La valoración de los activos de renta fija renta fija

Monografías de Juan Mascareñas sobre Finanzas Corporativas ISSN: 1988-1878

9 - La Valoración de los Activos Financieros de Renta Fija

La valoración de los activos de La valoración de los activos de renta fijarenta fija

© Juan Mascareñas

Universidad Complutense de Madrid

Versión inicial: mayo 1991 - Última versión: marzo 2013

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P1 = 12,51,1212 + 12,5

(1,1212)(1,1357) + 12,5 + 100(1,1212) (1.1357) (1.1551) = 97,45 ! !



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P1 = 12,5(1 + 1r3)

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+ 12,5 + 100(1 + 1r3)3

= 97,45 !" 1r3 = 13,59%!

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r = 12,50 + 97,45100 - 1 = 9,95% !

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2 El precio de mercado podría superar el precio de amortización anticipada si aún queda suficiente tiempo para la

fecha en que ésta se pueda producir. Lo mismo que les ocurre a las obligaciones normales cuyo precio de mercado puede superar el precio de reembolso si su rendimiento desciende por debajo del tipo de interés del cupón.



7

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!

104,5 = 10(1 + rv)j

j=1

5

" + 100(1 + rv)5

# rv = 8,8476% !

!M#&!#*&#!-,.#?!(-!&('.+)+('*#!@,$*,!-,!%&+)(&,!H(<@,!.(!;('<+)+('*#!D&,E!$(!<,-</-,&=!

.(!-,!$+>/+('*(!H#&),B!!

!

104,5 = 10(1 + ra )

+ 112(1 + ra)2

" ra = 8,4215%!

!C#)#!&,!g!&;!*#),&()#$!<#)#!&('.+)+('*#!.(!-,!#9-+>,<+0'!`?XQNWS5!h*&,!H#&),!.(!

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!

10(1 + r )j

j=1

5

" + 100(1 + r )5 = 10

(1 + r ) + 112(1 + r )2 # r = 9,207% !

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11

Q&N.(Q!!a!U?YWS!(R?Q!a!X?XYS!!

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U&N.!a!U&N!p!DR?NQW!r!R?UVW!(Q!!r!R?UVW!(X! r!R?NQW!(Y!E!

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Y&N.!a!Y&N!p!DR?RNWYQW!r!R?R_UVW!(Q!!r!R?QUXUVW!(X! r!R?UNQW!(Y! r!R?QUXUVW!(`! r!

R?R_UVW!(NR! r!R?RNWYQW!(NQ!E!

V&N.!a!V&N!p!DR?RRV`NQW!r!R?RWXY`VW!(Q!!r!R?NYXRYQW!(X! r!R?QVUXUVW!(Y! r!

R?QVUXUVW!(`! r!R?NYXRYQW!(NR! r!R?RWXY`VW!(NQ! r!R?RRV`NQW!(NX!E!!C#)#!$(!,%&(<+,!-,$!%&#9,9+-+.,.($!8/(!$(!,$#<+,'!,!<,.,!&($/-*,.#!$(!#9*+('('!)(F.+,'*(!-,!H0&)/-,!.(!-,!.+$*&+9/<+0'!9+'#)+,-!('!-,!8/(!%!J!DNF%E!+'.+<,'!-,$!%&#9,9+-+F.,.($!.(!,$<('$#!J!.($<('$#!DWRS?!('!'/($*&#!<,$#?!%,&,!,)9#$!;,-#&($EB!!!

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12

!O+>5NR!

!K+!-#$!*+%#$!,!%-,3#!+)%-7<+*#$!%,&,!-#$!%&+)(&#$!<+'<#!,2#$!$#'!-#$!$+>/+('*($!D$/:(*#$!*#.#$!(--#$!,!/',!;#-,*+-+.,.!.(-!NRS!,'/,-EB!

R&N!a!U?US!!!!N&N!a!X?RS!!!!Q&N!a!X?WS!!!!U&N!a!X?VWS!!!!X&N!a!X?_WS!!!!!!!!a!NRS!!%/(.(!;(&$(!$/!(;#-/<+0'!('!-,$!H+>/&,$!NR!J!NN!D-,!-7'(,!;(&.(!+'.+<,!-,!(;#-/<+0'!.(!-#$!*+%#$!,'/,-($!,!%-,3#E5!!

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!

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2,0%

3,0%

4,0%

5,0%

6,0%

7,0%

8,0%

9,0%

0 1 2 3 4 5

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13

!;";#$%#B%/01%+),-#.&#/0'#70-0'#01.)-%1)0'#^',!;(3!8/(!.+$%#'()#$!.(!-,!(;#-/<+0'!.(!-#$!*+%#$!,'/,-($!,!%-,3#!%#.()#$!%&#F<(.(&!,!;,-#&,&!/',!()+$+0'!.(!9#'#$5!K/%#'>,)#$!8/(!.+$%#'()#$!.(!/',!()+$+0'!,!-,!8/(!-(!8/(.,'!(1,<*,)('*(!<+'<#!,2#$!.(!;+.,?!</J#!;,-#&!'#)+',-!($!.(!NRR!(/F&#$!J!8/(!%,>,!/'!</%0'!,'/,-!.(-!Y?WS5!!!C#)('3,)#$!%#&!(-!H+',-!.(!$/!;+.,!D(-!8/+'*#!,2#E!.#'.(!$/!%&#%+(*,&+#!&(<+9+&=!(-!'#)+',-!.(-!9#'#!DNRR!bE!)=$!(-!</%0'!DY?W!bE5!^',!;(3!@(<@#!($*#!<,-</-,&()#$!$/!;,-#&!,-!H+',-!.(-!,2#!X!*('+('.#!('!</('*,!<,.,!/'#!.(!-#$!*+%#$!.(!+'*(&A$!,!%-,3#!.(!8/(!.+$%#'()#$!%,&,!($(!,2#!J!-,!%&#9,9+-+.,.!.(!,$<('$#!J!.($<('$#!DWRS!('!<,.,!<,$#E]!,$7!%#&!(:()%-#?!%,&,!(-!*+%#!.(!+'*(&A$!.(-!`?`XS!(-!;,-#&!.(-!9#'#!,-!H+',-!.(-!,2#!X!$(&=!+>/,-!,!D;(&!H+>/&,!NUEB!!

!

P4 = 0,5 (100 +6,5) + 0,5 (100 +6,5)1,0884 = 97,8! !

!

!O+>5NU!

!K+!&(%(*+)#$!(-!<=-</-#!%,&,!*#.#$!-#$!*+%#$!,!%-,3#!.(-!</,&*#!,2#!#9*('.&()#$!-#!8/(!,%,&(<(!('!-,!H+>/&,!NX5!!!

!O+>5NX!

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14

!O+>5NW!

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!O+>5NY!

!6@#&,! $(>/+&()#$!<#'!(-!%&#<($#!.(!)/-*+%-+<,&! <,.,!;,-#&!.(-!9#'#!)=$!(-! </%0'!%#&! $/! %&#9,9+-+.,.?! *,'*#! ('! (-! <,$#! .(! ,$<('$#! .(-! *+%#! .(! +'*(&A$! <#)#! .(!.($<('$#?! J! -#!,<*/,-+3,&()#$!,-! ,2#!,'*(&+#&! ,-! *+%#!,!%-,3#!8/(! <#&&($%#'.,! D;(&!H+>/&,!NNE5!4-!;,-#&!#9*('+.#!$(!<#)%,&,&=!,@#&,!<#'!(-!%&(<+#!.(!&($<,*(!DNRN?W!bE!



15

%,&,!;(&!8/A!;,-#&!e(1F</%0'f!H+>/&,&=!('!(-!*(&<(&!,2#!D;A,$(!-,!H+>/&,!NVE5!C#)#!$(!#9$(&;,!(-!&($<,*(!*('.&=!-/>,&!('!*&($!.(!-#$!</,*&#!($<(',&+#$!)#$*&,.#$5!!

!O+>5NV!

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!O+>5N`!

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16

!

109,9 = 6,5(1 + r)j

j=1

5

" + 100(1 + r)5 # r = 4,26% !

!)+('*&,$!8/(!('!(-!<,$#!.(-!9#'#!&($<,*,9-(!($!+>/,-!,B!!

!

107,2 = 6,5(1 + r')j

j=1

5

" + 100(1 + r')5 # r' = 4,84%!

!($!.(<+&?!(-!.+H(&('<+,-!('*&(!,)9,$!($!.(!W`!%/'*#$!9=$+<#$?!8/(!($!(-!%&(<+#!,!%,>,&!%#&! (-! ()+$#&! D('! *A&)+'#$!.(! &('*,9+-+.,.E! ,! <,)9+#!.(!.+$%#'(&!.(! -,!#%<+0'!.(!&($<,*(5!4$!.(<+&?!($! -,! &('*,9+-+.,.!(1*&,!8/(!.($(,'!&(<+9+&! -#$!%&#%+(*,&+#$!.(! -#$!9#'#$!&($<,*,9-($5!!

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!4'!-,!H+>/&,!N_!$(!,%&(<+,'!-,$!%&(;+$+9-($!H(<@,$!.(!&($<,*(!,'*+<+%,.#!$(>G'!-#$!.,F*#$!),'(:,.#$!('!(-!(:()%-#!D$0-#!('!(-!<,$#!.(!8/(!-#$!*+%#$!.(!+'*(&A$!,'/,-($!,-F<,'<('!$/!)=1+)#!'+;(-!('!<,.,!%(&+#.#!,'/,-!-,!()+$+0'!'#!$(&=!&($<,*,.,E5!!!! !

L"#$%#B%/01%+)L" #$%#B%/01%+),,-#.&#/%'#07/)C%+)0-&'#-#.&#/%'#07/)C%+)0-&'# +0-B&1()7/&'+0-B&1()7/&' !!!

^',!/6.(=$*(+,"*/,:%0'(6.%V?!($!/',!#9-+>,<+0'!<-=$+<,!8/(!%(&)+*(!,! $/!%&#%+(*,&+#!

<,)9+,&-,!%#&!/'!'G)(&#!.(*(&)+',.#!.(!,<<+#'($!#&.+',&+,$!./&,'*(!/'!%-,3#!.(!*+()%#!.(*(&)+',.#!J!,!/'!%&(<+#!%&(H+:,.#5!4$!/'!*+%#!.(!'B'3./"SB60(-/!%#&8/(!$(!<#)%#'(!.(!.#$!,<*+;#$!H+','<+(&#$!<#)#!$#'B!/',!#9-+>,<+0'!'#&),-!D8/(!%&#%#&F<+#',!/'#$!H-/:#$!.(!<,:,!H+:#$E!J!/',!#%<+0'!8/(!%(&)+*(!%(&)/*,&!A$*,!G-*+),!%#&!/'!

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



17

'G)(&#!.(*(&)+',.#!.(!,<<+#'($!#&.+',&+,$! D8/(!%&#%#&<+#','!/'#$! H-/:#$!.(!<,:,!;,&+,9-($!+'<-/+.,$!-,$!>,','<+,$!.(!<,%+*,-!($%(&,.,$E5!!4'!(-!)#)('*#!.(!-,!()+$+0'!.(!-,$!#9-+>,<+#'($!<#';(&*+9-($?!-,!<#)%,27,!()+$#&,!<#9&,&=!(-!%&(<+#!.(!()+$+0'!)('#$! -#$!<#$*($! +)%-+<,.#$!('!A$*,5!4$*#$!G-*+)#$!$(!.('#)+','!*/&'%&"-%"%)(&(+,! DA./'$'(/,"*/&'&E!8/(!('>-#9,'!*,'*#! -,$!<#)+$+#'($!,!-#$!9,'<#$!.(!+';(&$+0'!8/(!,<*G,'!<#)#!+'*(&)(.+,&+#$!u<#$*($!(1%-7<+*#$F?!<#)#!(-!.($<('$#!.(-!%&(<+#!.(!)(&<,.#!.(!-#$!#*&#$!,<*+;#$!H+','<+(&#$!8/(!-,!()%&($,!@,J,!()+*+.#!,'*(&+#&)('*(!J!8/(!$(!;('!%(&:/.+<,.#$!.(!,->/',!),'(&,!,-!&(,-+3,&$(!($*,!'/(;,!()+$+0'!F<#$*($!+)%-7<+*#$F5!4$!+)%#&*,'*(!&($,-*,&!8/(!A$*(!($!(-!G'+<#!+'$*,'F*(!('!(-!8/(!-,!()%&($,!&(<+9(!.+'(&#!.(!-#$!+';(&$#&($?!%/($*#!8/(!$+!A$*#$!$(!.(<+F.('!,!<#';(&*+&!$/$!*7*/-#$!.(!.(/.,!%#&!*7*/-#$!.(!%&#%+(.,.!'#!%,>,&='!',.,!,! -,!

()%&($,!()+$#&,!,-!*&,*,&$(!.(!/',!$+)%-(!%(&)/*,!.(!,<*+;#$!H+','<+(&#$`5!!

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F!J."0$'(/"-%"*/,:%0&(+,T!4$!(-!'G)(&#!.(!,<<+#'($!#&.+',&+,$!%#&!-,$!8/(!$(!%/(F.(!+'*(&<,)9+,&!-,!#9-+>,<+0'!DH+>/&,&=!('!-,!($<&+*/&,!.(!()+$+0'E5!M/(.(!$(&!+'.+<,.#!(1%-7<+*,)('*(!D/'!&,*+#!.(!W!+'.+<,&=!-,!%#$+9+-+.,.!.(!<,)9+,&!/',!#9-+>,<+0'!%#&!<+'<#!,<<+#'($!#&.+',&+,$E?!#!+)%-7<+*,)('*(?!,!*&,;A$!-%."10%!*(/"-%"*/,:%0&(+,?!$+'!)=$!8/(!.+;+.+&!(-!;,-#&!'#)+',-!.(!-,!#9-+>,<+0'!<#'F;(&*+9-(!('*&(!(-!%&(<+#!.(!<#';(&$+0'B!

!

!

ratio de conversión = valor nominal de la obligación convertibleprecio de conversión !

!K+?!%#&!(:()%-#?!.+$%#'()#$!.(!/',!#9-+>,<+0'!<#';(&*+9-(!</J#!;,-#&!'#)+F',-!($!.(!NRR!(/&#$!J!8/(!*+('(!/'!%&(<+#!.(!<#';(&$+0'!.(!NR!(/&#$!%#&!,<F<+0'?!%#.()#$!<,-</-,&!H=<+-)('*(!$/!&,*+#!.(!<#';(&$+0'B!NR!,<<+#'($!#&.+F',&+,$5!h*&,$!;(<($!(-!%&(<+#!.(!<#';(&$+0'!%/(.(!($*,&!('!H/'<+0'!.(-!;,-#&!.(!<#*+3,<+0'!.(!-,$!,<<+#'($!#&.+',&+,$!)=$!/'!.($</('*#!8/(!,<*G,!<#)#!

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

_!^'!,)%-+#!($*/.+#!$#9&(!-,$"/6.(=$*(/,%&"*/,:%0'(6.%&!.($.(!(-!%/'*#!.(!;+$*,! -(>,-!%/(.(!<#'$/-*,&$(!('!C6Cmh"?!o#$A!45B!2%0%*S/"-%."N%0*$-/"-%"U$./0%&![5!TJl+'$#'5!i,.&+.5!N__Q5!M=>$5B!QWNFQVY



18

('./-3,'*(!%,&,!(-!+';(&$#&?!%#&!(:()%-#B!e<#*+3,<+0'!)(.+,!.(!-,$!,<<+#'($!#&.+',&+,$! .(-! G-*+)#! *&+)($*&(!)('#$! /'! NWSf5! K+! .+<@,! <#*+3,<+0'! H/($(?!%#&!(:()%-#?!.(!NR!(/&#$!(-!&,*+#!.(!<#';(&$+0'!.(!-,$!#9-+>,<+#'($!<#';(&*+F9-($!$(&7,!+>/,-!,B!!

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-,$!8/(!$(!*+('(!.(&(<@#!,!<#';(&*+&!-,!#9-+>,<+0'5!K(!%/(.(!<,-</-,&!)/-*+%-+F<,'.#!(-!&,*+#!.(!<#';(&$+0'!%#&!(-!;,-#&!.(!)(&<,.#!.(!-,!,<<+0'5!6$7?!%#&!(:()%-#?! $+! (-! &,*+#!.(! <#';(&$+0'!($! +>/,-! ,! NN?VY! ,<<+#'($! J! (-! %&(<+#!.(!)(&<,.#!.(!<,.,!/',!.(!A$*,$!($!.(!NR!(/&#$?!(-!;,-#&!.(!<#';(&$+0'!$(&=!+>/,-!,!NNV?Y!(/&#$!D<#)%=&($(!<#'!(-!;,-#&!'#)+',-!.(!-,!#9-+>,<+0'!8/(!($!.(!NRR!(/&#$E5!

F!U$./0"-%".$"/6.(=$*(+,"/0-(,$0($5!4$!(-!%&(<+#!,-!8/(!$(!;('.(&7,!-,!#9-+>,<+0'!<#';(&*+9-(!$+!<,&(<+($(!.(!A$*,!G-*+),!<,&,<*(&7$*+<,5!K(!<,-</-,!.(*(&)+','F.#!(-!;,-#&!,<*/,-+3,.#!.(!-#$!</%#'($!J!(-!%&+'<+%,-!.(!/',!()+$+0'!.(!#9-+F>,<+#'($!#&.+',&+,$!/*+-+3,'.#!<#)#!*,$,!.(!,<*/,-+3,<+0'!(-!&('.+)+('*#!)7F'+)#!&(8/(&+.#!%#&!(-!)(&<,.#!%,&,!/',!()+$+0'!.(!#9-+>,<+#'($!#&.+',&+,$!.(-!)+$)#!%-,3#!J!&+($>#5!

F!J."1%0B/-/"-%"*/,:%0&(+,5!4$!(-!%-,3#!,! -#! -,&>#!.(-!</,-!($!%#$+9-(!(:(&<(&!(-!.(&(<@#!.(!<#';(&$+0'!DH+>/&,&=!('!-,!($<&+*/&,!.(!()+$+0'E5!

F!#$"10()$"-%"*/,:%0&(+,5!4$!-,!.+H(&('<+,!('*&(!(-!%&(<+#!.(!)(&<,.#!.(!-,!#9-+F>,<+0'!<#';(&*+9-(!J!(-!),J#&!.(!-#$!.#$!%&(<+#$!$+>/+('*($B!!

,E!(-!;,-#&!.(!)(&<,.#!.(!-,!#9-+>,<+0'!#&.+',&+,!9E!(-!;,-#&!.(!<#';(&$+0'5!

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

+',-*(&,.,NR5!

!L">#9%D0-&'#2%1%#&*)()1#07/)C%+)0-&'#+0-B&1()7/&'#^'#!.(! -#$!)#*+;#$! 9=$+<#$! %#&! -#$! 8/(! $(! ()+*('! -,$! #9-+>,<+#'($! <#';(&*+9-($! ($!8/(!-#$!*+%#$!.(!+'*(&A$!8/(!%,>,'!$#'!+'.(%('.+('*($!.(-!'+;(-!.(!&+($>#!.(!-,!+'$*+F*/<+0'!()+$#&,5!4H(<*+;,)('*(?!/'!,/)('*#!.(-!'+;(-!.(! &+($>#!.(! -,!()%&($,!%&#F

10 ^'!(:()%-#!.(!$+$*(),!.(!%&#*(<<+0'!<#'*&,!-,!.+-/<+0'!%&#./<+.,!,-!(1+$*+&!/',!,)%-+,<+0'!.(!<,%+*,-!./&,'*(!(-!%(&7#.#!.(!<#';(&$+0'!.(!-,!#9-+>,<+0'!%/(.(!<#'$/-*,&$(!('!-,!#9&,!.(!o#$A!45!C6Cmh"?!8/(!H+>/&,!('!-,!9+9-+#>&,H7,!D%=>+',$B!QYWFQVRE5



19

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v,! @,! 8/(.,.#! <-,&#! 8/(! -,! #9-+>,<+0'! <#';(&*+9-(! $(! <#)%#'(! .(! /',! #9-+>,<+0'!'#&),-!J!/',!#%<+0'!.(!<#';(&$+0'?!%#&!(--#!$/!;,-#&!*(0&+<#!($!$/%(&+#&!#!+>/,-!,-!.(!/',!#9-+>,<+0'!'#&),-5!4$*#!$(!%/(.(!<#)%&#9,&!,-!#9$(&;,&!<#)#!*,'*#!-#$!</%#'($!<#)#!(-!&('.+)+('*#!@,$*,!(-!;('<+)+('*#!.(!-,$!#9-+>,<+#'($!<#';(&*+9-($!$/(-('!$(&!



20

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

!O+>5QR!I,!&(-,<+0'!('*&(!-#$!%&(<+#$!.(!)(&<,.#!.(!-,!#9-+>,<+0'!<#';(&*+9-(!J!.(!-,!,<<+0'!#&.+',&+,!

!4-!)#.(-#!.(!;,-#&,<+0'!.(!/',!#9-+>,<+0'!<#';(&*+9-(! <#)9+',!(-! ;,-#&! *(0&+<#!.(!/',!#9-+>,<+0'!'#&),-?!<#'!(-!;,-#&!*(0&+<#!.(!-,$!,<<+#'($!,!('*&(>,&!,!<,)9+#!.(!.+F<@,!#9-+>,<+0'!('!(-!)#)('*#!.(!-,!<#';(&$+0'5!C#)#!$,9()#$?!(-!;,-#&!*(0&+<#!.(!-,!#9-+>,<+0'!#&.+',&+,!;+('(!.,.#!%#&!-,!(1%&($+0'!DC!($!(-!;,-#&!.(-!</%0']!M

'!($!(-!%&(F

<+#!.(!&(()9#-$#]!J!l!($!-,!*,$,!.(!&('.+)+('*#!)7'+),!&(8/(&+.,EB!!

!

P0 = C(1 + k)jj=1

n

" + Pn(1 + k)n !

!)+('*&,$!8/(!(-!:$./0"-%"*/,:%0&(+,!('!-,!H(<@,!.(!-,!)+$),!DPC*E!.(%('.(&=!.(-!%&(F<+#!.(!)(&<,.#!,<*/,-!.(! -,$! ,<<+#'($!.(! -,! <#)%,27,! DKRE?! .(! -,! *,$,!($%(&,.,!.(!<&(<+)+('*#!,'/,-!J!,</)/-,*+;#!.(!-,$!)+$),$!D>E!J!.(-!&,*+#!.(!<#';(&$+0'!D\CEB!!

PC*!a!KR!1!DNr>E*!1!\C!

!6$7!8/(!<#)9+','.#!,)9,$!(1%&($+#'($!J!$/$*+*/J('.#?!('! -,! H(<@,!.(!<#';(&$+0'?!-#$!</%#'($!8/(! &($*,'!%#&!%,>,&!)=$!(-!%&(<+#!.(! &(()9#-$#!%#&!(-!),J#&!.(! -#$!.#$!$+>/+('*($!;,-#&($B!(-!;,-#&!.(!<#';(&$+0'!DPC*E!J!(-!;,-#&!.(-!9#'#!#&.+',&+#!('!-,!H(<@,!.(!<#';(&$+0'! DM*E?!#9*('.&()#$!(-!;,-#&! *(0&+<#!.(! -,!#9-+>,<+0'!<#';(&*+9-(!('!(-!)#)('*#!,<*/,-B!!



21

!

P0 = C(1 + k)jj=1

t" + Máx [VCt ; Pt]

(1 + k)t !

!!

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q!@,$*,!(-!H+',-!.(!-,!;+.,!.(!-,!()+$+0'!($!.(-!`S! ,'/,-5! C,-</-,&()#$! (-! %&(<+#! *(0&+<#! .(! -,! #9-+>,<+0'! <#';(&*+9-(! $/%#'+('.#!8/(!$/!%&#%+(*,&+#!*#),!-,!.(<+$+0'!.(!<#';(&*+&-,?!#!'#?!('!,<<+#'($!('! -,!%&+)(&,!H(<@,!%#$+9-(5!!4-!%&(<+#!,<*/,-!.(!-,!#9-+>,<+0'!'#&),-!D'#!$(!#-;+.(!8/(!@,!*&,'$</&&+.#!/'!,2#!.($F.(!-,!()+$+0'?!-/(>#!,!-,!#9-+>,<+0'!-(!8/(.,'!'/(;(!,2#$!.(!;+.,E!($!+>/,-!,B!!

!

P0 = 10(1,09)jj=1

9" + 100

(1,09)9 = 106 euros!

!4-!%&(<+#!.(!-,!#9-+>,<+0'!'#&),-!,-!H+',-!.(-!</,&*#!,2#!u<#'*,'.#!.($.(!,@#&,F!$(&=!+>/,-!,!D&(</(&.(!X&W!a!`SEB!!

!

P4 = 10(1,08) + 10

(1,08)2 + 10(1,08)3 + 10

(1,08)4 + 110(1,08)5 = 108 euros !

!6@#&,!%&#<(.(&()#$!,!<,-</-,&!(-!;,-#&!.(!<#';(&$+0'!.(!-,!()+$+0'!.('*&#!.(!</,*&#!,2#$?!8/(!($!-,!%&+)(&,!.(!-,$!H(<@,$!%#$+9-($!%,&,!<#';(&*+&!-,!()+$+0'B!!

VC4 = S0 x (1+g)t x RC = 34 x (1,07)

4 x 3 = 133,7 euros

!I/(>#!(-!%&(<+#!+'*&7'$(<#!.(!-,!#9-+>,<+0'!<#';(&*+9-(!$(&=B!!

!

P0 = 10(1,09)jj=1

4" + Máx(108;133,7)

(1,09)4 = 127,11 euros !

!J?!<#)#!$(!,%&(<+,?!(-!;,-#&!.(!-,!#%<+0'!.(!<#';(&$+0'!($!+>/,-!,!QN?NN!(/&#$!DNQV?NN!F!NRYE5!

!!!



22

VTWT8"#$":$./0$*(+,"$"'0$:M&"-%".$"6(,/)($."6!($*,$!,-*/&,$!@,!8/(.,.#!<-,&#!8/(!(-!;,-#&!*(0&+<#!.(!-,!#9-+>,<+0'!<#';(&*+9-(!($!+>/,-!,!-,!$/),!.(-!;,-#&!*(0&+<#!.(!-,!#9-+>,<+0'!(8/+;,-('*(!)=$!-,!#%<+0'!.(!<#)F%&,!.(!,<<+#'($!8/(!-(!,<#)%,2,5!!51&+)0#.&#/%#07/)C%+),-#+0-B&1()7/&#I##

51&+)0#.&#/%#07/)C%+),-#-01*%/#A#J%/01#-&(0#.&#/%#02+),-#.&#+0*21%!!I,!;,-#&,<+0'!.(! -,!#%<+0'!('! -,!%&=<*+<,!($!.+H7<+-!.(!,%-+<,&!.(9+.#!,! -,$!$+>/+('*($!&,3#'($B!!

,E!4-!%&(<+#!.(!<#';(&$+0'!$/(-(!,/)('*,&!<#'!(-!*+()%#!-#!8/(!8/+(&(!.(<+&!8/(!(-!%&(<+#!.(!(:(&<+<+#!.(!-,!#%<+0'!<,)9+,5!6.()=$?!%/(.(!@,9(&!.+FH(&('*($!H(<@,$!.(!<#';(&$+0'?!($!.(<+&?!;,&+,$!H(<@,$!.(!;('<+)+('*#!.(!-,!#%<+0'5!

9E!I,$!,<<+#'($!%/(.('!%,>,&!.+;(&$#$!.+;+.('.#$!,!-#!-,&>#!.(!-,!;+.,!.(!-,!#9-+>,<+0'!<#';(&*+9-(! -#!8/(!<#)%-+<,!,G'!)=$!(-!,'=-+$+$!.(! -,!;,-#&,F<+0'!.(!-,!#%<+0'5!

<E!i/<@,$! #9-+>,<+#'($! <#';(&*+9-($! $#'! *,)9+A'! ,)#&*+3,9-($! ,'*+<+%,.,F)('*(!,!>/$*#!.(-!()+$#&!.(!-,$!)+$),$5!

!K+'!()9,&>#?!;,)#$!,!,%-+<,&!(-!)#.(-#!9+'#)+,-!.(!;,-#&,<+0'!.(!#%<+#'($?!<#'!-,!)#.+H+<,<+0'!.(!i(&*#'!%,&,!(-! <,$#!.(! &(%,&*#!.(!.+;+.('.#$?!,!/'!(:()%-#!%,&,!;(&! /'!)A*#.#! ,%&#1+),.#! .(! ;,-#&,<+0'! .(! -,! #%<+0'! .(! <#)%&,! +)%-7<+*,! ('! -,$!#9-+>,<+#'($!<#';(&*+9-($5!M,&,!(--#!/*+-+3,&()#$!-#$!.,*#$!.(-!(:()%-#!,'*(&+#&!,!-#$!8/(! ,2,.+&()#$! ,->/'#!)=$! <#)#B! -,! *,$,! .(! &('.+)+('*#! $+'! &+($>#! %,&,! </,*&#!,2#$!.(!%-,3#!($!.(-!VS?!-,!;#-,*+-+.,.!,'/,-!)(.+,!.(!-,!,<<+0'!#&.+',&+,!($!.(-!QWS!J!-,!*,$,!.(!&('.+)+('*#!$#9&(!.+;+.('.#$!('!H#&),!<#'*+'/,!($!.(-!US5!!v,! $,9()#$!8/(!(-! ;,-#&! *(0&+<#!.(! -,!#9-+>,<+0'!'#&),-! D'/(;(!,2#$!.(!;+.,!J!/'!&('.+)+('*#!@,$*,!(-! ;('<+)+('*#!.(-!_SE!($!.(!NRY!(/&#$?!)+('*&,$!8/(!(-!%&(<+#!($%(&,.#!.(-!9#'#!.('*&#!.(!</,*&#!,2#$!$(&=!+>/,-!,B!!!

!

P5 = 10(1,08) + 10

(1,08)2 + 10(1,08)3 + 10

(1,08)4 + 110(1,08)5 = 108 euros!

!C#';(&*+&! -,$!#9-+>,<+#'($!.('*&#!.(!</,*&#!,2#$! +)%-+<,&7,!$,<&+H+<,&!/'#$!H-/:#$!.(!<,:,!;,-#&,.#$!('!NR`!(/&#$!,!<,)9+#!.(!*&($!,<<+#'($?!#!-#!8/(!($!-#!)+$)#?!UY!(/F&#$!%#&!,<<+0'5!4$*(!G-*+)#!($!(-!%&(<+#!.(!(:(&<+<+#!.(!-,!#%<+0'!.(!<#)%&,5!!!4'!-,!H+>/&,!QN!$(!)/($*&,!(-!=&9#-!9+'#)+,-!&(%&($('*,*+;#!.(-!;,-#&!,<*/,-!.(!-,$!,<F<+#'($5!4-!=&9#-!$(!@,!<#'$*&/+.#!)/-*+%-+<,'.#!<,.,!;,-#&!%#&!/'!<#(H+<+('*(!.(!,$F<('$#!F/F!8/(!($!+>/,-!,!(!!1!(>!a!(R?QW!1!(R?RV!a!N?UVVN!J!%#&!/'#!.(!.($<('$#!F.F!8/(!

($!+>/,-!,!(F!!1!(>!a!(FR?QW!1!(R?RV!a!R?ÙWU5!h9$A&;($(!8/(!('!(-!<=-</-#!.(!,)9#$!<#(H+F<+('*($!$(!@,!+'<-/+.#!-,!*,$,!.(!<&(<+)+('*#!,'/,-!)(.+#!,</)/-,*+;,!8/(?!&(<#&.(F



23

)#$?!(&,!.(-!VS5!4-!=&9#-!$(!(1*+('.(!./&,'*(!</,*&#!,2#$5!Z,)9+A'!$(!%/(.(!,%&(F<+,&!<#)#!('!<,.,!,2#!$(!.(*&,(!(-!US!.(-!;,-#&!.(!-,!,<<+0'!('!<#'<(%*#!.(!&(%,&*#!.(!.+;+.('.#$5!!

!O+>5!QN!w&9#-!9+'#)+,-!.(!-,!(;#-/<+0'!.(-!%&(<+#!.(!-,!,<<+0'!./&,'*(!-#$!%&01+)#$!</,*&#!,2#$!

!^',!;(3!.+9/:,.#!(-!=&9#-!9+'#)+,-?! J! *('+('.#!('!</('*,!8/(! -,!%&+)(&,! H(<@,!.(!<#';(&$+0'!($!.('*&#!.(!</,*&#!,2#$?!%&#<(.(&()#$!,!<#)%,&,&!(-!;,-#&!8/(?!('!.+F<@,! H(<@,!.(! <#';(&$+0'?! *('.&=! -,! ,<<+0'! <#'! &($%(<*#! ,-! 8/(! *('.&=!(-! %&(<+#!.(!(:(&<+<+#5!K+!,8/A-!($!),J#&!8/(!A$*(!<,-</-,&()#$! -,!.+H(&('<+,?!8/(!($!(-!;,-#&! +'F*&7'$(<#!.(! -,!#%<+0'!.(!<#)%&,?!('!<,$#!<#'*&,&+#?!(-!&($/-*,.#!$(&=!<(&#5!6$7?!%#&!(:()%-#?!(-!;,-#&!.(!NR`?QY!b!$(&=!<#)%,&,.#!<#'!(-!%&(<+#!.(!(:(&<+<+#!UY!b!J!<#)#!,8/A-!($!),J#&!<,-</-,&()#$!(-!;,-#&!+'*&7'$(<#!.(!-,!#%<+0'!VQ?QY!b5!P(,!('!-,!H+>/&,!QQ!D('!;(&.(E!<#)#!.+<@#!;,-#&!,%,&(<(!(-!%&+)(&#!('!-,!<#-/)',!.(!)=$!,!-,!.(&(F<@,5!K+'!()9,&>#?!$+!(-!;,-#&!.(-!$/9J,<('*(!H/($(!QX?_R!b!,-!$(&!+'H(&+#&!,!-#$!UY!b!.(-!%&(<+#!.(!(:(&<+<+#!(-!;,-#&!+'*&7'$(<#!($!'/-#NN5!T(!($*,!),'(&,!#9*('.&()#$!-#$!;,F-#&($!+'*&7'$(<#$!.(!-,!#%<+0'!.(!<#)%&,!.('*&#!.(!</,*&#!,2#$!8/(!H+>/&,'!('!;(&.(!('!-,!%&+)(&,!<#-/)',!.(!-,!.(&(<@,!.(!-,!H+>/&,!QQ5!!!

11 \(</(&.(!8/(!-,!#%<+0'!%&#%#&<+#',!/'!.(&(<@#?!'#!/',!#9-+>,<+0'?!J!(-!;,-#&!.(!/'!.(&(<@#!$0-#!%/(.(!$(&!%#$+*+;#!#!'/-#?!'/'<,!'(>,*+;#!D%#&8/(!-#$!.(&(<@#$!$(!(:(&<('!#!'#!$(!(:(&<('!J!$(!$/%#'(!8/(!$(!(:(&<('!$0-#!</,'.#!$(!#9*+('(!/'!9('(H+<+#!#!$,*+$H,<<+0'E5



24

!O+>5QQ!P,-#&!.(!-,!#%<+0'!.(!<#)%&,!

!6@#&,!<,-</-,&()#$?!.(!H#&),!&(</&$+;,?!(-!;,-#&!.(!-,!#%<+0'!('!<,.,!/'#!.(!-#$!%(F&7#.#$!)#;+A'.#'#$!.(!.(&(<@,!@,<+,!+38/+(&.,5!M,&,!(--#!%&+)(&,)('*(!.(9(&()#$!<,-</-,&!(-!;,-#&!.(!-,$!e%&#9,9+-+.,.($!'(/*&,-($!,-!&+($>#f!8/(!+'.+<,'!-,$!%#'.(&,F<+#'($!8/(!(-!)#.(-#!,$+>',!,!<,.,!,$<('$#!J!,!<,.,!.($<('$#5!6$7?! -,!%&#9,9+-+.,.!.(!,$<('$#!$(&=?!$+()%&(!8/(!-,!;#-,*+-+.,.!$(!),'*('>,!<#'$*,'*(?!+>/,-!,B!!

!

p =1 + rf " du - d = 1,07 - 0,8353

1,3771 - 0,8353= 0,4332 = 43,32%

!)+('*&,$!8/(!-,!.(!<,.,!.($<('$#!$(&=!+>/,-!,!NF%!a!WY?Y`S5!M#&!*,'*#?!(-!%&+)(&!;,F-#&!.(!-,!#%<+0'!.(!<#)%&,!('!(-!,2#!*(&<(&#!$(&=!+>/,-!,!#9*('(&!-,!)(.+,!%#'.(&,F.,!.(!-#$!.#$!;,-#&($!8/(!%/(.(!*#),&!('!(-!</,&*#!,2#?!,<*/,-+3,.,!/'!%(&+#.#!,-!*+F%#!.(!+'*(&A$!$+'!&+($>#!<#'*+'/#B!!

!

cuuu = 72,26 " p + 29,66 " (1 -p)(1,07) = 44,97 !

!(-!$+>/+('*(!;,-#&!$(&7,B!!

!

cuud = 29,66 " p + 3,83 " (1 -p)1,07 = 14,04 !

!J!(-!*(&<(&!;,-#&!.(!($(!,2#!$(&7,B!!

!

cudd = 3,83 " p + 0 " (1 -p)1,07 = 1,55 !

!



25

J!(-!G-*+)#!;,-#&!.(!($(!,2#!*#),!/'!;,-#&!'/-#!%#&8/(!$/!;,-#&!,!H+',-($!.(-!</,&*#!,2#!*,)9+A'!-#!($5!4-!)+$)#!%&#<(.+)+('*#!/*+-+3,&7,)#$!%,&,!<,-</-,&!(-!;,-#&!*(0&+F<#!.(!-,!#%<+0'!.(!<#)%&,!,-!H+',-!.(-!$(>/'.#!,2#!DQW?YX!b]!Y?WR!b]!J!R?YU!bE?!,-!H+',-!.(-!%&+)(&#!DNU?Ù!b!J!Q?_Y!bE?! --(>,'.#!,$7!,-!)#)('*#!,<*/,-!</J#!;,-#&!$(!<,-</F-,&7,!.(!+>/,-!),'(&,B!!

!

c = 13,83 " p + 2,96 " (1 -p)1,07 a!V?NV!b!

!M#&!*,'*#?!V?NV!b!($!(-!;,-#&!.(!-,!#%<+0'!.(!<#)%&,!.(!/',!G'+<,!,<<+0'!J!<#)#!<,.,!#9-+>,<+0'!<#';(&*+9-(!<#'<(.7,!(-!.(&(<@#!,!,.8/+&+&!*&($!,<<+#'($!$/!;,-#&!$(&7,!(-!*&+%-(?!($!.(<+&?!QN?WN!b5!!!I/(>#!(-!%&(<+#!.(-!*7*/-#!<#';(&*+9-(!$(&=!+>/,-!,B!

!MR!a!NRY!r!QN?WN!a!NQV?WN!b5!

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b?! *('>,! /',! %&+),!.(! <#';(&$+0'! .(-!URS!J!/'!/)9&,-!.(!<#';(&$+0'!<#'*+'>('*(!.(-!NQRS?!%/(.(!<#';(&*+&$(!,!NUb!$0-#!$+!-,!,<<+0'!<#*+3,!%#&!('<+),!.(!NW?Yb!Da!NUb!1!N?QE!,!-#!-,&>#!.(!/'!%(&+#.#!.(*(&)+F',.#!D%#&!(:()%-#?!./&,'*(!QR!.(!-#$!UR!.7,$!,'*(&+#&($!,-!H+',-!.(-!*&+)($*&(E5!

12 M,&,!($*#!($!<#';('+('*(!.+$%#'(&!.(!-#$!*+%#$!.(!+'*(&A$!,!%-,3#!+)%-7<+*#$!($%(&,.#$!,!%,&*+&!.(! -,!H(<@,!.(!<#';(&$+0'?!8/(!$(!#9*('.&='!,!*&,;A$!.(!-,!($*&/<*/&,!*()%#&,-!.(!-#$!*+%#$!.(!+'*(&A$5



26

!4-! $(>/'.#!*+%#!.(!<#';(&*+9-($!<#'*+'>('*($!$/(-(!$(&!()+*+.#!%#&!,->/'#$!9,'<#$!.(!<,&,!,!,/)('*,&!$/!*/0%"*$1('$.!DZ+(&!NE!</,'.#!A$*(!<,(!%#&!.(9,:#!.(!/',!<+H&,!.(*(&)+',.,!D(-!/)9&,-!.(!<#';(&$+0'E5!M#&!(:()%-#?!('!(-!<,$#!.(-!I-#J.$!j,'l?!$+!(-!/)9&,-!.(!$/!*/0%"*$1('$.!*+(&!N!$(!$+*G,!%#&!.(9,:#!.(-!WS!$(!<#';(&*+&7,'!-#$!C#C#$!<#'! -#!8/(!.+<@#!;,-#&! $,-*,&7,!@,$*,!(-! Y?WS5!h9$A&;($(!8/(!('!($*,!)#.,-+.,.!(-!8/(!.+$%,&,!-,!<#';(&$+0'!($!/'!.,*#!<#'*,9-(!.(-!()+$#&!.(!-#$!*7*/-#$!J!'#!(-!%&#F%+(*,&+#!.(!-#$!)+$)#$5!!!!Q"#$%#07/)C%+),-#+0-B&1()7/&#R#1&'+%(%7/&#%-()+)2%.%*&-(&#!i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v!($*#!G-*+)#!$(!<#'$+>/(!&($<,*,'.#!-,$!#9-+F>,<+#'($!*,'!%&#'*#!<#)#!-,!#%<+0'!.(!<#';(&$+0'!$(,!.(-!*+%#!e.('*&#!.(!.+'(&#f?!($*#!($?!</,'.#!(-!;,-#&!.(!<#';(&$+0'!$/%(&(!,-!%&(<+#!.(!,)#&*+3,<+0'!,'*+<+%,.,5!4'*#'<($?!%,&(<(!-0>+<#!$/%#'(&?!8/(!-#$!#9-+>,<+#'+$*,$!<#';(&*+&='!$/$!*7*/-#$!%#&F8/(!(-!;,-#&!.(!<#';(&$+0'!($?!,-!)('#$?!*,'!>&,'.(!<#)#!(-!%&(<+#!.(!,)#&*+3,<+0'5!!P(,)#$! /'! (:()%-#5! I,! (;#-/<+0'! .(-! ;,-#&! .(! /',! ()%&($,! ./&,'*(! -#$! %&01+)#$!.#$!,2#$!;+('(!.(H+'+.,!%#&!(-!=&9#-!9+'#)+,-!8/(!,%,&(<(!('!-,!H+>/&,!QU5!!

!O+>5!QU!

!K/%#'>,!8/(!-,!()%&($,!@,!()+*+.#!$0-#!/',!,<<+0'!J!/',!#9-+>,<+0'!<#';(&*+9-(5!x$F*,!*+('(!/'!;,-#&!'#)+',-!.(!UR!J!.(9(!$(&!&(()9#-$,.#!,-! H+',-!.(-!,2#!Q5!I,!#9-+F



27

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