY
'O
5
WT
,QWKHOLPLWRIVPDOODQJOHVWKHWD
)URPJHRPHWU\
T 5
Y
*LYLQJWKHYHORFLW\DV
*LYHQWKLVDQG
I
PDVVOHQJWK
PY
'O
5
'O
W
P
O
IL[HGE\WKHGULYLQJVRXUFHWKHZDYHOHQJWKLV
Y
I
:DYH(TXDWLRQLQ2QH'LPHQVLRQ
7KHZDYHHTXDWLRQLVDVHFRQGRUGHUSDUWLDOGLIIHUHQWLDOHTXDWLRQZKLFKLILWGHULYHV
IURPWKHSK\VLFVRIDSDUWLFXODUSUREOHPFDUULHVWKHLPSOLFDWLRQWKDWDSURSDJDWLQJZDYH
VROXWLRQH[LVWV
$ZDYHHTXDWLRQIRUWKHWUDQVYHUVHZDYHRQDVWULQJUHVXOWVIURP1HZWRQ
V ODZ
¦ )
)˜ 6LQ T )
˜ 6LQ T PD
,QWKHOLPLWDV'[EHFRPHVLQILQLWHVLPDODQGIRUVPDOODQJOHVT
w \
) >7DQ T 7DQ T @ w[ ˜ P wW
w \
) >w7DQ T @ w[ ˜ P wW
w\
) w ˜
w[
w \
w[ w \
w[ ˜ P wW
w \
Y wW w \
' [P wW
(QHUJ\7UDQVSRUW
7KHHQHUJ\RIWKHZDYHLVLQWKHIRUPRINLQHWLFHQHUJ\DVVWULQJPDVVHOHPHQWVPRYH
ZLWKWUDQVYHUVHYHORFLW\
X [ W Z$&RVN[ ZW DQGLQWKHIRUPRI
HODVWLFSRWHQWLDOHQHUJ\DVWKHVWULQJLVVWUHWFKHG 3( v >6WUHWFKLQJ @ G . PG[>Z$&RVN[ ZW @
$ Z PG[ &RV N[ ZW N[ ZW LV]HUR7KLVLVWKHSRLQWDWZKLFKWKH
VWULQJLVFURVVLQJWKHHTXLOLEULXPSRVLWLRQDQG \ [ W LV]HUR
.LQHWLFHQHUJ\LVDPD[LPXPLI
7KHVWULQJLVDOVRPD[LPDOO\H[WHQGHGZKHQ
\ [ W LV]HURDQGWKHUHIRUHWKH
SRWHQWLDOHQHUJ\LVDPD[LPXPDVWKHPDVVHOHPHQWPG[SDVVHVWKURXJKHTXLOLEULXP
:KHQWKHVWULQJPDVVHOHPHQWLVDWPD[LPXPDPSOLWXGHLWVWUDQVYHUVHYHORFLW\LV]HUR
DQGWKHVWULQJH[WHQVLRQLV]HUR.LQHWLFDQGSRWHQWLDOHQHUJLHVDUH]HURDWWKLVORFDWLRQ
7LPHYDULDWLRQRIWKHNLQHWLFHQHUJ\LV
G.
GW
G[
$ Z P &RV N[ ZW GW
$ Z PY&RV N[ ZW $YHUDJLQJWKLVWRREWDLQWKHDYHUDJHUDWHRINLQHWLFHQHUJ\WUDQVSRUWHG
G.
$ Z PY&RV N[ ZW GW 7KHDYHUDJHRIWKHFRVLQHVTXDUHGIXQFWLRQLV
&RV N[ ZW N[ ZW
/HW X
"
&RV N[ ZW GW
³
7 Î
&RV N[ ZW GW
³
%
7
&
'
&RV N[ ZW GX
Z7
ZGW
!#
³!
Z7
$
&RV X GX
*+()
,
§ X 6LQX ·
¨ Z7 © ¹̧ *+
!#"
&RV X
³! GX
G.
0 0
$ Z PY
GW -./ 7KHDYHUDJHUDWHIRUSRWHQWLDOHQHUJ\WUDQVSRUWHTXDOVWKHUDWHRINLQHWLFHQHUJ\WUDQVSRUW
G.
G 3( GW 12
3
GW 123
7KHDYHUDJHSRZHUWUDQVPLWWHGE\WKHZDYHLVWKHQ
7 7
3456
$ Z PY
P Y
\P Z
7KHGHSHQGHQFHKHUHLV
'HSHQGRQWKHPDWHULDODQGWKHVWULQJWHQVLRQ
'HSHQGRQWKHGULYLQJIRUFHWKDWSURGXFHVWKHZDYH
7KHHQHUJ\WUDQVSRUWHGLVSURSRUWLRQDOWRWKHVTXDUHRIWKHZDYHDPSOLWXGH
,QWHQVLW\
,QWHQVLW\LVWKHHQHUJ\FURVVLQJSHUSHQGLFXODUO\WKURXJKDXQLWVXUIDFHDUHDSHUXQLWWLPH
,
3
$8
,QRQHGLPHQVLRQVXFKDVZLWKDWUDQVYHUVHZDYHPRYLQJDORQJDURSHWKHXQLWDUHDLV
IL[HGLQWLPH>FURVVVHFWLRQRIWKHURSH@DQGERWKWKHLQWHQVLW\RIWKHZDYHDQGLWV
DPSOLWXGHDUHFRQVWDQWQHJOHFWLQJORVVHV
)RUDSRLQWVRXUFHLQD'LVRWURSLFPHGLXPZDYHVSURSDJDWHRXWZDUGVSKHULFDOO\DQG
,
3
9
SU
ULVGLVWDQFHIURPWKHVRXUFH
:
7KLVIDOOVRIIDVU
;
:LWKSRZHU HQHUJ\WLPHLQWHQVLW\LVSURSRUWLRQDOWRWKHVTXDUHRIDPSOLWXGH$ , v $
<
)XUWKHULQWHQVLW\LVSURSRUWLRQDOWRU VXFKWKDW
$
v
U
$PSOLWXGHIDOOVRIIDVU DZD\IURPWKHVRXUFH>1RORVVHV@
3
/DVWO\VLQFHWKHSRZHURIWKHZDYHLV $9(
=
$ Z PY
,QWHQVLW\LVSURSRUWLRQDOWRI IRUPHFKDQLFDOZDYHV
,v I
6XSHUSRVLWLRQ,QWHUIHUHQFH3KDVRUV
7KHSULQFLSOHRIVXSHUSRVLWLRQVWDWHVWKDWIRUWZRRUPRUHZDYHVRYHUODSSLQJLQWKHVDPH
UHJLRQRIVSDFHWKHUHVXOWLQJZDYHLQWKDWUHJLRQFDQEHIRXQGE\DQDOJHEUDLFVXPRIWKH
LQGLYLGXDOZDYHV
'HSHQGLQJRQSKDVHGLIIHUHQFHVEHWZHHQWKHZDYHVLHWKHSRLQWVLQWLPHDWZKLFK
HDFKZDYHFUHVWVPD\YDU\WKHUHVXOWLQJZDYHPD\KDYHDQDPSOLWXGHJUHDWHUWKDQDQ\RI
WKHLQFLGHQWZDYHVRULWPD\EHOHVVWKDQWKHLQFLGHQWZDYHDPSOLWXGHVDQGSRVVLEO\]HUR
3KDVRU$GGLWLRQ
3KDVRUVDUHYHFWRUVZLWKPDJQLWXGHHTXDOWRZDYHDPSOLWXGHDQGZKLFKURWDWHFORFNZLVH
DERXWWKHRULJLQZLWKDQDQJXODUYHORFLW\HTXDOWRWKHZDYHVDQJXODUIUHTXHQF\Z
3URMHFWLRQRIDSKDVRURQWRWKHYHUWLFDOD[LVJLYHVWKHVDPHDQDO\WLFIRUPDVDSURSDJDWLQJ
ZDYH
\ [ W \PD[ 6LQN[ ZW ZW
\C 7KHSURMHFWLRQRQWRWKHYHUWLFDOD[LVKDVUDQJH
\ BA>?@
Z
\PD[ \ \ PD[
7RVXSHUSRVHDVHFRQGZDYHRIGLIIHULQJSKDVHLWVSKDVRULVDGGHGWRWKHILUVWUHVXOWLQJLQ
DZDYHKDYLQJWKHIRUP
\ [ W \ P 6LQN[ ZW E \
P ,VWKHDPSOLWXGHRIWKHVXSHUSRVLWLRQZDYH
E ,VWKHSKDVHVKLIWUHODWLYHWR \ [ W ([DPSOH
\PD[ 6LQN[ ZW \PD[ 6LQN[ ZW M G
K
DE
HI
I!ÎWKH ZDYH
ODJV
WKHFZDYH I
I
IÎWKH ZDYH
OHDGV
WKHJZDYH
\ [ W \POLMN
I
\ UTQRS
&RQVLGHULQWHUIHUHQFHEHWZHHQWZRZDYHVZLWKWKHVDPHDPSOLWXGHDQGZDYHOHQJWK
PRYLQJLQWKHVDPHGLUHFWLRQ
\ [ W \ [ W \P 6LQN[ ZW \P 6LQN[ ZW M 2ULHQWLQJWKHILUVW3KDVRUDORQJWKHKRUL]RQWDOVLPSOLILHVWKHYHFWRUDGGLWLRQSUREOHP
\V
\V
I
7KHUHVXOWLQJZDYHLV
\ Z
\ [ W \ P 6LQN[ ZW E :KHUH
XY
XW
> \Z &RVM @ > \Z6LQM @
\ \
6LQFH &RV
[
[
[
\\ &RVM &RV M 6LQ M ]M
\\ &RVM &RVM
\ ^ \^&RV
M
7KHSKDVHRIWKHVXSHUSRVLWLRQZDYHLV
b` \a6LQM
b` 6LQI
E 7DQ
7DQ
M
\a &RVM &RV _
7DQ
E
dc
7DQ 7DQ
M
M
7KHUHVXOWLQJVXSHUSRVLWLRQLV
\ [ W \ e6LQN[ ZW E \e&RV
7KHDPSOLWXGHLVDPD[LPXPLII
7KHDPSOLWXGHLV]HURIRUI
M
M
&RV
M
&RV _
b`6LQ
Mª
M º
Z
6LQ
N[
W
«¬
»¼
FRUUHVSRQGLQJWRIXOO\FRQVWUXFWLYHLQWHUIHUHQFH
S
SFRUUHVSRQGLQJWRIXOO\GHVWUXFWLYHLQWHUIHUHQFH
,QWHUPHGLDWHGHVWUXFWLYHLQWHUIHUHQFHWDNHVSODFHZKHQI
I
S
S
,QWHUPVRIDVKLIWLQZDYHOHQJWKEHWZHHQWKHWZRZDYHVWKHIXOO\FRQVWUXFWLYH
6KLIW
LQWHUIHUHQFHÎ]HURVKLIW
M
S
O
6KLIW
)RUIXOO\GHVWUXFWLYHLQWHUIHUHQFHWKHZDYHOHQJWKVKLIWLV
$VDQXPHULFH[DPSOHOHW
\f
g
M
S
O
O
PPDQGI )LQGDPSOLWXGHW\SHRILQWHUIHUHQFHDQGZDYHOHQJWKRIIVHWEHWZHHQWKHWZRZDYHV
\ h \h&RV
6KLIW
M
S
O
M
O
PP&RV
S
S
PP ! ,QWHUPHGLDWH
O
3KDVRUH[DPSOH/HW
\i [ W PP 6LQN[ ZW S
\j [ W PP 6LQN[ ZW kl
\qpmno
2ULHQWLQJZDYHDORQJWKH;D[LVDQGGUDZLQJWKH 3KDVRUODJJLQJDWGHJUHHV
I
\vurst
\ ‚ƒ
S |
S w
€
|
|
\ {}~ \ {}~&RV \ {xyz6LQ \ …†‡
S „
S „
&RV 6LQ PP
E
ˆ
7DQ ‰
9HUWLFDO
+RUL]RQWDO
ˆ
7DQ ‰
6LQ
S
S
&RV
UDG
\ [ W ODJV \ [ W E\UDGDQGWKHVXSHUSRVLWLRQLV
\ [ W PP 6LQN[ ZW 6WDQGLQJ:DYHV
&RQVLGHUWKHLQWHUIHUHQFHEHWZHHQWZRWUDYHOLQJZDYHVRIHTXDOZDYHOHQJWKDQG
DPSOLWXGHEXWWUDYHOLQJLQRSSRVLWHGLUHFWLRQV
,QWKLVFDVHDVWDQGLQJZDYHLVFUHDWHGWKDWRVFLOODWHVLQSODFHZLWKRXWSURSDJDWLRQKHQFH
WKHWHUP
VWDQGLQJ
7KHLQWHUIHUHQFHUHVXOWVLQSRLQWVDORQJWKHVWULQJZKHUHWKHVWULQJLVVWDWLRQDU\WHUPHG
QRGHVDQGSRLQWVZKLFKRVFLOODWHZLWKPD[LPXPDPSOLWXGHUHIHUUHGWRDVDQWLQRGHV
1RWHWKDWQRGHVDQGDQWLQRGHVDUHVWDWLRQDU\7KHUHLVDORFDOHQHUJ\WUDQVIHUEHWZHHQWKH
QRGHVRIDVWDQGLQJZDYHEXWVLQFHQRHQHUJ\IORZVEH\RQGHDFKQRGDOSRLQWVWDQGLQJ
ZDYHVGRQRWWUDQVIHUHQHUJ\OLNHWUDYHOLQJZDYHV
7KHORFDWLRQRIQRGHDQGDQWLQRGHSRLQWVPD\EHGHWHUPLQHGDVIROORZV
\ [ W \ [ W \ [ W \P 6LQN[ ZW \P 6LQN[ ZW \P 6LQN[ ZW 6LQN[ ZW \ [ W \P 6LQN[&RVZW $PSOLWXGH
1RGHVRFFXUZKHQ
\ P 6LQN[
N[ QS ZLWKQ
N[ Q
$QWLQRGHVRFFXUZKHQ
S
^«`
ZLWKQ ^«`
)LJXUHVKRZLQJRQHF\FOHIRUDVWDQGLQJZDYHZLWKDQWLQRGHV
7UDQVYHUVH6WDQGLQJ:DYHV5HVRQDQFH
)RUDVWULQJEHWZHHQWZRIL[HGERXQGDULHVVWDQGLQJZDYHUHVRQDQFHLVVHWXSDORQJWKH
VWULQJGXHWRWKHERXQGDU\UHIOHFWHGZDYHVLQWHUIHULQJDORQJWKHVWULQJ
7KHIUHTXHQF\DWZKLFKDUHVRQDQFHRFFXUVGHSHQGVRQWKHGLVWDQFHEHWZHHQWKH
ERXQGDULHVDQGWKHZDYHSURSDJDWLRQYHORFLW\VWULQJWHQVLRQDQGPDVVSHUXQLWOHQJWK
$KDOILQWHJUDOQXPEHURIZDYHOHQJWKVFRQWDLQHGZLWKLQWKHERXQGDU\SRLQWVZLOO
SURGXFHDUHVRQDQWVWDQGLQJZDYHFRQGLWLRQ
Q
I
O
/
Y
O
Q
:LWKQ ^«`
Y
/
:KHUHQ ^«`
‹
Q Î)XQGDPHQWDOPRGHRUŠ+DUPRQLF



‘
Ύ2YHUWRQHRU +DUPRQLF
Q Ό2YHUWRQHRUŽ+DUPRQLF
Q
7KHFROOHFWLRQRIDOORVFLOODWLQJPRGHVLVUHIHUUHGWRDVDKDUPRQLFVHULHV
(DFKSDUWLFXODUUHVRQDQFHLVFDOOHGDQRUPDOPRGHRIWKHV\VWHP
7KHIXQGDPHQWDOIUHTXHQF\LV
I“
Y
’
O
/
’
O
/
’
O
/
’
O
Y
O“ZKHUH Y LVWKHZDYHYHORFLW\)RUDVWULQJ
W
P
/
KDUPRQLFKDVZDYHOHQJWKGHWHUPLQHGE\WKHFRQGLWLRQ
”•
7KHQ
/
WKHFRUUHVSRQGLQJQRUPDOPRGHIUHTXHQFLHVDUH
I˜
Q O–DQG
Y
QI —
O˜