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The Derivation of New Model of Equity Duration

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					WORKING PAPERS SERIES WP01-11

The Derivation of New Model of Equity Duration
Richard Lewin and Steve Satchell

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Abstract:
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THE DERIVATION OF A NEW MODEL OF EQUITY DURATION
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1. Introduction:! !
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6

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7

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!

CD + @ H 3 ? = 0 7? !2! ! CD + > H 3 3 =D
1

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?=

7? CD + @ H !2! C> * @ H

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! N1-!*'5$18!4+8%!+.!(/,'*$+1!CD9_H!@(!'))$3(!'*M!

41 ? = 41 7? + 41CD + @ H * 41C> * @ H 2! !!!!!!!!!!!CD9FH9!

! `0!-$..()(1*$'*$18!*#$%!)(%,4*2!@(!+7*'$1!*#(!%*'1-')-!PPR!-,)'*$+1!.+)6,4'*$+1M!
!

P PPR = *
!

- 41 ? D = !2!! hA($7+@$*B!"3,/.<2!(/,'*$+1!ei! (> * @ ) -!>

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!! !! "#(!&)(;(-$18!(/,'*$+1!@$44!7(!)(;+81$%(-!'%!*#(!PPR!+.!T+)-+1!CDEGUH!*#'*!#'%! 7((1!;)$*$;$%(-!.+)!$*%!4';5!+.!&)';*$;'4!;+1%$-()'*$+1%2!6+%*!1+*'740!*#(!'%%,6&*$+1!*#'*!-$3$-(1-! 8)+@*#!$%!6+-(44(-!'%!'1!$1.$1$*(!8(+6(*)$;!&)+8)(%%$+19!

!

8

3. Calculating Equity Duration from the Rubinstein Model:
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

!

!

!

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! N%!$6&4$(-!$1!(/,'*$+1!CD9>H2! &3 !;'1!7(!%&4$*!$1*+!'!)('4!0$(4-!;+6&+1(1*2!-(.$1(-!#()(!'%! > 2!'1-!'1! (/,$*0!&)(6$,6!)(4'*(-!*+!;+1%,6&*$+19! !
?3 = 0
0 =D 1

*3 C 73 +0A3 +0 H !2! CD + > H 0 *3 CA3 +0 H
1

!

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! "#,%2!70!*'5$18!4+8%!+.!!CU9UHM!!
41 ?3 = 41 0
0 =D

*3 C 73 +0A3 +0 H !2!! CD + > H 0 *3 CA3 +0 H

!

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! ! I1!'--$*$+12!70!-$..()(1*$'*$18!CU9SH2!@(!;'1!1+@!-(.$1(!-,)'*$+1!'%!.+44+@%M! ! 1 - 41 ?3 *3 C 73 +0A3 +0 H = 00 ?3 !2! ! ! !!!!!!!!! !!! !!!!!! ! 0 +D -!> *3 CA3 +0 H 0 =D CD + > H

!

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! !! !! ^(! @$44! 1+@! '--! '1! (<*)'! %(*! +.! '%%,6&*$+1%2! $1! %$6$4')! 6'11()! *+! *#(! *)('*6(1*! &,)%,(-! $1! P'6'1*! '1-! X'*;#(44! CDEE>H9! ! "#,%2! *#(! 4+8%! +.! JK! '88)(8'*(! ;+1%,6&*$+1! '1-! JK! '88)(8'*(! -$3$-(1-%! ')(! -(.$1(-! '%! 7($18! )'1-+6! @'45%! @$*#! -)$.*2! %,;#! *#'*! *#(0! @$44! g+$1*40! .+44+@! '!7$3')$'*(!)'1-+6!@'45!@$*#!-)$.*9!!^(!.,)*#()!-(.$1(!+,)!)(&)(%(1*'*$3(!'8(1*!*+!#'3(!'! &+@()!,*$4$*0!.,1;*$+12!;+1%$%*$18!+.!*#(!.+44+@$18!'%%,6&*$+1M! !
! 9

Assumption 1:
B C- 3 H = - 3D* 4 2!!!!!!!!B 3C- 3 H = - 3* 4 = A3 2!!!!!!!!B 33C- 3 H = * 4- 3*CD+ 4 H 2 ! D* 4

!! `0! %,7%*$*,*$18! N%%,6&*$+1! D! $1*+! (/,'*$+1! CU9eH2! @(! '))$3(! '*! *#(! .+44+@$18! )(&)(%(1*'*$+12!@#()(!'!)(&)(%(1*%!-,)'*$+1M! !
'=
1 - 41 ?3 *3 C 73 +0 -3*+4 H 0 = 00 ?3 !2! 0 +D -!> *3 C-3*+4 H 0 =D CD + > H 0

!

!

!!!!!!!!!!!CU9>H9!

! f+@!6'5$18!*#(!'%%,6&*$+1!*#'*!4!k!?2!@#$;#!$%!*#(!)$%5!1(,*)'4!;'%(2!*#(1!CU9>H!@$44!7(;+6(M! ! 1 * C7 H ' = 0 0 3 3 +00+D ?3 !2! ! ! !!!!!!!!!! !!!!!!!!!!!CU9GH9! 0 =D CD + > H ! !! `0! '%%,6$18! *#'*! *3 C 73 +0 H = 73 CD + @ H 0 2! @#()(! 8! $%! *#(! (<&(;*(-! 8)+@*#! )'*(! $1!
73 CD + @ H D !2!*+!.$1-!+1;(!'8'$1!*#'*! ' = !2!'%!$1!CD9EH9! (> * @ ) C> * @ H

-$3$-(1-%2!@(!;'1!%+43(!.+)! ?3 = !

!! R+)(! 8(1()'440! #+@(3()2! @(! 1((-! *+! %&(;$.0! ;()*'$1! '%%,6&*$+1%! '7+,*! *#(! g+$1*! &)+7'7$4$*0! -(1%$*0! .,1;*$+1! C&-.H! +.! 73! '1-! -32! @#$;#! @(! @$44! 1+@! '%%,6(! .+44+@%! '! 7$3')$'*(! 4+81+)6'4!)'1-+6!@'459! ! Assumption 2:! 41C 73 H = 5 ' + 41C 73 *D H + ) '3 2! ! ! ! ! CU'H9!
41C-3 H = 5 2 !+ 41C-3 *D H + ) 23 2! ! ; (" @#()(! C) '3 2 ) 23 H3 j C 9C?2?H32 & '' &" ' 2' :

!

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" '2 %8 # 9! " 22 #6 $7

Proposition 1:! ! !! ! ! !

! I.!N%%,6&*$+1%!D!'1-!U!#+4-2!*#(1!*#(!&)$;(!+.!'1!'%%(*!?*!$%!8$3(1!70M! !!!!! 41 (?3 ) = 41 (73 ) * 41CD + > * < H + 41 < !2!!
D ; 8 @#()(! !< !!= (<&!95 ' + " '' * 4" 2' 6 !2! U : 7 ! -41 (?* ) D = ! 2!!! !!!!!"#,%2! ' = * -!> D+ > *<

!

!

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!

!

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! @#()(! > !)(&)(%(1*%!)('4!0$(4-%2!'1-! < !< D + > 9 !hO)++.M!%((!N&&(1-$<!Di9!
! 10

!! "#(! '7+3(! ;'4;,4'*$+1%! ;'1! 7(! ;+1%$-()(-! '! .+)6! +.! *#(! T+)-+1! T)+@*#! R+-(42! @#()(! < ! ;+))(%&+1-%! *+! *#(! (D + @ ) ! *()62! '1-! )(&)(%(1*%! *#(! (<&(;*(-! &()! &()$+-! )$%5='-g,%*(-! )(4'*$3(!-$3$-(1-!8)+@*#9!!X$1;(! 4 = ? !"@(!%((!*#'*!$.! " 2' > ? 2!@#$;#!@(!@+,4-!(<&(;*2!'1!$1;)('%(! $1! )(4'*$3(! )$%5! '3()%$+1! 4('-%! *+! '! -(;)('%(! $1! < ! '1-! #(1;(! '! -(;)('%(! $1! (/,$*0! -,)'*$+19!! c+@(3()2! @(! 6,%*! 7(! '@')(! +.! *#(! (/,$*0! &)(6$,6! &,BB4(2! '%! $-(1*$.$(-! 70! R(#)'! '1-! O)(%;+**! CDEF>H2! @#$;#! %,88(%*%! *#'*! .+)! &+@()! ,*$4$*02! )$%5! &)(6$'! ')(! .')! *++! 4+@! *+! (<&4'$1! (6&$)$;'4! 3'4,(%9! ! N4*()1'*$3(402! 4 "1((-%! *+! 7(! /,$*(! 4')8(! *+! 8(1()'*(! '&&)+&)$'*(! )$%5! &)(6$'! +.! *#(! 6'81$*,-(! (%*$6'*(-! $1! 6'*,)(! .$1'1;$'4! 6')5(*%9! ! N;;+)-$18402! @(! %(4(;*! *+! ,%(! '1! '4*()1'*$3(! 4$1(')! )$%5=*+4()'1;(! ,*$4$*0! .,1;*$+12! *#(! cNVN! %&(;$.$;'*$+12! @#()(70! @(! )(-(.$1(! B C-3 H "'%! .+44+@%M!! ! Assumption 3:!
B C-3 H = C-3 * -3 HD* 4 D * 4 m!!4 > ?2!!- 3 > -3 m!!-3 > ? 9_!

! !! c()(! -3 ! )(&)(%(1*%! 7'%(! ;+1%,6&*$+12! +)! *#(! 6$1$6,6! %,7%$%*(1;(! 4(3(4! +.! ;+1%,6&*$+1! )(/,$)(-! *+! &)+3$-(! *#(! 1(;(%%$*$(%! $1! 4$.(9! ! "#$%! $%! '1'4+8+,%! *+! @#'*! R')<! ;+1%$-()(-!*#(!&)+4(*')$'*a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o!

>!o!?9!!"#$%!'&&)+';#!$%!;+1%$%*(1*!@$*#!+*#()!(6&$)$;'4!&)+;(-,)(%!$1!*#(!;+1%,6&*$+1!4$*()'*,)(!'%! '-+&*(-! 70! [+1%*'1*$1$-(%! CDEE?H! '1-! X,1-')(%'1! CDEFEH9! ! `(.+)(! @(! (3'4,'*(! *#$%! 6+-(4! (6&$)$;'4402!@(!%#+,4-!;+1%$-()!)(%&(;$.0$18!*#(!-'*'!8(1()'*$18!&)+;(%%!+.! -3 9!!N;;+)-$18402!@(!
@$44!'%%,6(! -3 !$%!-(.$1(-!'%!.+44+@%M! ! Assumption 4: !
3 3 !!! !C-3 * -3 H!!!!!=!!(-? * -3 )!!!!(<&!(!5 2 3 + 0 ) 23 + D ) 9! 3 3 !!I1!'--$*$+12! C-3 * -3 H * 4 = (-? * -3 ) (<&!(45 2 3 + 4 0 ) 23 + D )2!
*4

!

!!!!!!!!!!!CS9SH9!

3 3 3 @#()(! ) 23 j C C?2 " 22 H 2!'1-! " 2' @$44!7(;+6(! " 2' 9!

'

!! !

! 3 V(.+)6,4'*$18!&)+&+%$*$+1!D!@$*#!N%%,6&*$+1%!S!'1-!e2!@(!(<&(;*! 5 2 !*+!7(!)(-,;(-!

3 )(4'*$3(!*+! 5 2 '1-! " 22 !*+!7(!4')8()!*#'1! " 22 9!!!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
_

!N%%,6&*$+1!S!7(;+6(%!N%%,6&*$+1!D2!'1-!4$5(@$%(!N%%,6&*$+1!e!@$44!7(;+6(!N%%,6&*$+1!U7!$.! - = ? 9!

!

11

Proposition 2: ! ! !! !I.!N%%,6&*$+1%!S!'1-!e!#+4-2!*#(1!*#(!1(*!(..(;*!+.!*#(%(!;#'18(%!+1!(/,'*$+1!CU9EH2! CS9?H!'1-! < !$1!&')*$;,4')2!$%!*#'*! < !7(;+6(%! < 3 9!
D 3 @#()(! < 3 = (<&!h5 ' + " '' * 4" 2' i ! ! U D !! ! !!!!!!!!@#$4%*! ' 3 = ! D+ > *<3
!

! !

!!!!!!!!!!!CS9eH9! !!!!!!!!!!!CS9>H9!

!

!

3 "#,%!$.! " 2' !7(;+6(%!4')8()!.+)!'10!8$3(1! 4 2! < 3 !7(;+6(%!%6'44()!'1-! ' 3 !7(;+6(%!

%6'44()9! ! "#(! '7+3(! ')8,6(1*! (..(;*$3(40! (%*'74$%#(%! '! 4$15! 7(*@((1! *#(! (/,$*0! &)(6$,6! &,BB4(! '1-!*#(!4')8(!3'4,(%!.+,1-!.+)!(/,$*0!-,)'*$+19!!"#()(.+)(2!*+!)($1.+);(!*#$%!&+$1*2!@(!@$44!'-+&*!*#(! .+44+@$18! 1,6()$;'4! (<'6&4(! $1! +,)! @+)59! ! I.! 4 = U2 5 ' = ?9?e C$81+)$18! $1.4'*$+1H2! !" '' = ?9??D 2!
" 2' = ?9??U !'1-! > = ?9?> 9!
@!!' = D D9?> * (<&C?9?e + ?9???> * ?9??eH ! D D = ? _F!0(')% !!!!!!!= D9?> * (<&C?9?SG>H ?9?DUF

!

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12

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F - 41 ?3 *3 C 73 +0 -3* 4 H +0 = 00 ?3 !2! ! ' !!= 0 +D -!> *3 C-3* 4 H 0 =D CD + > H +0

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! 13

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D ; 8 ^#()(! !< !!= (<&!95 ' + " '' * 4" 2' 6 2!'%!7(.+)(!$1!(/,'*$+1!CS9DH2! U : 7 !
F f+@2!#+@(3()2 !41 (?3 )!!=!!41 (73 ) * 41CD + > * < H + 41 (< ) * F 41 (D + > ) + 41 (D + > ) * < F 2!

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F *D

! ! ^(!1+*(!.+)!;+13()8(1;(! < < D + > 2!'%!F B 1 2!7,*!.+)!.$1$*(!F;!1+!%,;#!;+1-$*$+1!$%!)(/,$)(-9! !! ! J1-()! N%%,6&*$+1! S2! < ! @$44! '8'$1! 7(;+6(! < 3 ! '%! $1! (/,'*$+1! CS9eH9! ! A$5(@$%(2!

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m! @(! &)+3$-(! '1! (..$;$(1*! 6(*#+-! .+)! 8(1()'*$18! %,7%(/,(1*! (/,$*0! -,)'*$+1! (%*$6'*(%9! ! ^(! *#,%! $44,%*)'*(! '1! '&&4$;'*$+1! +.! *#$%! '&&)+';#! @$*#$1! +,)! )(%,4*%2! 70! $1;4,-$18!;+6&')'*$3(!)(;'4;,4'*$+1%!+.!*#(!;+))(%&+1-$18!(/,$*0!-,)'*$+1!&')'6(*()%!$1!"'74(!S2! @#()(!@(!#'3(!*'5(1!F!k!D??!0(')%!'%!'!&4',%$74(!$44,%*)'*$3(!'%%,6&*$+19! !

!

14

4. Empirical Estimation of Equity Duration in the UK Stock Market:
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m! %((! `'))!'1-!['6&7(44!CDEE_H9!!P,)$18! DEEF! '1-! DEEE2! ;+13(1*$+1'4! 8$4*%! 0$(4-(-! 7')(40! 6+)(! *#'1! Sb! +3()! ;+6&')'74(! $1-(<=4$15()%9!! "#$%! #$%*+)$;'440! 4+@! )('4! 0$(4-! %,88(%*%! *#'*! $13(%*+)%! 7(4$(3(! *#(! )(;(1*40! $1-(&(1-(1*! `'15! +.! :184'1-! @$442! +3()! *#(! 4+18! ),12! %,;;((-! $1! 6'$1*'$1$18! $1.4'*$+1! @$*#$1! *#(! T+3()16(1*a%! %(4.= $6&+%(-! *')8(*! )'18(! +.! U9>b9!!L,)!#$%*+)$;'4!%'6&4(!&()$+-2!#+@(3()2!#'%!@$*1(%%(-!%,;;(%%$3(! &+4$*$;'4! )(8$6(%! *#'*! ;+1%$%*(1*40! .'$4(-! *+! ';#$(3(! '10*#$18! ;4+%(! *+! *#$%! %+6(@#'*! '67$*$+,%! *')8(*!=!$1-((-!)('4!)'*(%!'8'$1!'46+%*!)(';#(-!Fb!'%!)(;(1*40!'%!DEE?9!!"#(%(!4+@!;,))(1*!0$(4-%! '&&(')! ;+1%$%*(1*! @$*#! '1! +3())$-$18! (<&(;*'*$+1! *#'*! *#(! JK! @$44! (1*()! *#(! :,)+&('1! %$184(! ;,))(1;0! $1! *#(! $66$1(1*! .,*,)(2! @#$;#! 6'0! 0(*! '44+@! ;+13(1*$+1'4! 8$4*! 0$(4-%! *+! .'44! .,)*#()! $1! 4$1(!@$*#!*#($)!:,)+&('1!;+,1*()&')*%9!!T4+7'4!$1*()(%*!)'*(%!.+)!7+*#!;+13(1*$+1'4!'1-!$1-(<=4$15(-! 7+1-%!')(!'4%+!'*!#$%*+)$;'440!4+@!4(3(4%!&)(%(1*402!$1;)('%$18!*#($)!3,41()'7$4$*0!*+!'10!)(%,)8(1;(! $1!;+66+-$*0!+)!@'8(!&)$;(!$1.4'*$+12!@#$;#!;+,4-!&)+3+5(!'!;+=+)-$1'*(-!$1*()1'*$+1'4!)'*(!)$%(! +3()!*#(!4+18()!*()69!! !! `,$4-$18! ,&+1! *#(! ;+13(1*$+1'4! '&&)+';#! $1! *#(! (6&$)$;'4! 4$*()'*,)(2! +,)! ;+1%,6&*$+1!6('%,)(!#'%!7((1!-()$3(-!.)+6!1+1=-,)'74(!'1-!%()3$;(!$1-,%*)0!(<&(1-$*,)(!$1!*#(! JK9! ! "#$%! $6&4$;$*40! '%%,6(%! *#'*! ,*$4$*0! $%! %(&')'74(! ';)+%%! *#$%! .+)6! +.! ;+1%,6&*$+1!'1-!+*#()! %+,);(%!+.!,*$4$*09!!"+!'44+@!.+)!*#(!1+1=%(&')'7$4$*0!$1!,*$4$*0!+3()!*$6(2!@(!#'3(!$1;+)&+)'*(-!'! %$6&4(! '-g,%*6(1*! *+! '44+@!.+)!#'7$*!.+)6'*$+12!+)!*#(!&+%$*$3(!(..(;*!+.!*+-'0a%!;+1%,6&*$+1!+1! *+6+))+@a%! 6')8$1'4! ,*$4$*0! +.! ;+1%,6&*$+19! ! L,)! &()! ;'&$*'! '-g,%*6(1*! .';*+)! 6'5(%! ,%(! +.! *#(! JK!&+&,4'*$+1!%()$(%!'%!;+6&$4(-!70!*#(!L..$;(!+.!f'*$+1'4!X*'*$%*$;%!CLfXH9!!^(!#'3(!6'-(!,%(! +.!'1!$6&4$;$*!&)$;(!-(.4'*+)!%()$(%!*+!+7*'$1!+,)!$1.4'*$+1!6('%,)(%2!'1-!*+!.,)*#()!'-g,%*!+,)!-'*'! *+!)(.4(;*!+,)!-(%$)(!*+!@+)5!$1!)('4!*()6%9!!`0!,%(!+.!(/,'*$+1!CS9DH2!CS9UH!'1-!CS9SH2!@(!;'1!-()$3(! 3 (%*$6'*(%!.+)! " 2' 2!!5 ' ! '1- !" '' !.)+6!+,)!JK!6')5(*!-'*'9!!^(!*#(1!,%(!+,)!*#(+)(*$;'4!.)'6(@+)5! *+!(6&$)$;'440!(%*$6'*(!(/,$*0!-,)'*$+1!&')'6(*()%!-$)(;*40!.)+6!JK!6')5(*!-'*'2!,%$18!(/,'*$+1! CS9>H! '1-! CS9EH9! ! "#(! ;#')';*()$%*$;%! +.! +,)! ,1-()40$18! (6&$)$;'4! -'*'! %(*! ')(! %,66')$%(-! 70! *#(! .+44+@$18!*'74(M! Table 1:
Quarterly Data 1963-99

Empirical Datasheet Summary Statistics:
FT-All Share Yield Inflation Rate FT-All Share Real Return

R('1! X*'1-')-!P(3$'*$+1! X5(@1(%%! K,)*+%$%! R$1$6,6! R'<$6,6!
!

?9?eGG! ?9?DUF! D9G__e! E9FeSE! ?9?USE! ?9DD_D!
15

?9?GG_! ?9?eGE! D9eS?_! e9>_GG! ?9?DUD! ?9UU?>!

?9?_e?! ?9UUG?! =D9>?>G! F9F?DU! =?9E>>E! ?9_eE>!

Table 1:!C;+1*$1,(-H Empirical Datasheet Summary Statistics: ! Quarterly Data 1963-99 Real Dividend Growth Real Treasury Yield Real Consumption Growth R('1! ?9?DS_! ?9?UEF! ?9?UDG! X*'1-')-!P(3$'*$+1! ?9?GDe! ?9?e?F! ?9?DEU! X5(@1(%%! =?9D_GG! =?9_FUe! ?9US>U! K,)*+%$%! U9S>FE! e9Se>F! U9_?S_! R$1$6,6! =?9DSE?! =?9D?GE! =?9?DFF! R'<$6,6! ?9DSeF! ?9DDG>! ?9?G__! ! Figure 1:

Real Consumption versus Real Dividends (1963-1999) 250 Rebased Series (Q1 1963 =100) 200 150 100 50 0

Real Consumption

Real Dividends

!! "#(!&)(;(-$18!$44,%*)'*$+1!+.!)('4!-$3$-(1-!'1-!)('4!;+1%,6&*$+1!8)+@*#!$1!Z$8,)(!D! #$8#4$8#*%!*#(!-$3()8(1;(!$1!-$3$-(1-!8)+@*#!-,)$18!*#(!DE_?%!'1-!DEF?%9!!V('4!-$3$-(1-%!.(44!$1! )(%&+1%(! *+! *#(! )'3'8(%! +.! $1.4'*$+1! $1! *#(! DE_?%2! .+44+@(-! 70! '! &()$+-! +.! )'&$-! )('4! -$3$-(1-! 8)+@*#! $1! *#(! DEF?%2! &')*40! $1! )(%&+1%(! *+! *#(! %,7%(/,(1*! 6')5(*! )(.+)6%! $1*)+-,;(-! ,1-()! *#(! "#'*;#()! '-6$1$%*)'*$+19! ! "#(! 8)'&#! $44,%*)'*(%! *#'*! )(=7'%(-! )('4! ;+1%,6&*$+1! #'%! 7((1! .')! 4(%%! 3+4'*$4(! *#'1! -$3$-(1-%2! '*! 4('%*! .)+6! '!#$%*+)$;'4!&()%&(;*$3(9!!"#(!-$3$-(1-!%()$(%!$44,%*)'*(%!*#(! $6&';*! +.! *#(! %,;;(%%$3(! -$3$-(1-! ;+1*)+4! )(8$6(%! '&&4$(-! -,)$18! *#(! DE_?%9! ! P$3$-(1-! ;+1*)+4%! @()(!.$)%*!$1*)+-,;(-!'%!&+4$;0!$1%*),6(1*%!$1!*#(!4'*(!DEG?%!'1-!@()(!;+1*$1,+,%40!(6&4+0(-!70! %,;;(%%$3(! 8+3()16(1*%! .)+6! P(;(67()! DE_U! ,1*$4! Y,40! DE_E9! ! "#(! )'*$+1'4(! 7(#$1-! $6&+%$18! ;,)7%! +1! -$3$-(1-! &'06(1*%! '%%,6(%! '! -(8)((! +.! %*+;5! 6')5(*! $1(..$;$(1;0! $1! *#(! '44+;'*$+1! +.! .,1-%2! '1-! *#()(70! %((5%! *+!$1;)('%(!)(*'$1(-!(')1$18%!$1!+)-()!*+!$6&)+3(!*#(!%,&&40!+.!+)8'1$;! ;+6&'10!.$1'1;$18!.+)!.,*,)(!$13(%*6(1*!+&&+)*,1$*$(%9!!!
! 16

19 63 19 Q1 65 19 Q1 67 19 Q1 69 19 Q1 71 19 Q1 73 19 Q1 75 19 Q1 77 19 Q1 79 19 Q1 81 19 Q1 83 19 Q1 85 19 Q1 87 19 Q1 89 19 Q1 91 19 Q1 93 19 Q1 95 19 Q1 97 19 Q1 99 Q 1

!! N1!$6&+)*'1*!.';*+)!*#'*!;+1*)$7,*(-!*+!*#(!4+18(3$*0!+.!*#(%(!6(;#'1$%6%!@'%!*#($)! $1#()(1*!;+1%$%*(1;0!@$*#!*#(!6';)+(;+1+6$;!+7g(;*$3(%!+.!*#(!-'09!!"#(%(!@()(!(%%(1*$'440!7'%(-! +1! $1;+6(! &+4$;$(%! *#'*! ';*(-! '%! )(%*)'$1*%! +1! *#(! )$%$18! 4(3(4! +.! (')1(-! $1;+6(%2! @$*#! -$3$-(1-%! ;+1*)+4%!*'1*'6+,1*!*+!)(%*)'$1*%!+1!,1(')1(-!$1;+6(2!'1-!*#,%!'!1'*,)'4!$-(+4+8$;'4!;+6&4(6(1*! *+@')-%! .+)6,4'*$18! ;+1%$%*(1*! (;+1+6$;! &+4$;09! ! c()(! @(! #$8#4$8#*! *#(! $6&4$;'*$+1%! +.! *#(%(! ;+1*)+4%!+1!+,)!+@1!-$3$-(1-!%()$(%2!7,*!)(.()!*#(!$1*()(%*(-!)('-()!*+!'!6+)(!;+6&4(*(!';;+,1*!+.! *#(%(!(..(;*%!'%!-(%;)$7(-!$1!T+,-$(!'1-!c'1%(1!CDEFFH9!! !! "#(!1(<*!%(;*$+1!;+1*'$1%!+,)!%,66')0!)(%,4*%!$1!"'74(%!U!'1-!S2!;'4;,4'*(-!,%$18! (/,'*$+1!CS9>H!'1-!CS9EH!)(%&(;*$3(409!!I1!*#(%(!*'74(%!@(!&)(%(1*!(6&$)$;'4!)(%,4*%!.+)!)('4!0$(4-%!+.! >! k! eb2! .+)! 7+*#! (/,$*0! -,)'*$+1! 6+-(4%2! @#()(! @(! #'3(! *'5(1! F! k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c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e2! &+)*)'0$18! -$..()(1;(%! $1! (/,$*0! -,)'*$+1! (%*$6'*(%! +7*'$1(-! .)+6! 7+*#! 6+-(4%9! ! "#$%! )(3('4%! *#(! &)';*$;'4! %$81$.$;'1;(!+.!;+1%$-()$18!*#(!4+18(3$*0!+.!4$%*(-!(/,$*0!%(;,)$*$(%!*#(6%(43(%2!70!;#'18$18!3'4,(%! .+)!F2!+,)!;#+%(1!6('%,)(!+.!*#(!'3()'8(!4$.(!%&'1!.+)!(/,$*0!%(;,)$*$(%9!!Z$1'4402!Z$8,)(!>!(<&4+)(%! ;#'18(%!$1!>2!'1-!*#()(70!*#(!4(3(4!+.!7'%(!;+1%,6&*$+1!,&+1!+,)!(/,$*0!-,)'*$+1!(%*$6'*(%9!!!

!

17

Table 2:
!

FT All Share [1963-98] Equity Duration Estimates for 4% Real Yields
"'CD ?9?DUFU! ?9??EUU! ?9??_FS! ?9??_?F! ?9??GGD! ?9??GUE! ?9??G?G! ?9??>FF! ?9??>_e! ?9??>GS! ?9??>>S! ?9??>eG! ?9??>SE! ?9??>SS! ?9??>UF! ?9??>Ue! ?9??>U?! ?9??>D_! ?9??>DS! ?9??>DD! 4=1 S>9EE! ee9F>! eE9>_! >U9>>! >e9G?! >G9DD! >_9UG! >F9D_! >F9E?! >E9>D! G?9?U! G?9eG! G?9FS! GD9DG! GD9e>! GD9_?! GD9ES! GU9DS! GU9SU! GU9eF! 4=2 UG9GE! Se9DU! SF9UG! e?9EU! eU9F?! ee9DF! e>9U>! eG9D?! eG9_E! e_9S_! e_9F>! eF9UG! eF9GU! eF9ES! eE9UD! eE9e>! eE9G_! eE9F_! >?9?e! >?9U?! 4=3 UD9U>! U_9>_! SD9DF! SS9>e! S>9UU! SG9e_! S_9eS! SF9UD! SF9Fe! SE9SG! SE9FD! e?9DE! e?9>U! e?9FD! eD9?G! eD9UE! eD9eE! eD9G_! eD9Fe! eD9EE! 4=4 DG9_>! UD9EE! U>9?e! U_9?_! UF9>U! UE9GD! S?9eG! SD9De! SD9GE! SU9DG! SU9>>! SU9FE! SS9DF! SS9ee! SS9G_! SS9F_! Se9?>! Se9UD! Se9SG! Se9eE! 4=5 DS9F>! DF9SU! U?9E>! UU9_U! US9EE! Ue9E>! U>9_?! UG9S?! UG9_E! U_9UD! U_9>G! U_9FG! UF9DU! UF9S>! UF9>>! UF9_S! UF9FE! UE9?e! UE9D_! UE9UE! 4=6 DD9FS! D>9_D! DF9?S! DE9>E! U?9_U! UD9>_! UU9Ue! UU9_F! US9UU! US9>E! US9ED! Ue9DF! Ue9eD! Ue9GU! Ue9F?! Ue9EG! U>9DD! U>9Ue! U>9SG! U>9e_! 4=7 D?9Se! DS9__! D>9Fe! D_9Ue! DF9U>! DE9?U! DE9GU! U?9DD! U?9>D! U?9Fe! UD9DS! UD9S_! UD9>F! UD9__! UD9Ee! UU9?F! UU9UU! UU9Se! UU9ee! UU9>e! 4=8 E9U?! DU9U_! De9De! D>9e?! DG9SU! D_9?D! D_9>G! DF9??! DF9S_! DF9G_! DF9ES! DE9DG! DE9S>! DE9>U! DE9GF! DE9FD! DE9ES! U?9?e! U?9De! U?9US! 4=9 F9UE! DD9?F! DU9__! DS9EU! De9_G! D>9e?! D>9E?! DG9SD! DG9Ge! DG9EU! D_9DG! D_9S_! D_9>>! D_9_D! D_9F>! D_9E_! DF9?F! DF9DF! DF9U_! DF9SG! 4 = 10 _9>G! D?9D?! DD9GG! DU9_U! DS9eE! De9?F! De9>e! De9ED! D>9UU! D>9eF! D>9_?! D>9FE! DG9?G! DG9UD! DG9SS! DG9e>! DG9>>! DG9G>! DG9_S! DG9FD!

> Value 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
!

Table 3:
> Value 0.95 0.90 0.85 0.80 0.75 0.70 0.65 0.60 0.55 0.50 0.45 0.40 0.35 0.30 0.25 0.20 0.15 0.10 0.05 0.00
!

Recalculation of Equity Duration Estimates for m = 100 years and k = 4%
"'CD ?9?DUFU! ?9??EUU! ?9??_FS! ?9??_?F! ?9??GGD! ?9??GUE! ?9??G?G! ?9??>FF! ?9??>_e! ?9??>GS! ?9??>>S! ?9??>eG! ?9??>SE! ?9??>SS! ?9??>UF! ?9??>Ue! ?9??>U?! ?9??>D_! ?9??>DS! ?9??>DD! 4=1 UE9DU! SU9eU! SS9_F! Se9>e! S>9?U! S>9S>! S>9G?! S>9_F! S>9ES! SG9?>! SG9D>! SG9US! SG9S?! SG9SG! SG9eU! SG9eG! SG9>?! SG9>e! SG9>_! SG9G?! 4=2 Ue9D_! UF9U_! S?9?_! SD9?E! SD9_>! SU9U?! SU9>e! SU9F?! SS9?D! SS9D_! SS9SD! SS9eS! SS9>S! SS9GD! SS9GE! SS9_>! SS9FD! SS9FG! SS9ED! SS9E>! 4=3 U?9S?! Ue9_S! UG9_E! U_9EE! UF9_F! UE9SS! UE9_e! S?9?>! S?9S?! S?9>D! S?9GF! S?9FU! S?9Ee! SD9?>! SD9De! SD9UU! SD9UE! SD9SG! SD9eU! SD9e_! 4=4 DG9eF! U?9FF! US9?E! Ue9eU! U>9S?! U>9ES! UG9e?! UG9__! U_9?_! U_9SD! U_9>D! U_9G_! U_9FU! U_9Ee! UF9?>! UF9D>! UF9Ue! UF9SD! UF9SF! UF9ee!
18

4=5 DS9_F! D_9FF! U?9?_! UD9eS! UU9S>! US9?U! US9>S! US9EU! Ue9Ue! Ue9>?! Ue9_U! Ue9E?! U>9?G! U>9U?! U>9SU! U>9eU! U>9>U! U>9G?! U>9GF! U>9_>!

4=6 DD9FD! D>9>e! D_9GS! DF9EG! DE9FF! U?9>G! UD9?_! UD9e_! UD9F?! UU9?_! UU9S?! UU9eE! UU9G>! UU9F?! UU9EU! US9?S! US9DS! US9UU! US9S?! US9S_!

4=7 D?9SS! DS9_D! D>9GG! DG9ES! D_9FU! DF9eF! DF9EF! DE9SF! DE9_D! DE9EF! U?9U?! U?9e?! U?9>G! U?9_D! U?9Fe! U?9E>! UD9?>! UD9De! UD9UU! UD9S?!

4=8 E9U?! DU9U>! De9?>! D>9U>! DG9?E! DG9_S! D_9UU! D_9G?! D_9EU! DF9DE! DF9eD! DF9G?! DF9_G! DF9E?! DE9?S! DE9De! DE9Ue! DE9SS! DE9eD! DE9eE!

4=9 F9UE! DD9?_! DU9_S! DS9F>! De9G>! D>9U>! D>9_D! DG9?E! DG9SE! DG9G>! DG9FG! D_9?>! D_9U?! D_9Se! D_9e_! D_9>_! D_9G_! D_9_G! D_9Fe! D_9ED!

4 = 10 _9>G! D?9D?! DD9Ge! DU9GF! DS9eS! DS9EE! De9ee! De9_E! D>9?F! D>9SU! D>9>S! D>9_D! D>9FG! D>9EE! DG9DD! DG9UU! DG9SD! DG9SE! DG9e_! DG9>e!

Figure 2:

Comparative Statics for k
50

Equity Duration Model for Firms with Infinite Lifespan Equity Duration (in years)
40

Equity Duration Model for Firms with Lifetime m

30

20

10

0

2.00%

2.50%

3.00%

3.50%

4.00%

4.50%

5.00%

5.50%

6.00%

6.50%

7.00%

Real Rate of Return (k)

!

Figure 3:
!

Comparative Statics for 4
50
Equity Duration (in years) Equity Duration Model for Firms with Infinite Lifespan Equity Duration Model for Firms with Lifetime m

40 30 20 10 0
1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5 8 8.5 9 9.5 Values for 4

!
! 19

Figure 4:

Comparative Statics for m
50
Equity Duration Model for Firms with Infinite Lifespan

Equity Duration (in years)

40

Equity Duration Model for Firms with Lifetime m

30

20

10

0 30
! Figure 5:

40

50

60

70

80

90

100

110

120

130

140

150

Equity Horizon m (in years)

50

Comparative Statics for >

Equity Duration (in years)

40

30

20

10

Equity Duration Model for Firms with Infinite Lifespan Equity Duration Model for Firms with Lifetime m

0

95
!

85

75

65

55
20

45

35

25

15

5

Values for >

5. Conclusions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b!*+!Ub!&()!'11,6! $1! *#(! JK2! %,88(%*$18! *#'*! *#()(! 1((-%! *+! 7(! '! )('%+1'74(! '&&)+<$6'*$+1! *+! *#(! '3()'8(! (/,$*0! %,)3$3'4!#+)$B+19!!^(!*#,%!;'4$7)'*(-!+,)!6+-(4!*+!)(.4(;*!*#(!)('4!$13(%*6(1*!(3$-(1;(!+.!(/,$*$(%! 7(#'3$18!'%!6+)(!.$1$*(!#+)$B+1!%(;,)$*$(%2!;#++%$18!D??!0(')%!'%!'1!'3()'8(!4$.(*$6(!7(1;#6')59!! ^(!')8,(-!*#'*!*#(!)(%,4*'1*!;'4;,4'*$+1%!')(!;+1%$%*(1*!@$*#!*#(!4$'7$4$*0!%*),;*,)(!+.!6+%*!&(1%$+1! .,1-%2!@#$;#!#'3(!,&&()!4$'7$4$*0!-,)'*$+1%!;+1%$-()'740!7(4+@!*#(!eE=0(')!6'<$6,6!;+1*)$7,*$+1! &()$+-!.+)!'!6'4(!@+)5()!$1!*#(!JK9!!`+*#!'3()'8(!;+1*)$7,*$+1!'1-!)(-(6&*$+1!&()$+-%!';)+%%!'44! &+4$;0#+4-()%! ')(! '46+%*! ;()*'$1! *+! 7(! +.! '! ;+1%$-()'740! %#+)*()! -,)'*$+1! ;#')';*()$%*$;%! '1-! +,)! -,)'*$+1!(%*$6'*(%!%,88(%*!'!.')!;4+%()!;+))(%&+1-(1;(!$1!'%%(*!6'*;#$189! !! "#(!-(8)((!*+!@#$;#!&(1%$+1!.,1-%!;'1!$66,1$%(!'!6,4*$='%%(*!&+)*.+4$+!)(4$(%!6+)(! 8(1()'440! ,&+1! *#(! -(.$1$*$+1! +.! (/,$*0! -,)'*$+1! ,%(-2! '1-! #(1;(! #+@! @(! $1*(1-! *+! 6+-(4! *+*'4! &+)*.+4$+!-,)'*$+1!*+!*#()(70!;+1*)+4!'!.,1-a%!+3()'44!$1*()(%*!)'*(!)$%59!!Z)+6!+,)!(6&$)$;'4!@+)52! @(!#'3(!;+1;4,-(-!*#'*!'!&+)*.+4$+a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c()(! @(! -)'@! +1! *#(! &')'44(4%! 7(*@((1! *#(! JK! '1-! *#(! JX! $13(%*+)! 6')5(*%2! *+! $44,%*)'*(! #+@! -$..()(1;(%! $1! -$3$-(1-! 0$(4-%!')(!$1.4,(1;(-!70!%&(;$.$;!.$%;'4!'1-!7'15),&*;0!4(8$%4'*$+19!! !! c$%*+)$;'4402!*#(!;(1*)'4!+7g(;*$+1!*+!'-+&*$18!*#(!*)'-$*$+1'4!PPR!.)'6(@+)5!'%!'! ,%(.,4! *++4! .+)! &(1%$+1! '1-! '%%(*! 6'1'8(6(1*! @'%! *#'*! $*! *(1-(-! *+! &)+3$-(! (<*)(6(40! 4+18! 6('%,)(%!+.!(/,$*0!-,)'*$+12!'%!-$%*$1;*!.)+6!&)';*$*$+1()!+7%()3'*$+1%9!!!

!

21

!! "#$%! %,88(%*(-2! ;+1*)')0! *+! '%%(*! 6'1'8(6(1*! ;+13(1*$+12! *#'*! *#(! .$<(-! $1;+6(! &+)*$+1! +.! '! .,1-! 6'1'8()a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g+)!&(1%$+1!.,1-%9!!! !! R+)(+3()2! +,)! '&&)+';#! &)+3$-(%! '1! $1#()(1*! ;+1%$%*(1;0! @$*#! *#(! ';*,')$'4! '%%(%%6(1*! +.! (/,$*0! .,1-! 3'4,'*$+19! ! "#$%! #'%! 7(;+6(! $1;)('%$1840! $6&+)*'1*! *#)+,8#! )(;(1*! 8+3()16(1*! $1$*$'*$3(%! *+! )(3$(@! *#(! &(1%$+1!4(8$%4'*$+1!@$*#!)(.()(1;(!*+!*#(!R01()%!;+66$**((! $1/,$)0!)(;+66(1-'*$+1%2!'1-!*#(!)(&+)*!.)+6!*#(!I1%*$*,*(!+.!N;*,')$(%9!!O(1%$+1!.,1-!';*,')$(%! '%%(%%!*#(!3'4,(!+.!'1!(/,$*0!$13(%*6(1*!&+)*.+4$+!70!&)+g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

!

22

Appendix 1: Derivation of the Equity Duration Model J1-()!Assumptions 1 and 2,!$*!.+44+@%!*#'*M!! ! 73 = 73 *D (<&C5 ' + ) '3 H
0

'1-!!!73 +0 = 73 (<&C5 ' 0 + 0 ) '3 + D H
D =D

D *3 (73 +0 ) = 73 (<&C5 ' 0 + " '' 0 H!!!!!CU9_H U
0

-3 +0 = -3 (<&C5 2 0 + 0 ) 23 + D H
D =D 0

!

-3* 4 = -3* 4 (<&C* 45 2 0 * 4 0 ) 23 + D H +0
D =D

*3 (-3*+4 ) = -3* 4 (<&C* 45 2 0 + 0

D U 4 " 22 0 H!!!CU9FH U

0 0 ; 8 %+!*#'*!!73 +0 -3*+4 = 73 -3* 4 (<&!9(5 ' * 45 2 )!0 + 0 ) '3 + D * 4 0 ) 23 + D 6!!CU9EH 0 D =D D =D : 7

J%$18!6+6(1*!8(1()'*$18!.,1;*$+1%2!*#(!)$8#*!#'1-!%$-(!+.!CU9_H2!;'1!7(!;'4;,4'*(-!%+!*#'*M!! ! 0 ; 8 *3 (73 +0 -3*+4 ) = 73 -3* 4 (<&!9(5 ' * 45 2 )!0 + (" '' * U 4" 2' + 4 U" 22 )6 !2!!! ! 0 U : 7 ! :/,'*$+1!CU9_H!'1-!CU9FH!;'1!7(!%,7%*$*,*(-!$1*+!(/,'*$+1!CS9?H2!*#,%2!@(!.$1-!*#'*M!
1 <73 <0 ( < % ' = 73 0 0 ?3 2!!!@#()(!?3 = 73 0 & 2 # = 0 +D CD + > H D+ > *< 0 =D ' D + > $ 0

CS9?H!

D D ; 8 '1-!@#()(!!< !!=!!(<&!9(5 ' * 45 2 ) + (" '' * U 4" 2' + 4 U" 22 ) + 45 2 * 4 U" 22 62 U U : 7 D ; 8 = (<&!95 ' + " '' * 4" 2' 6!!!!!CS9DH U : 7

!
c(1;(!!41 (?3 ) = 41 (73 ) * 41CD + > * < H + 41 < 2 '1-!!' = * -41 (?* ) D = !!CS9UH -!> D+ > *<

^(!1+*(!*#'*!.+)!;+13()8(1;(!< < D + > 9

!

23

Appendix 2: Extensions to Compute Option Duration ! ! !! ! N1!'-3'1*'8(!+.!+,)!&)+;(-,)(!$%!*#'*!@(!;'1!'&&40!*#$%!-$)(;*40!*+!&)$;$18!+&*$+1%! ;+1%$%*(1*40! @$*#! `4';5! '1-! X;#+4(%! CDE_SH9! ! "#,%! -,)'*$+1! ;'1! 7(! ;+6&,*(-! .+)! -()$3'*$3(%! '%! @(442! $1! %$6$4')! *)('*6(1*! *+! T')6'1! CDEF>H2! @#()(! *#$%! ;'1!7(!*#+,8#*!+.!'%!*#(!,%,'4!\K&"">]2! )#+! C H! @$*#! )(%&(;*! *+! $1*()(%*! )'*(%2! @#()(! L3 ! $%! *#(! +&*$+1! &)$;(! '1-! @#()(! ' = ! L3 9!! N%%,6$18!`4';5!'1-!X;#+4(%!#+4-%2!@#$;#!@(!#'3(!1+*!%#+@1!$1!+,)!&'&()M! ! P,)'*$+1! =
D5 CD + > H *D C (' U ) 2! ?3 C C ' D H * 5 CD + > H *D C C ' U H

!

!

!!!!!!!!!!!Ce9DH9!

!!

! ^#()(! 5! $%! *#(! (<();$%(! &)$;(! +.! *#(! ;'442! C (9) $%! *#(! %*'1-')-! 1+)6'4! ;,6,4'*$3(!

-$%*)$7,*$+1!.,1;*$+12! "$%!*#(!6'*,)$*02!'1-!'D!'1-!'U!')(!-(.$1(-!'%!.+44+@%E!! !
'D , 'U , 4+8(?3 5 ) + (> + D " U )!D U 2! " D 4+8(?3 5 ) + (> * D " U )!D U 2! " D

! !! "#'*! `4';5! '1-! X;#+4(%! #+4-%! .+44+@%! .)+6! *#(! .';*! *#'*! @(! ;'1! '%%,6(! *#'*! *#(! +&*$+1!&'0%!+,*! ?3 +D * 5 !'*!6'*,)$*0!$.! ?3 +D > 5 2!'1-!&'0%!B()+!$1!'44!+*#()!&()$+-%9! !! !!!!!!!!!!"#()(.+)(2! !L3 =
*3 h6'<C ?3 +D * M 2?HA3 +D i 2! ! CD + > HD *3 CA3 +D H

!

!!!!!!!!!!!Ce9UH9!

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!

24

Appendix 3: Extensions of the CAPM to Include a Duration Parameter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`(.+)(!@(!7(8$1!*#$%!(1-('3+,)2!@(!;+1%$-()!%+6(!-'*'!$%%,(%!)(4'*$18!*+!'1!(6&$)$;'4!3()$.$;'*$+1! +.!'10!%,;#!6+-(49!!I1$*$'4402!@(!;+,4-!4++5!*+!*'5(!%+6(!>!0(')%!+)!G?!6+1*#%!C$1!($*#()!@((5%!+)! -'0%H!+.!-'*'2!*'5$18!(/,$*0!'1-!7+1-!$1-$;(%!@($8#*(-!.+)!*#(!6')5(*2!.+)!(<'6&4(!70!?9F!'1-!?9U! *+! )(.4(;*! %*'1-')-! $1%*$*,*$+1'4! @($8#*$18%9! ! `+1-! -'*'! ;#')';*()$%*$;%! ;+,4-! *#(1! 7(! '%%,6(-! *+! 6$6$;! *#(! 7(#'3$+,)! +.! 7,44(*! 7+1-%9! ! "#(! 6('%,)(! +.! -,)'*$+1! .+)! &,)(! -$%;+,1*! 7+1-%2! $1! *#(! '7%(1;(! +.! ;+,&+1! &'06(1*%2! $%! (/,$3'4(1*! *+! *#($)! *()6! *+! 6'*,)$*09! ! ^(! ;+,4-! %,7%(/,(1*40! %&(;$.0!+,)!%(*!+.!'%%(*%2!@#()(!0!k!DtC2!'1-!.,)*#()!'%%,6(!.)+6!*#(!'7+3(!;+1-$*$+1!*#'*!+,)!C! '%%(*%!#'3(!'!@(44=-(.$1(-!-,)'*$+1!'0!2!C'8'$1!0!k!D9999CH2!6$6$;5$18!*#(!7(#'3$+,)!+.!B()+!;+,&+1! 7+1-%9!!^(!;'1!*#(1!(%*$6'*(! µ0, N,&, 1, ;, 40F!'1-!'0! .)+6!'3'$4'74(!-'*'2!@#()(! µ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µFN,&1,;, "F! '1-!'F!C@#()(!*#(!%,7%;)$&*!F!-(1+*(%!*#(!6')5(*!&+)*.+4$+H9!

A(*!,*$4$*0! (O ), = ,O (µ F * &1 2 " F 2 ' F )!2! @#()(! µ F = µ F * &1 = 0 5 D C µ D * &1 H 2!.+)!C!)$%50!'%%(*%2!!
D =D C

*#(!)$%54(%%!'%%(*!#'3$18!'!B()+!-,)'*$+12!
U @$*#! " F = C C

00 50 5 D" 0D 2 !'1-!-,)'*$+12! !' F = 0 5 D ' D !9!
0 =D D =D
D =D

C

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25

! c()(!@(!#'3(!'!)(&)(%(1*'*$3(!'8(1*!@$*#!*#)((=.,1-!6+1(0!%(&')'*$+19! ! -O = ? 9! N*!*#(!+&*$6,6 !-O = ?2 !;+))(%&+1-$18402! -50 ! X+2! !
-O -O -O = 9C µ 0 * &1 H + -5$ -µ F -" F ( D % -O & & U" U0 M 0" 0D # + -!' C' 0 H =!!? 2! # F ' F $ !
* -O -" F -O -µ F % # #!9( " 0F & # &"F ' # $ ( & % & #+ # & $ & ' * -O -' F -O -µ F % # #!(' ) !2! # 0 # $

( & "#()(.+)(2!)('))'18$182! µ 0 * & 1 = & & & '

!
( -µ % ( -µ %(' % Z$1'4402!@(!'))$3(!'*2! µ 0 * &1 = & F " F #!4 0F + & F ' F #!9& 0 # !2!! & -" # & -' # &' # ' F $ ' F $' F$ ; N-O 8 9 -!/ -!G 6 = X$1;(!'*!*#(!+&*$6,62! 9 6 !.+)!'10!/2!G!9! -O -!/ 6 9 : -!G 7 ^(!&)(%(1*!*#(!'7+3(!)(%,4*!$1!(/,'*$+1!C>9?H!'%!"#(+)(6!D9!

!

C>9?H!

! Theorem 1:! ! ! ! I1!'!)(&)(%(1*'*$3(!'8(1*!(/,$4$7)$,62!8$3(1!'!+1(!&()$+-!@+)4-!@$*#!)(*,)1%2! -,)'*$+1!'1-!3+4'*$4$*0!-(.$1(-!.+)!(';#!'%%(*2!(/,$4$7)$,6!(<&(;*(-!(<;(%%!)(*,)1%!')(!4$1(')!$1! ;+3')$'1;(! 4 0 2!'1-!$1!-,)'*$+1! ' 0 2!@#()(! 5D '1-! 5 U ')(!-(.$1(-!$1!*#(!&)++.!';;+)-$18409! $9(9! µ 0 * &1 = 5D 4 0F + 5 U ' 0 2!4(*2! 5D =
-µ F D -µ F 'F9 9!" F !'1-! 5 U = !9 -' F 'F -" F

!!

T$3(1!(/,'*$+1!C>9?H2!*#(!)(%,4*!.+44+@%!,&+1!%,7%*$*,*$+1!'1-!)('))'18(6(1*9! !"#(!*(%*'74(!$6&4$;'*$+1%!+.!*#$%!6+-(4!;+1;()1!*#(!%$81%!+.! 5D !'1-! 5 U 9!!Z)+6!*#(!

'7+3(! @(! (<&(;*! 5D > ? 2! %$1;(! -µ F -" F > ? 9! ! c+@(3()2! *#(! %$81! +.! 5 U $%! '67$8,+,%9! ! L1(! ;+1%$-()'*$+1!#()(!$%!0$(4-!;,)3(!)(4'*(-2!@#()(!@(!6$8#*!')8,(!*#'*! 5 U > ? !$.!*#(!0$(4-!;,)3(!$%! ,&@')-! %4+&$182! 7,*! 5 U < ? ! $.! $*! $%! $13()*(-9! ! X+! $1! .';*2! $13(%*+)a%! &)(.()(1;(%! .+)! '10! -,)'*$+1! ;+6&+1(1*!@$*#$1!*#($)!&+)*.+4$+%!@$44!7(!)(4'*(-!*+!*#(!(13$%'8(-!%#'&(!+.!*#(!0$(4-!;,)3(2!'%!@(44! '%!*#($)!)(%&(;*$3(!$13(%*6(1*!#+)$B+19!!L.!;+,)%(2!%#+,4-!+,)!)(&)(%(1*'*$3(!'8(1*!7(!'!%#+)*=*()6! $13(%*+)2!@(!6$8#*!(<&(;*! -µ F -' F < ? 2!7,*!$.!#(!@()(!'!4+18()=*()6!$13(%*+)2!@(!6$8#*!(/,'440! (<&(;*! -µ F -' F > ? 9!!I1!+,)!(<'6&4(2!*#(!)(&)(%(1*'*$3(!'8(1*a%!,*$4$*0!%&(;$.$;'*$+1!@$44!-(&(1-! ,&+1!*#(!(<&(;*(-!(<;(%%!)(*,)1%2!*#(!$6&4$(-!3+4'*$4$*0!'1-!*#(!-,)'*$+19!!"#(!*0&$;'4!JK!$13(%*+)! ;+,4-!7(!*#+,8#*!+.!'%!'!4')8(!&(1%$+1!.,1-!@#+%(!4$'7$4$*$(%!')(!)(4'*$3(40!4+18=*()69!!"#,%2!@(! 6$8#*!(13$%'8(!'!&)(.()(1;(!.+)!'%%(*%!;#')';*()$%(-!70!*0&$;'440!4+18()!-,)'*$+1%9!!
! 26

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c'1%')-DD9! ! Z+)! '! 6+)(! g+,)1'4$%*$;! ;+1;+;*$+1!+.!*#(!%'6(!8),(42!*#(!)('-()!$%!)(.())(-!*+!*#(!')8,6(1*%!&)(%(1*(-!$1!c,**+1a%!7++5%! CDEEGH!'1-!CDEE_H9!!I))(%&(;*$3(!+.!*#(!6'11()!$1!@#$;#!+,)!)('-()!6'0!(4(;*!*+!$1*()&)(*!*#(!%#+)*= *()6$%*! ;4'$6%2! '%! ;+1%$-()(-! $1! '10! +.! *#(%(! %+,);(%2! +,)! 6+-(4! &()6$*%! *#(%(! $%%,(%! *+! 7(! '--)(%%(-!+7g(;*$3(402!70!'%%+;$'*$18!*#(6!@$*#!'!3'4,(!.+)! 5 U 9!! !! I1!+)-()!*+!(<*(1-!*#(!(/,$4$7)$,6!;+1-$*$+1%!.+)!+,)!)(&)(%(1*'*$3(!'8(1*!#+@(3()2! @(! @+,4-! #'3(! *+! $13+5(! '! *#)((=.,1-! %(&')'*$+1! *#(+)(69! ! L,)! *#(+)0! 6'$1*'$1%! *#'*2! ;)+%%= %(;*$+1'440!*#(!)$%5!&)(6$,6!@$44!7(!)(4'*(-!*+!7(*'!'%!@(44!'%!*#(!-,)'*$+1!+.!*#(!'%%(*!)(4'*$3(!*+! *#(!-,)'*$+1!+.!*#(!6')5(*9!!^(!;'1!*#()(.+)(!(%*$6'*(! 4 0F !.)+6!#$%*+)$;'4!6')5(*!-'*'9!!c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
µF * &1 = 5D + 5 U 'F !9!

'44+@!.+)!*#(!.+44+@$18!(<&)(%%$+1M! "#,%2! (µ F * & 1 )!! = 5 D 4 0F + !
( % '0 !!!!!!!!! =!!5D & 40F!*! '0 # +!(µF * &1 ) !2! ! & # & # 'F 'F $ ' ! C>9DH9!

((µ F

* &1 )* 5 D )

'0 2! 'F

!

! "#()(.+)(2!$1!(/,'*$+1!C>9DH2!@(!#'3(!'))$3(-!'*!'1!'4*()1'*$3(!)(&)(%(1*'*$+19! ! Corollary 1:!!!C.+)!0,P,DQ<,F9H! ! ! ! ! "#(!(6&$)$;'4!;+1%(/,(1;(%!+.!(/,'*$+1!C>9DH!')(!*#'*!*#()(!%#+,4-!7(!%+6(!;)+%%= %(;*$+1'4!)(4'*$+1%#$&!7(*@((1!)$%5!&)(6$'2!*#(!7(*'%!'1-!*#(!)(4'*$3(!-,)'*$+12! ' 0 ' F 9!!I1-((-2!*#(! %,6!+.!*#(!*@+!;+(..$;$(1*%!%#+,4-!(/,'4!*#(!(<;(%%!)(*,)1%!'1-!@$44!#'3(!*(%*'74(!;+1%(/,(1;(%2! %,$*'74(!.+)!(6&$)$;'4!$13(%*$8'*$+19!

! d`,-8(*! V(%+4,*$+1%! '1-! :;+1+6$;! X$*,'*$+1a2! R/4(/&'2! _*#! Y,40! DEE_2! &&9GFF=GE?2! '1-! *#(! Z$1'1;(! `$442! R/4(/&'2,DG*#!Y,40!DEE_2!&&9e_>=e_G9!

!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! !
DD

!

27

Theorem 2: ! ! c()(! @(! 1+@! ;+1%$-()! @#'*! '! 6$1$6,6! 3')$'1;(! .)+1*$()! 6$8#*! 4++5! 4$5(! .+)! '! -,)'*$+1=;+1%*)'$1(-!&+)*.+4$+9!!"#$%!'1'40%$%!$%!)(6$1$%;(1*!+.!(')40!@+)5!70!Y('1!CDE_DH!'1-!CDE_SH2! '%!@(44!'%!I18()%+44!CDE_>H9!!N%!7(.+)(2!@(!#'3(!C!'%%(*%!@$*#!)'*(%!+.!)(*,)1! & 2!@#()(! & j C µ 2!EH !'1-! *#(! -,)'*$+1! +.! *#(! '%%(*%! $%! -(1+*(-! 70! ' 9! ! A(*! *#(! ;+1%*)'$1(-! (<&(;*(-! )(*,)1%! 7(! 2 ! '1-! ;+1%*)'$1(-!-,)'*$+1! ' ? 9!!L,)!.)+1*$()!$%!*#(!4+;,%!+.!&+$1*%! C 22 ' ? 2 " U H !@#()(! " U = 5 30 5 !'1-!%+! @(!%+43(!*#(!.+44+@$18!+&*$6$%'*$+1!&)+74(6!C@#()(!"!$%!'!3(;*+)!+.!+1(%HM! ! S04!!(C 5 30 5 H * . D!C 5 3" * DH * . U!C 5 3µ * 2 H * . S!C 5 3' * ' ? H ) !9!
<

! P$..()(1*$'*$182!@(!%((!*#'*!'%!7(.+)(2!@(!;+,4-!8(*!*#)((=.,1-!%(&')'*$+19!! ! N4*()1'*$3(402!*#$%!;'1!7(!@)$**(1!'%2! (D % S04!!& 5 30 5 * . 3C :35 * : ? H # 2! 5 'U $ @#()(! :!C'1!4 F S!!6'*)$<H = h"2 µ 2 ' i !'1-! : ? = CD2 22 ' ? Hu 9! ! The first order conditions imply that: ! n n n .+) 5 2!*#(!%+4,*$+1!7(;+6(%! E5 = :. !!'1-! :35 = : ? 9! !
n n n n "#()(.+)(2! 5 !k! E *D :. 2!'1-! :35 = : ? = C :3E *D :H + . 9!

!
n X+!! . = C :3E *D :H *D : ? 2!!

!
n '1-!*#()(.+)(! 5 = E *D :!C :3E *D :H *D : ? !9!

! I.!@(!-(1+*(!C*#(!S!70!D!6'*)$<H!> = C>D 2 > U 2 > S Hu = C :3E *D :H *D : ? !2!! !
n *#(1! 5 = >DE "!+!> U E *D µ !+!> SE *D' !9! !!
*D

!

!

C>9UH9!

! Remark 1:! !! I*!$%!&+%%$74(!*+!;+6&,*(!(<&)(%%$+1%!.+)!> 0 2!70!;+6&,*$18!*#(!$13()%(!+.!*#(!CS!70!SH!

6'*)$<! C :3E *D:H *D 9! ! ! Remark 2: ! !! :/,'*$+1! C>9UH! %#+@%! *#'*! *#(! 6$1$6,6! 3')$'1;(! &+)*.+4$+! %'*$%.$(%! *#)((=.,1-! %(&')'*$+19! ! N%! @(! 3')0! 2 ! '1-! ' ? 2! @(! @$44! *)';(! +,*! '! 6$1$6,6! 3')$'1;(! .)+1*$()! $1! *#)((! -$6(1%$+1%2! ((" 2 22 ' )!!%&';( ) 9!!! !
28

"#(!(/,'*$+1!.+)!*#(!(..$;$(1*!.)+1*$()!@$44!*#,%!7(!8$3(1!702! ! " U = :3 !C :3E *D :H *D : ? "9! ! ? !!

!

!

!

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! :/,'*$+1!C>9SH!$%!;4(')40!/,'-)'*$;!$1! " 2 22!'1-!' ? 9!!^#$4%*!$*!$%!3'4$-!*+!'%%,6(!*#'*!

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!

29

Addendum:

US & UK Dividend Payments, Fiscal and Bankruptcy Issues.

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!

30

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! 31

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! 32

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Abel2! A9 B92! \N%%(*! O)$;(%! ,1-()! c'7$*! Z+)6'*$+1! '1-! ['*;#$18! ,&! @$*#! *#(! Y+1(%(%]2! :F"&02/4, *2%4%F02,8"W0"$2!Q+49!F?2!f+9U2!R'0!DEE?2!&&9SF=eU9! Abel2! A9 B92! \"#(! :/,$*0! O)(6$,6! O,BB4(]2! 6U(04"((, 8"W0"$, N, 9"'"&/., 8"("&W", 6%/&', %1, ?X0./'".=X0/2!f+9>2!X(&*(67()xL;*+7()!DEED2!&&9S=De9 Altman2 E9! I92! \N! Z,)*#()! :6&$)$;'4! I13(%*$8'*$+1! +.! *#(! `'15),&*;0! [+%*! y,(%*$+1]2! )%U&4/., %1, 904/42"2!Q+49SE2!f+9e2!X(&*(67()!DEFe2!&&9D?G_=D?FE9! Arnott2!R9!D92 Hanson2 H9 N92!Leibowitz2 M9 L92 Sorensen2 E9 H92!\N!"+*'4!P$..()(1*$'4!N&&)+';#!*+! :/,$*0!P,)'*$+1]2!904/420/.,:4/.#(3(,)%U&4/.2!Q+49!e>2!f+9!>2!X(&*(67()xL;*+7()!DEFE2!&&9S?=S_9! Barr2!D9!G92!Campbell2!J9!Y92!\I1.4'*$+12!V('4!I1*()(%*!V'*(%2!'1-!*#(!`+1-!R')5(*M!N!X*,-0!+.!JK! f+6$1'4!'1-!I1-(<=A$15(-!T+3()16(1*!`+1-!O)$;(%]2!)%U&4/.,%1,S%4"3/&#,*2%4%F02(2!Q+49SE2!f+9S2! DEE_2!&&9SGD=SFS9 Bernheim2!B9!D92!\"'<!O+4$;0!'1-!*#(!P$3$-(1-!O,BB4(]2!8/4',)%U&4/.,%1,*2%4%F02(2!Q+49UU2!f+9e2! @$1*()!DEED2!&&9e>>=e_G9!! ! Bernstein2!R92!Tew2!B92!\"#(!:/,$*0!dz$(4-a![,)3(]2!)%U&4/.,%1,?%&31%.0%,S/4/@"F"432!Q+49DF2!f+9D2! Z'44!DEED2!&&9S>=SE9!! Betker2 B92!\R'1'8(6(1*a%!I1;(1*$3(%2!:/,$*0a%!`')8'$1$18!O+@()2!'1-!P(3$'*$+1%!.)+6!*#(!N7%+4,*(! O)$+)$*0!$1![#'&*()!DD!`'15),&*;$(%]2!)%U&4/.,%1,6U(04"((2!Q+49GF2!f+9U2!DEE>2!&&9DGD=DFS9! Bierwag2!G9O92!P,)'*$+1!N1'40%$%2!`'44$18()2!['67)$-8(2!R'%%';#,%(**%2!DEF_9 Bierwag2!G9O92!Kaufman2!G92!Toevs2!A92!\I66,1$%'*$+1!X*)'*(8$(%!.+)!Z,1-$18!R,4*$&4(!A$'7$4$*$(%]2! )%U&4/.,%1,904/420/.,/4',YU/4303/30W",:4/.#(0(2!Q+49DF2!f+9D2!DEFS2!&&9DDS=DUS9 Black2 F92!\"#(!P$3$-(1-!O,BB4(]2!)%U&4/.,%1,?%&31%.0%,S/4/@"F"432!Q+49U2!f+9U2!@$1*()!DE_G2!&&9>= F9 Black2 F92 Scholes2 M92 \"#(! O)$;$18! +.! L&*$+1%! '1-! [+)&+)'*(! A$'7$4$*$(%]2! )%U&4/., %1, ?%.0302/., *2%4%F#2!Q+49FD2!f+9U2!R'0xY,1(!DE_S2!&&9GS_=G>e9! Bodie2! Z92! \[+66+1! X*+;5%! '%! '! c(-8(! N8'$1%*! I1.4'*$+1]2!)%U&4/., %1, 904/42"2! Q+49SD2!f+9U2! R'0! DE_G2!&&9e>E=e_?9 Bodie2!Z92!Kane2 A92!Marcus2 A92!I13(%*6(1*%!CZX0&',"'030%4H2!P+@!Y+1(%=I)@$12!c+6(@++-2!I44$1+$%2! DEEG9!

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Campbell2! J9! Y92! Lo2!A9 W92!MacKinlay2!A9 C92!"#(!:;+1+6(*)$;%!+.!Z$1'1;$'4!R')5(*%2!O)$1;(*+1! J1$3()%$*0!O)(%%2!DEE_9! Casabona2! P9! A92! Fabozzi2! F9! J92! Francis2! J9! C92! \c+@! *+! N&&40! P,)'*$+1! *+! :/,$*0! N1'40%$%]2! )%U&4/.,%1,?%&31%.0%,S/4/@"F"432!Q+49D?2!f+9U2!@$1*()!DEFe2!&&9>U=>F9! Chan2! L9! K9! C9! \[+1%,6&*$+12! I1.4'*$+1! V$%52! '1-! V('4! I1*()(%*! V'*(%M! N1! :6&$)$;'4! N1'40%$%]2! )%U&4/.,%1,6U(04"((2!Q+49G_2!f+9D2!DEEe2!&&9GE=EG9! Chapman2! D9! A92! \c'7$*! Z+)6'*$+1! '1-! N88)(8'*(! [+1%,6&*$+1]2! *2%4%F"3&02/2! Q+49GG2! f+9>2! X(&*(67()!DEEF2!&&9DUUS=DUS?9! ! Christie2! W9! G92! \P$3$-(1-! z$(4-! '1-! :<&(;*(-!V(*,)1%]2!)%U&4/.,%1,904/420/.,*2%4%F02(2! Q+49UF2! f+9D2!DEE?2!&&9E>=DU>9! Constantinides2!G9!M92!\c'7$*!Z+)6'*$+1M!N!V(%+4,*$+1!+.!*#(!:/,$*0!O)(6$,6!O,BB4(]2,)%U&4/.,%1, ?%.0302/.,*2%4%F#2!Q+49EF2!f+9S2!DEE?2!&&9>DE=>eS9! Constantinides2 G9!M92!Ferson2!W9!E92!\c'7$*!O()%$%*(1;(!'1-!P,)'7$4$*0!$1!N88)(8'*(![+1%,6&*$+1M! :6&$)$;'4!"(%*%]2!)%U&4/.,%1,904/420/.,*2%4%F02(2!Q+49UE2!f+9U2!L;*+7()!DEED2!&&9DEE=Ue?9! Cornell2! B92! \:/,$*0! P,)'*$+12! T)+@*#! L&*$+1%2! '1-! N%%(*! O)$;$18]2! )%U&4/., %1, ?%&31%.0%, S/4/@"F"432!Q+49UG2!f+9S2!%&)$18!U???2!&&9D?>=DDD9 Damant2 D9C92 Satchell2 S9 E92!\"(%*$18!.+)!X#+)*!"()6$%6!$1!*#(!JK!X*+;5!R')5(*M!N![+66(1*]2, *2%4%F02,)%U&4/.2!Q+49D?>2!f+9eSU2!X(&*(67()!DEE>2!&&9DUDF=DUUS9! Damant2!D9C92!Hwang2!S92 Satchell2 S9 E92!\J%$18!'!R+-(4!+.!I1*(8)'*(-!V$%5!*+!N%%(%%!J9K9!N%%(*! N44+;'*$+1]2,:==.0"',S/3X"F/302/.,904/42"2!Q+49_2!f+9U2!U???2!&&9DU_=D>U9! Dimson2!E92 Marsh2 P92!\J9K9!Z$1'1;$'4!R')5(*!V(*,)1%2!DE>>=U???]2!)%U&4/.,%1,6U(04"((2!Q+49_e2! f+9D2!Y'1,')0!U??D2!&&9D=SD9! Elton2!E9J92!Gruber2!M9J92!R+-()1!O+)*.+4$+!"#(+)0!'1-!I13(%*6(1*!N1'40%$%2!>*#!:-$*$+12!^$4(02! DEE>9! Evans2! M9! D9! D92! \V('4! V'*(%2! :<&(;*(-! I1.4'*$+12! '1-! I1.4'*$+1! V$%5! O)(6$']2!)%U&4/., %1, 904/42"2! Q+49>S2!f+9D2!Z(7),')0!DEEF2!&&9DF_=UDF9! Fabozzi2 F9 J92 "#(! c'1-7++5! +.! Z$<(-! I1;+6(! X(;,)$*$(%! C9013X, *'030%4H2! I)@$1! O)+.(%%$+1'4! O,74$;'*$+12!DEE_9 Farrar2 J9 H92! Furey2! N9 E92! Hannigan2! B. M92! Z'))')a%! [+6&'10! A'@2! S)-! :-$*$+12! `,**()@+)*#%2! DEED9

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Idol2 C9 R92! \:/,$*0! -,)'*$+1! *+! '%%(%%! $1*()(%*=)'*(! )$%5]2! -&"'03, B40%4, *5"2U30W"2! Q+49S_2! f+9e2! Y,40xN,8,%*!DEE_2!&&9SU=SG9! Idol2 C9 R92!\NAR!IIIM!O)';*$;'4!$%%,(%!@$*#!(/,$*0!-,)'*$+1]2!-&"'03,B40%4,*5"2U30W"2!Q+49S_2!f+9>2! X(&*(67()xL;*+7()!DEE_2!&&9DG=DE9! Ingersoll2 J92! \R,4*$-$6(1%$+1'4! X(;,)$*0! O)$;$18]! )%U&4/., %1, 904/420/., /4', YU/4303/30W", :4/.#(0(2, Q+49D?2!f+9>2!DE_>2!&&9_F>=_EF9! ! Jean2 W9 H92! \"#(! :<*(1%$+1! +.! O+)*.+4$+! N1'40%$%! *+! "#)((! +)! R+)(! O')'6(*()%]2! )%U&4/., %1, 904/420/.,/4',YU/4303/30W",:4/.#(0(2!Q+49G2!f+9D2!Y'1,')0!DE_D2!&&9>?>=>D>9! Jean2 W9 H92!\R+)(!+1!R,4*$-$6(1%$+1'4!O+)*.+4$+!N1'40%$%]2!)%U&4/.,%1,904/420/.,/4',YU/4303/30W", :4/.#(0(2!Q+49F2!f+9S2!Y,1(!DE_S2!&&9e_>=eE?9! Jensen2! M9! C92 Meckling2 W9! H92 \"#(+)0! +.! *#(! Z$)6M! R'1'8()$'4! `(#'3$+,)2! N8(1;0! [+%*%! '1-! L@1()%#$&!X*),;*,)(]2!)%U&4/.,%1,904/420/.,*2%4%F02(2!Q+49S!f+9e2!&&9S?>=SG?9! Johnson2! L9 D92! \:/,$*0! P,)'*$+1M! N1+*#()! A++5]2! 904/420/., :4/.#(3(, )%U&4/.2! Q+49e>2! f+9U2! R');#xN&)$4!DEFE2!&&9_S=_>9! Johnson2! L9 D92! \[+13(<$*0! .+)! :/,$*0! X(;,)$*$(%M! P+(%! [,)3'*,)(! R'**()l]2! 904/420/., :4/.#(3(, )%U&4/.2!Q+49!eG2!f+9>2!X(&*(67()xL;*+7()!DEE?2!&&9_?=_S9 Johnson2!L9 D92!\"#(!V+4(!+.![+13(<$*0!$1!:/,$*0!O)$;$18]2!904/420/.,:4/.#(3(,)%U&4/.2!Q+49eF2!f+9>2! X(&*(67()xL;*+7()!DEEU2!&&9GE=_e9 Joyce2 M92! Lomax2 J92! \O'**()1%! +.! P(.',4*! $1! *#(! f+1=Z$1'1;$'4!O)$3'*(!X(;*+)%]2!6/4>,%1,*4@./4', YU/&3"&.#,6U.."3042!f+3(67()!DEED2!&&9>Se=>S_9! Kahneman2! D92 Tversky2! A92! ! \O)+%&(;*! "#(+)0M! N1! N1'40%$%! +.! P(;$%$+1! J1-()! V$%5]2! *2%4%F"3&02/2!Q+49e_2!R');#!DE_E2!&&9UGS=UED9 ! Kadiyala2!P92\"#(!V(4'*$+1!7(*@((1!*#(!R'81$*,-(!+.!T)+@*#!L&&+)*,1$*$(%!'1-!*#(!P,)'*$+1!+.! :/,$*0]2 ZX",)%U&4/.,%1,904/420/.,8"("/&2X2!Q+49!US2!f+9!S2!Z'44!U???2!&&9UF>=SD?9 Kraus2!A92!Litzenberger2!R9!H92!\N!X*'*(=O)(.()(1;(!R+-(4!+.!L&*$6'4!Z$1'1;$'4!A(3()'8(]2!)%U&4/., %1,904/42"2!Q+49UF2!X(&*(67()!DE_S2!&&9EDD=EUU9! ! Lanstein2! R92! Sharpe2! W9! F92! \P,)'*$+1! '1-! X(;,)$*0! V$%5]2, )%U&4/., %1, 904/420/., /4', YU/4303/30W", :4/.#(0(2!Q+49DS2!f+9e2!f+3(67()!DE_F2!&&9G>S=GGF9! Lasfer2!M9!A92!\:<=-'0!`(#'3$+)M!"'<!+)!X#+)*="()6!")'-$18!:..(;*%2!ZX",)%U&4/.,%1,904/42"2!Q+49! >?2!f+9!S2!Y,4!DEE>2!&&9F_>=FEF9!

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39

Leibowitz2 M9 L92! \"+*'4! O+)*.+4$+! P,)'*$+1M! N! f(@! O()%&(;*$3(! +1! N%%(*! N44+;'*$+1]2! 904/420/., :4/.#(3(,)%U&4/.2!Q+49eU2!f+9>2!X(&*(67()xL;*+7()!DEFG2!&&9DF=UE!'1-!&'8(!__9! ! Leibowitz2 M9 L92! \X&)('-=-)$3(1! P$3$-(1-! P$%;+,1*! R+-(4%]2 904/420/., :4/.#(3(, )%U&4/.2! Q+49>G2! f+9G2!f+3(67()xP(;(67()!U???2!&&9Ge=FD9! Leland2 H9! E92! \[+)&+)'*(! P(7*! Q'4,(2! `+1-! [+3(1'1*%2! '1-! L&*$6'4! ['&$*'4! X*),;*,)(2!)%U&4/., %1, 904/42"2!Q+49eE2!f+9e2!DEEe2!&&9DUDS=DU>U9! ! Leland2 H9! E92! Toft2! K9! B92! \L&*$6'4! ['&$*'4! X*),;*,)(2! :1-+8(1+,%! `'15),&*;0! '1-! *#(! "()6! X*),;*,)(!+.![)(-$*!X&)('-%]2!)%U&4/.,%1,904/42"2!Q+49>D2!f+9S2!Y,40!DEEG2!&&9EF_=D?DE9! Macaulay2! F9! R92! \X+6(! "#(+)(*$;'4! O)+74(6%!X,88(%*(-!70!*#(!R+3(6(1*!+.!I1*()(%*!V'*(%2! `+1-! z$(4-%2! '1-! X*+;5! O)$;(%! $1! *#(! J1$*(-! X*'*(%! X$1;(! DF>G]2!C/30%4/.,6U&"/U,%1,*2%4%F02,8"("/&2X2! f(@!z+)52!DESF9! Mehra2! R92! Prescott2! E9 C92! \"#(! :/,$*0! O)(6$,6M! N! O,BB4(]2! )%U&4/., %1, S%4"3/&#, *2%4%F02(2! Q+49D>2!DEF>2!&&9De>=DGD9! Michaely2 R92! Vila2 J-L92, \I13(%*+)%u! c(*()+8(1($*02! O)$;(%2! '1-! Q+4,6(! ')+,1-! *#(! :<=-$3$-(1-! P'0]2!)%U&4/.,%1,904/420/.,/4',YU/4303/30W",:4/.#(0(2!Q+49!S?2!f+9!U2!Y,1(!DEE>2!&&9D_D=DEE9! Miles2 D92!\"(%*$18!.+)!X#+)*!"()6$%6!$1!*#(!JK!X*+;5!R')5(*]2!*2%4%F02,)%U&4/.2!Q+49D?S2!f+9eUD2! f+3(67()!DEES2!&&9DS_E=DSES9! ! Miles2 D92! \"(%*$18! .+)! X#+)*! "()6$%6! $1! *#(! JK! X*+;5! R')5(*M! N! )(&40]2!*2%4%F02, )%U&4/.2! Q+49! D?>2!f+9!eSU2!X(&*(67()!DEE>2!&&9DUUe=DUU_9 Miller2 M9H92! ]`(#'3$+)'4! V'*$+1'4$*0! $1! Z$1'1;(M! ! "#(! ['%(! +.! P$3$-(1-%]2! )%U&4/., %1, 6U(04"((2! Q+49>E2!f+9e2!DEFG2!&&9e>D=eGF9 Miller2 M9H92! Modigliani2! F92! \"#(! [+%*! +.! ['&$*'42! [+)&+)'*$+1! Z$1'1;(2! '1-! *#(! "#(+)0! +.! I13(%*6(1*]2!:F"&02/4,*2%4%F02,8"W0"$2!Q+49eF2!f+9S2!Y,1(!DE>F2!&&9UGD=UE_9! Miller2!M9!H92!Shiller2!R9!J<2 \`(#'3$+)'4!V'*$+1'4$*0!$1!Z$1'1;(M!"#(!['%(!+.!P$3$-(1-%]!ZX",)%U&4/., %1,6U(04"((2!Q+49>E2!f+9e2!L;*+7()!DEFG2!&&9!e>D=e_U9 Modigliani, F., \A$.(![0;4(2!I1-$3$-,'4!"#)$.*2!'1-!*#(!^('4*#!+.!f'*$+1%]2!ZX",:F"&02/4,*2%4%F02, 8"W0"$2!Q+49_G2!f+9S2!Y,1(!DEFG2!&&9UE_=SUS9 Pickens2!T9 B92!\O)+.(%%$+1%!+.!'!X#+)*="()6()]!R/&W/&',6U(04"((,8"W0"$2!Q+49Ge2!I%%,(!S2!R'0=Y,1(! DEFG2!&&9_>=_E9! Redington2! F9 M92! \V(3$(@! +.! *#(! O)$1;$&4(! +.! A$.(=L..$;(! Q'4,'*$+1%]2!)%U&4/., %1, 3X", ]4(303U3", %1, :23U/&0"(2!Q+49_F2!DE>U2!&&9UFG=Se?9! ! 40

Rubinstein2!M92!\"#(!Q'4,'*$+1!+.!J1;()*'$1!I1;+6(!X*)('6%!'1-!"#(!O)$;$18!+.!L&*$+1%]2!ZX",6".., )%U&4/.,%1,*2%4%F02(2!Q+49!_2!DE_>2!&&9e?_=eU>9! ! Schwartz2!A92!\X(;,)$*0!I1*()(%*%!'1-!`'15),&*;0!O)$+)$*$(%M!N!V(3$(@!+.![,))(1*!"#(+)$(%]2!)%U&4/.,%1, +"@/.,E3U'0"(2!Q+49D?2!Y'1,')0!DEFD2!&&9D=S_9! Shefrin2!H9!M92!Statman2!M92!\:<&4'$1$18!I13(%*+)!O)(.()(1;(!.+)!['%#!P$3$-(1-%]2!)%U&4/.,%1, 904/420/.,*2%4%F02(2!Q+49DS2!f+9U2!Y,1(!DEFe2!&&9U>S=UFU9! Shefrin2!H9!M92!Thaler2!R9!H92!\N1!:;+1+6$;!"#(+)0!+.!X(4.=[+1*)+4]2 ZX",)%U&4/.,%1,?%.0302/., *2%4%F#2!Q+49!FE2!f+9U2!N&)$4!DEFD2!&&9SEU=e?_9 Shiller2! R9! J92! \[+6+3(6(1*! $1! X*+;5! O)$;(%! '1-![+6+3(6(1*%!$1!P$3$-(1-%]2!)%U&4/.,%1,904/42"2! Q+49ee2!f+9S2!Y,40!DEFE2!&&9_DE=_UE9! ! Simonson2!D9!G92!\c+@!L1(!A('-()!R'1'8(%!NxA!V$%5]2!B403"',E3/3"(,6/4>"&2!Q+49D?S2!f+9U2!N&)$4! DEES2!&&9>G=>_9 ! Sundaresan,! S92! \I1*()*(6&+)'440! P(&(1-(1*! O)(.()(1;(%! '1-! *#(! Q+4'*$4$*0! +.! [+1%,6&*$+1! '1-! ^('4*#]2!8"W0"$,%1,90(2/.,E3U'0"(2!Q+49!U2!DEFE2!&&9_S=FF9! Talmor2 E92!Titman2!S92!\"'<(%!'1-!P$3$-(1-!O+4$;0]2!904/420/.,S/4/@"F"432!Q+49DE2!f+9U2!%,66()! DEE?2!&&9SU=S>9 Webb2 D9 C92!\N1!:;+1+6$;!:3'4,'*$+1!+.!I1%+43(1;0!O)+;(-,)(%!$1!*#(!J1$*(-!K$18-+6M!P+(%!*#(! DEFG! I1%+43(1;0! N;*! X'*$%.0! "#(! [)(-$*+)%a! `')8'$1l]2! L51%&', *2%4%F02, ?/="&(2! Q+49eS2! f+9D2! Y'1,')0!DEED2!&&9DSE=D>_9! White2! M9! J92! \"#(! [+)&+)'*(! `'15),&*;0! P(;$%$+1]2!)%U&4/., %1, *2%4%F02,?"&(="230W"(2!Q+49S2! f+9U2! %&)$18!DEFE2!&&9DUE=D>D9! White2!M9!J92!\I1%+43(1;0!'1-!`'15),&*;0]2!$1!:'*@(442!Y92!R$48)+62!R92!f(@6'12!O92!"#(!f(@!O'48)'3(! P$;*$+1')0!+.!R+1(0!W!Z$1'1;(2!Q+49U2!R';6$44'1!CA+1-+1H2!DEEU2!&&9eDE=eUU9! Williams2 J9 B92! \:3'4,'*$+1! `0! *#(! V,4(! +.! O)(%(1*! ^+)*#]2! "#(! "#(+)0! +.! I13(%*6(1*! Q'4,(2! c')3')-!J1$3()%$*0!O)(%%2!DESF2!&&9>>=_>9! ! Winter2!S9!G92!\:;+1+6$;!df'*,)'4!X(4(;*$+1a!'1-!*#(!"#(+)0!+.!*#(!Z$)6]2!A/.",*2%4%F02,*((/#(2!DEGe2! &&9UU>=U_D9! ! Woodward2!G9!T92!\"#(!V('4!"#$18M!N!P01'6$;!O)+.$4(!+.!*#(!"()6!X*),;*,)(!+.!V('4!I1*()(%*!V'*(%! '1-!I1.4'*$+1!:<&(;*'*$+1%!$1!*#(!J1$*(-!K$18-+62!DEFU=FE]2!)%U&4/.,%1,6U(04"((2!Q+49GS2!f+9S2!DEE?2! &&9S_S=SEF9! ! Zingales2!L92!\X,)3$3'4!+.!*#(!Z$**(%*!+.!*#(!Z'**(%*l!!:<$*!'1-!Z$1'1;$18!$1!*#(!"),;5$18!I1-,%*)0]2! )%U&4/.,%1,904/42"2!Q+49>S2!f+9S2!Y,1(!DEEF2!&&9E?>=ESF9 ! 41

!

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List of other working papers: 2001
1. Soosung Hwang and Steve Satchell , GARCH Model with Cross-sectional Volatility; GARCHX Models, WP01-16 2. Soosung Hwang and Steve Satchell, Tracking Error: Ex-Ante versus Ex-Post Measures, WP01-15 3. Soosung Hwang and Steve Satchell, The Asset Allocation Decision in a Loss Aversion World, WP01-14 4. Soosung Hwang and Mark Salmon, An Analysis of Performance Measures Using Copulae, WP01-13 5. Soosung Hwang and Mark Salmon, A New Measure of Herding and Empirical Evidence, WP01-12 6. Richard Lewin and Steve Satchell, The Derivation of New Model of Equity Duration, WP0111 7. Massimiliano Marcellino and Mark Salmon, Robust Decision Theory and the Lucas Critique, WP01-10 8. Jerry Coakley, Ana-Maria Fuertes and Maria-Teresa Perez, Numerical Issues in Threshold Autoregressive Modelling of Time Series, WP01-09 9. Jerry Coakley, Ana-Maria Fuertes and Ron Smith, Small Sample Properties of Panel Timeseries Estimators with I(1) Errors, WP01-08 10. Jerry Coakley and Ana-Maria Fuertes, The Felsdtein-Horioka Puzzle is Not as Bad as You Think, WP01-07 11. Jerry Coakley and Ana-Maria Fuertes, Rethinking the Forward Premium Puzzle in a Nonlinear Framework, WP01-06 12. George Christodoulakis, Co-Volatility and Correlation Clustering: A Multivariate Correlated ARCH Framework, WP01-05 13. Frank Critchley, Paul Marriott and Mark Salmon, On Preferred Point Geometry in Statistics, WP01-04 14. Eric Bouyé and Nicolas Gaussel and Mark Salmon, Investigating Dynamic Dependence Using Copulae, WP01-03 15. Eric Bouyé, Multivariate Extremes at Work for Portfolio Risk Measurement, WP01-02 16. Erick Bouyé, Vado Durrleman, Ashkan Nikeghbali, Gael Riboulet and Thierry Roncalli, Copulas: an Open Field for Risk Management, WP01-01 2000 1. Soosung Hwang and Steve Satchell , Valuing Information Using Utility Functions, WP00-06 2. Soosung Hwang, Properties of Cross-sectional Volatility, WP00-05 3. Soosung Hwang and Steve Satchell, Calculating the Miss-specification in Beta from Using a Proxy for the Market Portfolio, WP00-04 4. Laun Middleton and Stephen Satchell, Deriving the APT when the Number of Factors is Unknown, WP00-03 5. George A. Christodoulakis and Steve Satchell, Evolving Systems of Financial Returns: AutoRegressive Conditional Beta, WP00-02 6. Christian S. Pedersen and Stephen Satchell, Evaluating the Performance of Nearest Neighbour Algorithms when Forecasting US Industry Returns, WP00-01

1999
1. Yin-Wong Cheung, Menzie Chinn and Ian Marsh, How do UK-Based Foreign Exchange Dealers Think Their Market Operates?, WP99-21

2. Soosung Hwang, John Knight and Stephen Satchell, Forecasting Volatility using LINEX Loss Functions, WP99-20 3. Soosung Hwang and Steve Satchell, Improved Testing for the Efficiency of Asset Pricing Theories in Linear Factor Models, WP99-19 4. Soosung Hwang and Stephen Satchell, The Disappearance of Style in the US Equity Market, WP99-18 5. Soosung Hwang and Stephen Satchell, Modelling Emerging Market Risk Premia Using Higher Moments, WP99-17 6. Soosung Hwang and Stephen Satchell, Market Risk and the Concept of Fundamental Volatility: Measuring Volatility Across Asset and Derivative Markets and Testing for the Impact of Derivatives Markets on Financial Markets, WP99-16 7. Soosung Hwang, The Effects of Systematic Sampling and Temporal Aggregation on Discrete Time Long Memory Processes and their Finite Sample Properties, WP99-15 8. Ronald MacDonald and Ian Marsh, Currency Spillovers and Tri-Polarity: a Simultaneous Model of the US Dollar, German Mark and Japanese Yen, WP99-14 9. Robert Hillman, Forecasting Inflation with a Non-linear Output Gap Model, WP99-13 10. Robert Hillman and Mark Salmon , From Market Micro-structure to Macro Fundamentals: is there Predictability in the Dollar-Deutsche Mark Exchange Rate?, WP99-12 11. Renzo Avesani, Giampiero Gallo and Mark Salmon, On the Evolution of Credibility and Flexible Exchange Rate Target Zones, WP99-11 12. Paul Marriott and Mark Salmon, An Introduction to Differential Geometry in Econometrics, WP99-10 13. Mark Dixon, Anthony Ledford and Paul Marriott, Finite Sample Inference for Extreme Value Distributions, WP99-09 14. Ian Marsh and David Power, A Panel-Based Investigation into the Relationship Between Stock Prices and Dividends, WP99-08 15. Ian Marsh, An Analysis of the Performance of European Foreign Exchange Forecasters, WP99-07 16. Frank Critchley, Paul Marriott and Mark Salmon, An Elementary Account of Amari's Expected Geometry, WP99-06 17. Demos Tambakis and Anne-Sophie Van Royen, Bootstrap Predictability of Daily Exchange Rates in ARMA Models, WP99-05 18. Christopher Neely and Paul Weller, Technical Analysis and Central Bank Intervention, WP9904 19. Christopher Neely and Paul Weller, Predictability in International Asset Returns: A Reexamination, WP99-03 20. Christopher Neely and Paul Weller, Intraday Technical Trading in the Foreign Exchange Market, WP99-02 21. Anthony Hall, Soosung Hwang and Stephen Satchell, Using Bayesian Variable Selection Methods to Choose Style Factors in Global Stock Return Models, WP99-01

1998
1. Soosung Hwang and Stephen Satchell, Implied Volatility Forecasting: A Compaison of Different Procedures Including Fractionally Integrated Models with Applications to UK Equity Options, WP98-05 2. Roy Batchelor and David Peel, Rationality Testing under Asymmetric Loss, WP98-04 3. Roy Batchelor, Forecasting T-Bill Yields: Accuracy versus Profitability, WP98-03 4. Adam Kurpiel and Thierry Roncalli , Option Hedging with Stochastic Volatility, WP98-02 5. Adam Kurpiel and Thierry Roncalli, Hopscotch Methods for Two State Financial Models, WP98-01


				
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Description: The Derivation of New Model of Equity Duration
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