extensions of the capm to include a duration parameter - CiteSeerX

WORKING PAPERS SERIES WP01-11

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

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Table 2:

FT All Share [1963-98] Equity Duration Estimates for 4% Real Yields

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> 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

"'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!

!

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 Model for Firms with Lifetime m

Equity Duration (in years)

40

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 Equity Duration Model for Firms with Infinite Lifespan

50 Equity Duration (in years)

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 (in years)

Equity Duration Model for Firms with Infinite Lifespan

40

Equity Duration Model for Firms with Lifetime m

30

20

10

0 30

40

50

60

70

80

90

100

110

120

130

140

150

Equity Horizon m (in years) ! Figure 5:

Comparative Statics for >

50

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

45

Values for > !

20

35

25

15

5

5. Conclusions: ! !! ^(! 7(8'1! +,)! &'&()! 70! ';51+@4(-8$18! '! ;)$*$;$%6! +.! *#(! *)'-$*$+1'4! -$3$-(1-! -$%;+,1*!6+-(42!@#$;#!&)+3$-(%!.+)!(
-3 +0 = -3 ( H D+ > *< 0 =D ' D + > $

D D ; 8 '1-!@#()(!!< !!=!!( D+ > *
]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! $%! *#(! ( * DU " U )!D 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' HD *3 CA3 +D H

!

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

! !! "#(! '7+3(2! ,1-()! +,)! '%%,6&*$+1%2! $%! (/,$3'4(1*! *+! V,7$1%*($1a%! 6+-(4D?9! ! I*! $%! %*)'$8#*.+)@')-!*+!(U9! !N%!%(*!+,*!$1![#'&*()!>!+.!c,'18!'1-!A$*B(17()8()!CDEFFH9!!

D?

!

24

Appendix 3: Extensions of the CAPM to Include a Duration Parameter! ! ! !! I1!*#$%!$1$*$'4!N&&(1-$ U 2 > S Hu = C :3E *D :H *D : ? !2!! ! *D

*#(1! 5n = >DE "!+!> U E *D µ !+!> SE *D' !9! !!

!

!

C>9UH9!

! Remark 1:! !!

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

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

-" -2 !$%!&+%$*$3(2!$*!6'0!1+*!1(;(%%')$40!7(!*#(!;'%(!*#'*! -" -!' ? !;'1!7(!%$81(-9!!"#,%2!*#()(!$%!1+!

8(+6(*)$;'440!-(*()6$1(-!6')5(*!&+)*.+4$+2!7,*!)'*#()!'!)(8$+19!!^(!;+,4-!*#(1!(H2!@#$;#! %,88(%*%!*#'*!%+6(!$13(%*+)%!%5$6!-$%*)$7,*$+1%!')+,1-!*#(!(b2! )'*#()! *#'1! [T"2!&'0'74(!+1!&)+;((-%!$1!(=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!

!

36

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$(4-! '1-! :=DU>9! Constantinides2!G9!M92!\c'7$*!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!^$4(02! DEE>9! Evans2! M9! D9! D92! \V('4! V'*(%2! :2!f+9S2!',*,61!DEEG2!&&9FG=D?D9! Garman2!M9!B92!\"#(!P,)'*$+1!+.!L&*$+1!O+)*.+4$+%]2!)%U&4/.,%1,904/420/.,*2%4%F02(2!Q+49De2!f+9U2! Y,1(!DEF>2!&&9S?E=SD>9!! ! Goetzmann2! W9! N92! Jorion2! P92! \N! A+18()! A++5!'*!P$3$-(1-!z$(4-%]2!)%U&4/.,%1,6U(04"((2! Q+49GF2! f+9e2!DEE>2!&&9eFS=>?F9! Gordon2 M9 J92! "#(! I13(%*6(1*2! Z$1'1;$182! '1-! Q'4,'*$+1! +.! *#(! [+)&+)'*$+12! c+6(@++-! I)@$12! DEGU9! Gordon2! R9!H92!Malkiel2 B9!G92!\[+)&+)'*$+1!Z$1'1;(]2!$1!c+@!"'2!f+9D2!.'44!DEEF2!&&9eS=>?9! Hicks2!J9!R92!Q'4,(!'1-!['&$*'42!L2!&&9__=_E9! Hutton2 W92!"#(!X*'*(!^(a)(!I12!Q$1*'8(!O,74$;'*$+1%2!A+1-+12!DEEG9! Hutton2 W92 "#(!X*'*(!"+![+6(2!Q$1*'8(!O,74$;'*$+1%2!A+1-+12!DEE_9! !

38

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(;,)$*0! O)$;$18]! )%U&4/., %1, 904/420/., /4', YU/4303/30W", :4/.#(0(2, Q+49D?2!f+9>2!DE_>2!&&9_F>=_EF9! ! Jean2 W9 H92! \"#(! :=eE?9! Jensen2! M9! C92 Meckling2 W9! H92 \"#(+)0! +.! *#(! Z$)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(2!X(&*(67()xL;*+7()!DEE?2!&&9_?=_S9 Johnson2!L9 D92!\"#(!V+4(!+.![+13(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!\: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)$;(%2! '1-! Q+4,6(! ')+,1-! *#(! :F2!&&9UGD=UE_9! Miller2!M9!H92!Shiller2!R9!JE2!f+9e2!L;*+7()!DEFG2!&&9!e>D=e_U9 Modigliani, F., \A$.(![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

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