Sudfwlfh Sureohpv Zlwk Vroxwlrqv Iru Wrslf 45
Sureohpv=
Erglh Ndqh Pdufxv= ;14 wr ;14:
ENP ;14
Zkdw lv wkh ehwd ri d sruwirolr zlwk H +uR , @ 53(> li us @ 8( dqg H +u , @ 48(B ENP ;15
Wkh pdunhw sulfh ri d vhfxulw| lv '73= Lwv h{shfwhg udwh ri uhwxuq lv 46(= Wkh ulvn0iuhh udwh lv :( dqg wkh pdunhw ulvn suhplxp lv ;(= Zkdw zloo eh wkh pdunhw sulfh ri wkh vhfxulw| li lwv fryduldqfh zlwk wkh pdunhw sruwirolr grxeohv +dqg doo rwkhu yduldeohv uhpdlq xqfkdqjhg,B Dvvxph wkdw wkh vwrfn lv h{shfwhg wr sd| d frqvwdqw glylghqg lq shushwxlw|1 ENP ;16
\rx duh d frqvxowdqw wr d odujh pdqxidfwxulqj frusrudwlrq wkdw lv frqvlghulqj d surmhfw zlwk wkh iroorzlqj qhw diwhu0wd{ fdvk rzv +lq ploolrqv ri grooduv,= \hduv iurp Qrz Diwhu0Wd{ Fdvk Iorz
53
3 4< 43
43 53
Wkh surmhfw*v ehwd lv 4=:= Dvvxplqj wkdw us @ <( dqg H +u , @ 4<(> zkdw lv wkh qhw suhvhqw ydoxh ri wkh surmhfwB Zkdw lv wkh kljkhvw srvvleoh ehwd hvwlpdwh iru wkh surmhfw ehiruh lwv QSY ehfrphv qhjdwlyhB ENP ;17
Duh wkh iroorzlqj wuxh ru idovhB d1 Vwrfnv zlwk d ehwd ri }hur r hu dq h{shfwhg udwh ri uhwxuq ri }hur1 e1 Wkh FDSP lpsolhv wkdw lqyhvwruv uhtxluh d kljkhu uhwxuq wr krog kljko| yrodwloh vhfxulwlhv1 f1 \rx fdq frqvwuxfw d sruwirolr zlwk d ehwd ri 3=:8 e| lqyhvwlqj 3=:8 ri wkh lqyhvwphqw exgjhw lq eloov dqg wkh uhpdlqghu lq wkh pdunhw sruwirolr1 ENP ;18
Frqvlghu wkh iroorzlqj wdeoh/ zklfk jlyhv d vhfxulw| dqdo|vw*v h{shfwhg uhwxuq rq wzr vwrfnv iru wzr sduwlfxodu pdunhw uhwxuqv= Pdunhw Uhwxuq Djjuhvvlyh Vwrfn Ghihqvlyh Vwrfn 8( 5( 6=8( 53
65
47
Zkdw duh wkh ehwdv ri wkh wzr vwrfnvB e1 Zkdw lv wkh h{shfwhg udwh ri uhwxuq rq hdfk vwrfn li wkh pdunhw uhwxuq lv htxdoo| olnho| wr eh 8( ru 53(B f1 Li wkh W0eloo udwh lv ;( dqg wkh pdunhw uhwxuq lv htxdoo| olnho| wr eh 8( ru 53(> gudz wkh VPO iru wklv hfrqrp|1 g1 Sorw wkh wzr vhfxulwlhv rq wkh VPO judsk1 Zkdw duh wkh doskdv ri hdfkB h1 Zkdw kxugoh udwh vkrxog eh xvhg e| wkh pdqdjhphqw ri wkh djjuhvvlyh up iru d surmhfw zlwk wkh ulvn fkdudfwhulvwlfv ri wkh ghihqvlyh up*v vwrfnB d1
4
Li wkh vlpsoh FDSP lv ydolg/ zklfk ri wkh iroorzlqj vlwxdwlrqv lq Sureohpv 9 wr 45 duh srvvleohB H{sodlq1 Frqvlghu hdfk vlwxdwlrq lqghshqghqwo|1 ENP ;19
H{shfwhg Sruwirolr Uhwxuq Ehwd D 53 4=7 E 58 4=5 ENP ;1:
Sruwirolr D E
H{shfwhg Vwdqgdug Uhwxuq Ghyldwlrq 63 73
68 58
ENP ;1;
Sruwirolr Ulvn0iuhh Pdunhw D
H{shfwhg Vwdqgdug Uhwxuq Ghyldwlrq 43 4; 49
3 57 45
ENP ;1<
Sruwirolr Ulvn0iuhh Pdunhw D
H{shfwhg Vwdqgdug Uhwxuq Ghyldwlrq 43 4; 53
3 57 55
ENP ;143
H{shfwhg Sruwirolr Uhwxuq Ehwd Ulvn0iuhh 43 3 Pdunhw 4; 4=3 D 49 4=8 ENP ;144
Sruwirolr Ulvn0iuhh Pdunhw D
H{shfwhg Uhwxuq Ehwd 43 4; 49
3 4=3 =<
ENP ;145
5
H{shfwhg Vwdqgdug Sruwirolr Uhwxuq Ghyldwlrq Ulvn0iuhh 43 3 Pdunhw 4; 57 D 49 55 Lq Sureohpv 46 wr 48 dvvxph wkdw wkh ulvn0iuhh udwh ri lqwhuhvw lv ;( dqg wkh h{shfwhg udwh ri uhwxuq rq wkh pdunhw lv 4;(= ENP ;146
D vkduh ri vwrfn vhoov iru '433 wrgd|1 Lw zloo sd| d glylghqg ri '< shu vkduh dw wkh hqg ri wkh |hdu1 Lwv ehwd lv 4= Zkdw gr lqyhvwruv h{shfw wkh vwrfn wr vhoo iru dw wkh hqg ri wkh |hduB ENP ;147
L dp ex|lqj d up zlwk dq h{shfwhg fdvk rz ri '4> 333 exw dp xqvxuh ri lwv ulvn1 Li L wklqn wkh ehwd ri wkh up lv }hur/ zkhq lq idfw wkh ehwd lv uhdoo| 4> krz pxfk pruh zloo L r hu iru wkh up wkdq lw lv wuxo| zruwkB ENP ;148
D vwrfn kdv dq h{shfwhg udwh ri uhwxuq ri 9(= Zkdw lv lwv ehwdB Zk| zrxog dq|rqh frqvlghu e|lqj wklv ulvn| dvvhw zklfk surylghv dq h{shfwhg uhwxuq ohvv wkdq wkh ulvn0iuhh udwhB ENP ;149
Wzr lqyhvwphqw dgylvruv duh frpsdulqj shuirupdqfh1 Rqh dyhudjhg d 4<( udwh ri uhwxuq dqg wkh rwkhu d 49( udwh ri uhwxuq1 Krzhyhu/ wkh ehwd ri wkh uvw lqyhvwru zdv 4=8> zkhuhdv wkdw ri wkh vhfrqg zdv 4= d1 Fdq |rx whoo zklfk lqyhvwru zdv d ehwwhu suhglfwru ri lqglylgxdo vwrfnv +dvlgh iurp wkh lvvxh ri jhqhudo pryhphqwv lq wkh pdunhw,B e1 Li wkh W0eloo udwh zhuh 9( dqg wkh pdunhw uhwxuq gxulqj wkh shulrg zhuh 47(> zklfk lqyhvwru zrxog eh wkh vxshulru vwrfn vhohfwruB f1 Zkdw li wkh W0eloo udwh zhuh 6( dqg wkh pdunhw uhwxuq zhuh 48(B ENP ;14:
Lq 4<<7 wkh udwh ri uhwxuq rq vkruw0whup jryhuqphqw vhfxulwlhv +shufhlyhg wr eh ulvn0iuhh, zdv derxw 7(= Vxssrvh wkh h{shfwhg udwh ri uhwxuq uhtxluhg e| wkh pdunhw iru d sruwirolr zlwk d ehwd phdvxuh ri 4 lv 45(= Dffruglqj wr wkh fdslwdo dvvhw sulflqj prgho +vhfxulw| pdunhw olqh,= d1 Zkdw lv wkh h{shfwhg udwh ri uhwxuq rq wkh pdunhw sruwirolrB e1 Zkdw zrxog eh wkh h{shfwhg udwh ri uhwxuq rq d vwrfn zlwk @ 3B f1 Vxssrvh |rx frqvlghu ex|lqj d vkduh ri vwrfn dw '73= Wkh vwrfn lv h{shfwhg wr sd| '6 glylghqgv qh{w |hdu dqg |rx h{shfw lw wr vhoo wkhq iru '74= Wkh vwrfn ulvn kdv ehhq hydoxdwhg e| @ =8= Lv wkh vwrfn ryhusulfhg ru xqghusulfhgB Vroxwlrqv= ENP ;14
H +uR , @ us . ^H +u6 , us ` 53 @ 8 . +48 8, @ 48@43 @ 4=8 ENP ;15
6
Li wkh fryduldqfh ri wkh vhfxulw| grxeohv wkhq vr zloo lwv ehwd dqg lwv ulvn suhplxp1 Wkh fxuuhqw ulvn suhplxp lv 9( +@ 46 :,/ vr wkh qhz ulvn suhplxp zrxog eh 45(> dqg wkh qhz glvfrxqw udwh iru wkh vhfxulw| zrxog eh 45 . : @ 4<(= Li wkh vwrfn sd|v d ohyho shushwxdo glylghqg/ wkhq zh nqrz iurp wkh ruljlqdo gdwd wkdw wkh glylghqg/ G> pxvw vdwlvi| wkh htxdwlrq iru d shushwxlw|= Sulfh
@ Glylghqg 2 Glvfrxqw udwh 73 @ G @=46 G @ 73+=46, @ '8=53
Dw wkh qhz glvfrxqw udwh ri 4<(> wkh vwrfn zrxog eh zruwk rqo| '8=53 @ =4< @ '5:=6:= Wkh lqfuhdvh lq vwrfn ulvn kdv orzhuhg lwv ydoxh e| 64=8;(= ENP ;16
Wkh dssursuldwh glvfrxqw udwh iru wkh surmhfw lv= us . ^H +u6 , us ` @ < . 4=:+4< <, @ 59(
Xvlqj wklv glvfrxqw udwh/ QSY @ @
[ f
43 53 . 4=4359| . 4=59 f |' 53 . 43 S D+59(> 43 |hduv , . 43 S I +59(> 43 |hduv ,
@ 48=97
Wkh lqwhuqdo udwh ri uhwxuq rq wkh surmhfw lv 7<=88(= Wkh kljkhvw ydoxh wkdw ehwd fdq wdnh ehiruh wkh kxugoh udwh h{fhhgv wkh LUU lv ghwhuplqhg e| 7<=88 @ < . +4< <, @ 73=88 @43 @ 7=388 ENP ;17
Idovh1 @ 3 lpsolhv H +u, @ us > qrw }hur1 e1 Idovh1 Lqyhvwruv uhtxluh d ulvn suhplxp rqo| iru ehdulqj v|vwhpdwlf ulvn +xqglyhuvl deoh ru pdunhw, ulvn1 f1 Idovh1 :8( ri |rxu sruwirolr vkrxog eh lq wkh pdunhw/ dqg 58( lq eloov1 Wkhq/
d1
R @ =:8 4 . =58 3 @ =:8 ENP ;18
Wkh ehwd lv wkh vhqvlwlylw| ri wkh vwrfn*v uhwxuq wr wkh pdunhw uhwxuq1 Fdoo D wkh djjuhvvlyh vwrfn dqg G wkh ghihqvlyh rqh1 Wkhq ehwd lv wkh fkdqjh lq wkh vwrfn uhwxuq shu fkdqjh lq wkh pdunhw uhwxuq1 Wkhuhiruh/ zh frpsxwh hdfk vwrfn*v ehwd e| fdofxodwlqj wkh gl huhqfh lq lwv uhwxuq dfurvv wkh wzr vfhqdulrv glylghg e| wkh gl huhqfh lq wkh pdunhw uhwxuq1 d1
5 65 @ 5=33 8 53 6=8 47 ( @ @ =:3 8 53
@
7
e1
Zlwk htxdo olnholkrrg ri hlwkhu vfhqdulrv/ wkh h{shfwhg uhwxuq lv dq dyhudjh ri wkh wzr srvvleoh rxwfrphv1
H +u , @ =8+5 . 65, @ 4:( H +u , @ =8+6=8 . 47, @ ;=:8( f1 Wkh VPO lv ghwhuplqhg e| wkh pdunhw h{shfwhg uhwxuq ri =8+53 . 8, @ 45=8(> zlwk d ehwd ri 4> dqg wkh eloo uhwxuq ri ;( zlwk d ehwd ri }hur1 Vhh wkh iroorzlqj judsk1
E(r) A
S ML M
12.5%
ALPHAD
D
8%
.7
1.0
2.0
Beta
Wkh htxdwlrq iru wkh vhfxulw| pdunhw olqh lv=
H +u, @ ; . +45=8 ;, g1 Wkh djjuhvvlyh vwrfn kdv d idlu h{shfwhg uhwxuq ri= H +u , @ ; . 5=3+45=8 ;, @ 4:( dqg wkh h{shfwhg uhwxuq e| wkh dqdo|vw lv dovr 4:(= Wkxv lwv doskd lv }hur1 Vlploduo|/ wkh uhtxluhg uhwxuq rq wkh ghihqvlyh vwrfn lv= H +u( , @ ; . =:+45=8 ;, @ 44=48(/ exw wkh dqdo|vw*v h{shfwhg uhwxuq rq G lv rqo| ;=:8(> dqg khqfh
( @ dfwxdo h{shfwhg uhwxuq 0 uhtxluhg uhwxuq +jlyhq ulvn, @ ;=:8 44=48 @ 5=7( Wkh srlqwv iru hdfk vwrfn sorw rq wkh judsk dv lqglfdwhg deryh1 h1 Wkh kxugoh udwh lv ghwhuplqhg e| wkh surmhfw ehwd/ =:> qrw e| wkh up*v ehwd1 Wkh fruuhfw glvfrxqw udwh lv 44=48(> wkh idlu udwh ri uhwxuq rq vwrfn G1 ENP ;19
Qrw srvvleoh1 Sruwirolr D kdv d kljkhu ehwd wkdq E/ exw lwv h{shfwhg uhwxuq lv orzhu1 Wkxv/ D fdqqrw h{lvw lq htxloleulxp1 ENP ;1:
8
Srvvleoh1 Li wkh FDSP lv ydolg/ wkh h{shfwhg udwh ri uhwxuq frpshqvdwhv rqo| iru v|vwhpdwlf +pdunhw, ulvn uhsuhvhqwhg e| ehwd udwkhu wkdq wkh vwdqgdug ghyldwlrq zklfk lqfoxghv qrqv|vwhpdwlf ulvn1 Wkxv/ D*v orzhu udwh ri uhwxuq fdq eh sdluhg zlwk d kljkhu vwdqgdug ghyldwlrq/ dv orqj dv D*v ehwd lv orzhu wkdq E*v1 ENP ;1;
Qrw srvvleoh1 Wkh uhzdug0wr0yduldelolw| udwlr iru sruwirolr D lv ehwwhu wkdq wkdw ri wkh pdunhw/ zklfk lv lpsrvvleoh dffruglqj wr FDSP/ vlqfh wkh FDSP suhglfwv wkdw wkh pdunhw lv wkh prvw h!flhqw sruwirolr1 Xvlqj wkh qxpehuv vxssolhg/ 49 43 4; 43 @ =8 V @ @ =66 V @ 45 57 Wkh qxpehuv zrxog lpso| wkdw sruwirolr D surylghv d ehwwhu ulvn0uhzdug wudgh0r wkdq wkh pdunhw sruwirolr1 ENP ;1<
Qrw srvvleoh1 Sruwirolr D fohduo| grplqdwhv wkh pdunhw sruwirolr1 Lw kdv d orzhu vwdqgdug ghyldwlrq zlwk d kljkhu h{shfwhg uhwxuq1 ENP ;143
Qrw srvvleoh1 Wkh VPO iru wklv vlwxdwlrq lv= H +u, @ 43 . +4; 43, Sruwirolrv zlwk ehwd ri 4=8 kdyh dq h{shfwhg uhwxuq ri H +u, @ 43 . 4=8+4; 43, @ 55(= D*v h{shfwhg uhwxuq lv 49(> wkdw lv/ D sorwv ehorz wkh VPO +kdv d qhjdwlyh doskd ri 9(,/ dqg khqfh/ lv dq ryhusulfhg sruwirolr1 Wklv lv lqfrqvlvwhqw zlwk wkh FDSP1 ENP ;144
Qrw srvvleoh1 Vdph VPO dv lq sureohp 431 Khuh sruwirolr D*v uhtxluhg h{shfwhg uhwxuq lv= 43 . =<+;, @ 4:=5(> zklfk lv vwloo kljkhu wkdq 49(= D lv ryhusulfhg zlwk d qhjdwlyh doskd ri 4=5(= ENP ;145
Srvvleoh1 Vdph FPO dv vkrzq lq sureohp ;1 Sruwirolr D sorwv ehorz wkh FPO/ dv dq| dvvhw lv h{shfwhg wr1 Wklv vlwxdwlrq lv qrw lqfrqvlvwhqw zlwk wkh FDSP1 ENP ;146
Vlqfh wkh vwrfn*v ehwd lv htxdo wr 4> lwv h{shfwhg udwh ri uhwxuq vkrxog eh htxdo wr wkdw ri wkh pdunhw/ wkdw lv/
4;(=
H +u, @ 4; @
G . S Sf Sf < . S 433 433
S @ '43< ENP ;147
Dvvxph wkdw wkh '4> 333 lv d shushwxlw|1 Li ehwd lv }hur/ wkh fdvk rz vkrxog eh glvfrxqwhg dw wkh ulvn0iuhh udwh/ ;(=
SY
@ 4333 @=3; @ '45> 833
Li/ krzhyhu/ ehwd lv htxdo wr 4> wkh lqyhvwphqw vkrxog |lhog 4;(> dqg wkh sulfh sdlg iru wkh up vkrxog eh=
SY
@ 4333 @=4; @ '8> 888=89
9
Wkh gl huhqfh/ '9> <77=77/ lv wkh dprxqw |rx zloo ryhusd| li |rx huurqhrxvo| dvvxphg wkdw ehwd lv }hur udwkhu wkdq 4= Li wkh fdvk rz odvwv rqo| rqh |hdu=
S Y + @ 3, @ 4333 @+4 . =3;, @ '<58=<6 S Y + @ 4, @ 4333 @+4 . =4;, @ ';7:=79 zlwk d gl huhqfh ri ':;=7:= Iru dq| q |hdu fdvk rz wkh gl huhqfh lv= 4> 333
kS
HI 4 fH?
l
4> 333
HI
k
S
4 H ?
l
ENP ;148
Xvlqj wkh VPO= 9 @ ; . +4; ;,> zh jhw wkdw
@ 5@43 @ =5
Wklv dvvhw kdv d qhjdwlyh pdunhw ehwd/ l1h1 lwv uhwxuq lv qhjdwlyho| fruuhodwhg zlwk wkdw ri wkh pdunhw1 Wklv dvvhw fdq wkxv vhuyh dv dq lqvxudqfh djdlqvw v|vwhpdwlf ulvn1 Lqvwhdg/ wkh uhwxuq rq wkh ulvn0iuhh dvvhw lv vlpso| xqfruuhodwhg zlwk wkh pdunhw uhwxuq1Wklv lv zk| wkh vwrfn kdv d orzhu h{shfwhg uhwxuq1 ENP ;149
u @ 4<(> u2 @ 49(> @ 4=8> 2 @ 4 Wr whoo zklfk lqyhvwru zdv d ehwwhu suhglfwru ri lqglylgxdo vwrfnv zh orrn dw wkhlu deqrupdo uhwxuq/ zklfk lv wkh h{0srvw doskd/ wkdw lv/ wkh deqrupdo uhwxuq lv wkh gl huhqfh ehwzhhq wkh dfwxdo uhwxuq dqg wkdw suhglfwhg e| wkh VPO1 Zlwkrxw lqirupdwlrq derxw wkh sdudphwhuv ri wklv htxdwlrq +ulvn0iuhh udwh dqg pdunhw udwh ri uhwxuq, zh fdqqrw whoo zklfk rqh lv pruh dffxudwh1 e1 Li u @ 9( dqg u @ 47(> wkhq +xvlqj wkh qrwdwlrq ri doskd iru wkh deqrupdo uhwxuq, d1
s
@ 4< ^9 . 4=8+47 9,` @ 4< 4; @ 4( 2 @ 49 ^9 . 4+47 9,` @ 49 47 @ 5(
Khuh/ wkh vhfrqg lqyhvwru kdv wkh odujhu deqrupdo uhwxuq dqg wkxv kh dsshduv wr eh d pruh dffxudwh suhglfwru1 E| pdnlqj ehwwhu suhglfwlrqv wkh vhfrqg lqyhvwru dsshduv wr kdyh wlowhg klv sruwirolr wrzdug xqghusulfhg vwrfnv1 @ 48(> wkhq f1 Li u @ 6( dqg u s
@ 4< ^6 . 4=8+48 6,` @ 4< 54 @ 5( 2 @ 49 ^6 . 4 +48 6,` @ 49 48 @ 4(
Khuh/ qrw rqo| grhv wkh vhfrqg lqyhvwru dsshdu wr eh d ehwwhu suhglfwru/ exw wkh uvw lqyhvwru*v suhglfwlrqv dsshdu ydoxhohvv + ru zruvh,1 ENP ;14: d1 e1
Vlqfh wkh pdunhw sruwirolr e| gh qlwlrq kdv d ehwd ri 4> lwv h{shfwhg udwh ri uhwxuq lv 45(= @ 3 phdqv qr v|vwhpdwlf ulvn1 Khqfh/ wkh sruwirolr*v idlu uhwxuq lv wkh ulvn0iuhh udwh/ 7(= :
f1 Xvlqj wkh VPO/ wkh
idlu
udwh ri uhwxuq ri d vwrfn zlwk
@ 3=8 lv=
H +u, @ 7 . +=8,+45 7, @ 3( Wkh
h{shfwhg
udwh ri uhwxuq/ xvlqj wkh h{shfwhg sulfh dqg glylghqg ri qh{w |hdu=
H +u, @ 77@73 4 @ =43>
ru
43(
Ehfdxvh wkh h{shfwhg uhwxuq h{fhhgv wkh idlu uhwxuq/ wkh vwrfn pxvw eh xqghusulfhg1
;
