Evaluacin AlternativaEvaluacin AlternativaTEORA DE LA INVERSIN- TEMA 5. RENTABILIDAD Y RIESGO DE LAS CARTERAS DE INVERSIN - TEMA 6. EL MODELO DE MARKOWITZProfesor! E"o S# M$%&e" More#oA"&'#o! A#(o#$o Lore#(e C&es()Gr&*o A CAG+(2011/2012),Evaluacin AlternativaNDICE-Tema 5: Rentaili!a! " rie#$% !e la# cartera# !e inver#in&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&'&'(5'1'- El c%m)%rtamient% !el c%n#umi!%r " la *uncin!e utili!a!&&&&&&&&&&&&&&&&&&&&&&&&&&&&&''' (5'2'- Rentaili!a! " rie#$%&&&&&&&&&&&&&&&&&'&&&'' +5'('- ,enta-a# !e la !iver#i.cacin&&&&&&&&&&&&&&&''' 1/-Tema +: El m%!el% 0ar1%2it3&&&&&&&&&&&&&&&&&&&&&'' 25+'1'- Intr%!uccin&&&&&&&&&&&&&&&&&&&&&&&&&'' 25+'2'- C%n-unt% variale " c%n-unt% e.ciente' Curva# !e in!i*erencia&&&&&&&&&&&&&&&&&&&'' 2++'('- 4a cartera )tima !el inver#%r'&&&&&&&&&&&&&&&' (/+'5'- Intr%!uccin !el t6tul% #in rie#$%: el m%!el% !e T%in'''''''''''''''''''''''''''''''''''''''''''''''''''''''''' (/-Evaluacin Alternativa7ili%$ra*6a&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&''52.Evaluacin AlternativaTE0A 5: Rentaili!a! " Rie#$% !e la# Cartera# !e Inver#in5'1'- El c%m)%rtamient% !el inver#%r " la *uncin !e utili!a!'A#(es$(&/$0#1er$es%o2 *r($e#1o1e"3$*0(es$s1e4&e"os$#1$5$1&os ($e#e# 5ers$0# " r$es%o )*refere#/$ *or $#5ers$0# /o# %r1o 1e r$es%o #(es 4&e o(r /o# '6or r$es%o 6 '6orre#(7$"$11+2 6 1es1e *(o 1e 5$s( r/$o#"2 e" $#5ersor $#(e#(r8'9$'$:r " &($"$11 es*er1 1e s& r$4&e:. ;# /"s$(&"os s$%&e# 1$s(r$7&/$0# #or'"se *&e1ee9*resr e# f/$0# 1e " 'e1$ 6 1e " 5r$#: 1e "os re#1$'$e#(os1e "os (>(&"os e# /&es($0#!E[U(p)]=[E((p), 2(p)] (1)Mr?o@$(: ),A5-2 BC+- seD"0 " #e/es$11 1e /o#s$1err " f/$0# 1e&($"$111e"os $#5ersores *r1e(er'$#r "/o'7$#/$0#0*($'e#(rere#(7$"$11-r$es%o. Pre""o#(er$es%o11o2 e" 1e/$sor7&s/r8 '9$'$:r e" re#1$'$e#(o es*er1o 6 '$#$'$:r e" r$es%o!EEvaluacin AlternativafE(p) 80 9 f2( p) :0(2)Es #e/esr$o 4&e re"$/e'os ser$e 1e reFe9$o#es #8"$s$s 6/o#s$1er/$o#es-!,.- L &($"$11 4&e "e re*or( " $#5ersor " re#(7$"$11 6 e" r$es%o se*&e1e# 'e1$r /o# f/$o#es 1e &($"$11 (o(" 6 'r%$#".-.- E" $#5ersor s$e'*re *&e1e *ro#/$rse /o# re"/$o#es 1e$#1$fere#/$ o *refere#/$ e#(re 1os es(1os /o#/re(os 1e re#(7$"$11-r$es%o2 &e #o *&e1e *re/$sr " /(> 1e (" $#1$fere#/$...- Ls/&r5s1e$#1$fere#/$e#Teor>1eCr(er2 " (r7=r/o#re#(7$"$11 6 r$es%o2 so# s/e#1e#(es /o# r$es%o e# 7s/$ss 6re#(7$"$11 e# or1e#1s!Ilu#tracin 1L e9*res$0# #">($/ 1e " f'$"$ 1e /&r5s 1e $#1$fere#/$ 5e#1r>/($#1$/e 1e &($"$11 re#(7$"$11-r$es%o2 (Ep;p) !(Ep;p)!6 G);+ re*rese#( " f/$0# 1e &($"$11.I#(ro1&/$e#1o e"e'e#(os o7=e($5os2 1e7e'os $#(ro1&/$r e" /o#/e*(o 1ere"/$0# 'r%$#" 1e s&s($(&/$0#2 4&e ser> e" ">'$(e 1e " re"/$0# *ore" /o/$e#(e e# "os $#(er/'7$os re#(7$"$11-r$es%o *r 4&e "s($sf//$0# 1e" $#5ersor *er'#e:/ $#5r$7"e!(Epp)= dEpdpR(E p; p)=limn To1os "os e9*er(os e# %es($0# 1e /r(ers $#1$/# 4&e2 *r 4&e "&($"$11 1e" $#5ersor /o# 5ers$0# " r$es%o *er'#e:/ /o#s(#(e2 "5r$/$0# $#/re'e#(" re#(7$"$11-r$es%o 1e7e ser /re/$e#(e! $#$#/re'e#(o 11o 1e" #$5e" 1e r$es%o2 e" $#5ersor /o# 5ers$0# " r$es%oe9$%$r8 $#/re'e#(r '8s 4&e *ro*or/$o#" " re#(7$"$11 1e" (>(&"o. ;#e=e'*"o *o1r> ser s$ $#5ersor 5e $#/re'e#(1 " re#(7$"$11 1e /r(er 1e" .2AH " 52CH ),BI *(os 78s$/os+ es(#1o 1$s*&es(os&'$r $#/re'e#(o1e" r$es%o1e" .2EH" B25H)5,I*(os78s$/os+2 *or o(ros 5,I *(os 78s$/os 1e &'e#(o e# e" r$es%o e9$%$r8 /o'*e#s/$0# 1e '8s 1e ,BI *(os 78s$/os s$ se *re(e#1e 4&e"s /o'7$#/$o#es re*or(e# e" '$s'o #$5e" 1e &($"$11.De 4&> se 1$/e 4&e "os re#1$'$e#(os 'r%$#"es so# /re/$e#(esre*rese#(8#1ose *or s /&r5s s$'$"res" $"&s(r/$0# ,.E# e" /so 1e 4&e "os re#1$'$e#(os 'r%$#"es f&ese# /o#s(#(es "s/&r5s se s&s($(&$r># *or re/(s! 6Evaluacin AlternativaIlu#tracin 2Es(o #os o7"$% 0Pos(&"1o )7+! Re#1$'$e#(os 'r%$#"es /re/$e#(es o /o#s(#(es e# e"r$es%o! " &'e#(re" r$es%oe" &'e#(o1ere#(7$"$11/re/e#1e '#er 1$re/('e#(e *ro*or/$o#" o '8s 4&e 1$re/('e#(e*ro*or/$o#".d2E pd()2 0Por "o 4&e n" tiene l#ica (&"os 1e " e'*res i *or *re/$o $(r$o ($i"+ s$ ($e#e#"es*er#:1e 4&e s&7 s& /o($:/$0# 6 "s*&e1 5e#1er *or*re/$o $(r$o~P$(+ '8s e"e51o. A1e'8s e" $#5ersor ($e#e e#/&e#( 1$5$1e#1o *or //$0# $#/$er(o (~d it)%s$e#1o " re#(7$"$11"e(or$ )*or 'o5erse 1e#(ro 1e" es*/$o 1e "s *ro77$"$11es+!i= P~(it Pio)+~di tPio(5)D0#1e &i re*rese#( " re#(7$"$11 1e (>(&"o e9*res1o e# (#(o *or o o *or /$e#(o 6 P(it Pio)~ re*rese#( " *"&s5"> *or " 5e#( 1e" (>(&"o.Por "o (#(o*r$or$se 1$/e 4&e " re#(7$"$11 1e (>(&"o es 5r$7"e "e(or$ 4&e *&e1e (o'r 1$fere#(es 5"ores e# f/$0# 1es&s *ro77$"$11es 6 4&e se *&e1e# /"/&"r 1os es(1>s($/os! ,.- " 'e1$ o re#1$'$e#(o es*er1o 1e" (>(&"o (E(&i))! E(i)= j=1mRijPj(5)S$e#1o Ri' " re#(7$"$11 1e" /($5o i e# e" es(1o 1e " #(&r"e: ' 6$' es " *ro77$"$11 1e 4&e o/&rr e" es(1o 1e " #(&r"e: '.-.- " 5r$#: 1e 1$/3o re#1$'$e#(o!((&i)%4&e '$1e " 5r$7$"$111e " re#(7$"$11 1e" (>(&"o i. Pr $#(er*re(r es(e es(1>s($/o /o#(r'6or seJs(2 's 1$s*ersoses(r8#"os *os$7"es5"ores 1e"AEvaluacin Alternativare#(7$"$112 s$e#1o '8s $#/$er( 6 rr$es%1 " re#(7$"$11 1e"/($5o!2(i) = E[(i E(i))2] = E(i)2 [E(i)]2 =R(ijE(i))j=1m2Pj (6)See9*res/o'oe" s&'(or$o1e"1$fere#/$1e/&1r1ose#(re/1 5"or *os$7"e 1e " re#(7$"$11 6 " re#(7$"$11 'e1$2 (o1o e""o'&"($*"$/1o *or "s res*e/($5s *ro77$"$11es.T#(o"re#(7$"$11'e1$/o'o"5r$#:es(8#e9*res1se#*or/e#(=e o (#(o *or o. L 1es5$/$0# (>*$/ 1e "os re#1$'$e#(os/o'o 'e1$1 1e 1$s*ers$0# ('7$J# es(r8 e9*res1 e# *or/e#(=e 6se /"/&" /o'o " r>: /&1r1 1e " 5r$#:! (i) = i = 2( i)(;)Ls1es5$/$o#es#e%($5sso#"s4&e($e#e#4&e*reo/&*r$#5ersor /o# 5ers$0# " r$es%o 6 4&e e" re#1$'$e#(o 1e" (>(&"o"/#:r8 5"or 'e#or 1e "o es*er1o. ;($"$:r (#(o 5r$#: /o'o1es5$/$0# (>*$/ s0"o ($e#e se#($1o s$ " f/$0# 1e 1$s(r$7&/$0# 1e "osre#1$'$e#(os es s$'J(r$/2 /o'o *or e=e'*"o 1$s(r$7&/$0# #or'"6 4&e (#(o " *ro77$"$11 /&'&"1" 1ere/3 e $:4&$er1 1e"*(o 'e1$o es " '$s'. E# e" /so 1e 4&e " f/$0# se s$'J(r$/ &($"$:re'os e" /o#/e*(o1e se'$5r$#:!S= R(ijE(i))2j=1kPj(/),IEvaluacin AlternativaD0#1eRijso#"os5"ores *os$7"es*or 1e7=o1e"'e1$4&e*&e1e (o'r (>(&"o i e# "os ) es(1os 1e " #(&r"e:.Pr /"/&"r re#(7$"$11 6 r$es%o se *&e1e (o'r M#$/o (>(&"o o/r(er1e5"oreso/o#=(o1e/o'7$#/$o#es1$fere#(es1e(>(&"oso/($5os/"$/o 1e "$#1&s(r$&(o'o(r$:2 6#($ />/"$/o1e"$#1&s(r$1ere/'7$os2 $#5ersor/o#(>(&"ose#"s1os$#1&s(r$s*&e1e(e#erre#1$'$e#(os'8s es(7"es *or " 1$5ers$(&"os 6 o7"$%#" $#5ersor (r(r "os (>(&"os 1e for' /o#=(2 6 4&e "s /o5r$#:s#e%($5s o re1&/$1s 1$s'$#&6e# " 5r$#: 1e"re#1$'$e#(o 1e "/r(er e# s& /o#=(o.Re/or1e'os 4&e " /o5r$#: se /"/&" 1e " s$%&$e#(e for'!,-Evaluacin Alternativa(#$(i, ))=E[(i E(i)( ) E())]= j=1m(ij E(i))( )jE()))Pj(15))#o(! (e#er e# /&e#( 4&e " ser 1os (>(&"os e" 1e#o'$#1or 1e " f0r'&" 1e7e ser! #-,N,+D1o 4&e e" /oe(&"os e#f/$0#1es&/oe(&"ofor'*r(e1e/r(er1e7er8ser 5"or1ose%M# s& *or(/$0#" re#(7$"$11 6 "r$es%o 1e " '$s'. Es(os$%#$ s$ e" *eso 1e 1$/3o (>(&"o ('*o/o "o 3/e 6 4&e" *or(/$0# " r$es%o 1e " /r(er2 M# s$e#1o /o#s(#(e e" *eso 1e"(>(&"o2 s$ /'7$ se%M# /&"es se# "os (>(&"os 4&e "e /o'*D# 6 "s*o#1er/$o#es 1e es(os M"($'os e# e"". De*e#1$e#1o 1e 4&J*r8'e(ros *r 'e1$r re#(7$"$11-r$es%o &($"$/e'os 5'os #e/es$(r s 'e1$1s 1e r$es%o & o(rs. As> (e#e'os!E-e# Rentaili!a! Rie#$%(E) > ?2)) @$E)Q$+ @$ /o5)Q$ 2 Q*+(E) > ?)) @$E)Q$+ @$ /o5)Q$ 2 Q*+KR)Q*+(E) > @)) @$E)Q$+ @$S$,EEvaluacin AlternativaPo1e'os seD"r 4&e e" r$es%o 1e /r(er 1e*e#1e 1e " *or/$0# o*eso re"($5o 1e /1 /($5o )@$+2 1e " 1es5$/$0# (>*$/ 1e /1 (>(&"o6 1e " /o5r$#: e#(re "os re#1$'$e#(os 1e "os (>(&"os.S$ #or'"$:'os e" r$es%o 1e " /r(er $ 6 1e/$'os 4&e s& 1es5$/$0#(>*$/ )R*+ se /orres*o#1e /o# e" ,IIH 1e" r$es%o2 *o1re'os e5"&r "/o#(r$7&/$0#1e/1(>(&"oese,IIH1e" r$es%o1e"s$%&$e#(efor'!1=(p) + (p) =i=1Nwi conv( i , p) 2(p) = i=1Nwi,i = ,p(21)Por "o 4&e e" *r8'e(ro 4&e '$1e " *e#1$e#(e 1e " re%res$0# "$#e"e#(re "os re#1$'$e#(os *re5$s(os 1e" (>(&"o ie# "os e=es(Ep%2p)5$e#e1e(er'$#1o *or!@iAconv(i, p)2(p)(22))Me1$1 1e" r$es%o 1e (>(&"o e# "os e=es (Ep%2p)1 V'os5er *r 1e e=e'*"os /o# 1os (>(&"os 6 1$s($#(os es/e#r$os!E-em)l% 1: D%# t6tul%# T6tul% BiE(Ci) ?2(Ci) DiA 6IH ,IH BH EIH7 EIH ,5H 6H 6IHE" re#1$'$e#(o es*er1o 1e " /r(er 5e#1r8 11o *or "se9*res$o#es )5+ 6 ),I+!E(Ci)A j=1mRijPj,5Evaluacin AlternativaE(C))A i=1N2iE(Ci) A 0>+E0>1E0>5F0>5E0>15E0>+ A0>+ (+G)S&*o#$e#1o /o5r$#: 1e" .H *o1re'os /"/&"r e" r$es%o 1e "/r(er /o# " e9*res$0# ),,+!?2(C))A212?2(C1)F222?2(C2)F&F2N2?2(CN)F22122C%v(C1>C2)F 2212(C%v(C1> C()F & F22n-12nC%v(Cn-1> Cn) ?2(C))A0>+02E0>0/F0>502E0>0+F2E0>0+E0>05E0>0(E0>0/E0>0+A0>0520;55G 0>00( 0>001Re/es$0#20G20G 0>05 10G 0>010 0>002E(R) 10G ?20>01?10GTala !e am%# t6tul%#: cartera'Es/e#r$oP)9+ Des5.ADes5.BDes5.ATBP)9+TDes5.ATDes5.BE9*#s$0# (0G 22G -15G-0>0(( -0>00#1$/es 1e" '(r$:! Co5)U,2U-+2 Co5)U,2U.+2 Co5)U,2UE+W.. Co5)UAU,I+. A" sers$'J(r$/2 " se% *r(e 1e " (7" es /o*$ 1e " *r$'er.--Evaluacin Alternativa;($"$:re'os " f0r'&"!Co5r$#:s 1e " /r(er! NCOVAR)XBX5!XKX5YXBX6!XKX6+2 10#1eXBXH!XKX re*rese#(# "os re#1$'$e#(os 1e" (>(&"o U, e# "os *er$o1os/o#s$1er1os 6XBX6!XKX6"os /orres*o#1$e#(es " (>(&"oU-2 6s>s&/es$5'e#(e. A3or /"/&"re'os "os r$es%os 6 re#(7$"$11es 1e /1 (>(&"o e# "/r(er 6 e" 1e " /r(er e# s& /o#=(o!T6tul%#De#%(Bi)Dr%me!i%Rentaili!a!KumaDr%!uct%#CLlcul%!elRei#$%U, 5H EH I2III,B I2IIII,U- ,IH EH I2III-C I2IIII.U. 5H -H I2III,. I2IIII,UE 5H .H -I2IIIE- -I2IIII-U5 EH .H -I2IIIIB I2IIIIIU6 ,5H 6H I2III-C I2IIIIEUC ,5H CH I2IIII5 I2IIII,UB ,5H BH I2II,I5 I2III,6UA ,5H 6H I2II,A, I2III-AU,I ,,H .H I2III-C I2IIII.,IIH ,arian3a 0>00055Rentaili!a! !e la Cartera 5>50G-.Evaluacin Alternativa(R)):Rie#$% !e la Cartera: 2>((GDe#viacin T6)icaL /o"&'# Peso re*rese#( e" H 1e *r($/$*/$0# 1e /1 (>(&"o e# "/r(er. E# re"$11 es( /o"&'# 1e7er> ser #&es(r $#/0%#$( *r*o1er /"/&"r " /r(er /o# '6or re#(7$"$11 *os$7"e2 *ero e# es(e*(o#ose5/o#s$1err 1eesfor'. Ls$%&$e#(e/o"&'#!Pro'e1$o Re#(7$"$11 6 " 3e'os /o'e#(1o #(er$or'e#(e. L/&r(/o"&'#es "s&'1e*ro1&/(os 1e" *eso1e/1(>(&"o'&"($*"$/1o*or /1(&"o6 1e " /r(er. ;($"$:re'os " f0r'&"!NS;MAPROD;CTO)XLX.I!XZX.IYL.-!Z.-+ 10#1e XLX.I!XZX.Ire*rese#( e" *eso 1e /1 (>(&"o e# " /r(er 6 " e9*res$0# L.-!Z.-re*rese#( " (&"o U, 1e " '(r$: 1e 5r$#:s 6 /o5r$#:s.Se 1e7e /o#($#&r /o# (o1os "os (>(&"os 3s( /7r " (7".-EEvaluacin AlternativaL M"($' /o"&'# re*rese#( e" *ro1&/(o 1e" *eso 1e " /r(er *or "/o"&'# s&' 1e *ro1&/(os. L s&' )I2III5E+ re*rese#(r8 "5r$#: 1e " /r(er. L r>: 1e " 5r$#: #os 'os(rr8 "1es5$/$0# (>*$/ o r$es%o 1e " /r(er.5'('- ,enta-a# !e la !iver#i.cacin;# *r$'er *ro9$'/$0# se 5re"$:r /o# " 1$5ers$(&"os s$ e9/e*(&'os e" /so1e/orre"/$0#*erfe/((0+)%37r8#5e#(=s e#es1$5ers$*$/s 1e "os (>(&"os/o'7$#r.L 1$5ers$1 7r&s/ e# e"*re/$o 1e (>(&"o *&e1e /orres*o#1er />1 'e#os*ro#/$1 4&eo(r e$#/"&so e" &'e#(o e#e" *re/$o 1e(er/er. Co# /r(ers '8s 7&"(1s )4&$#/e2 5e$#(e (>(&"os o '8s+ "/o#(r$7&/$0# 'r%$#"" re1&//$0# 1e" r$es%o ser8 s&''e#(e*e4&eD. ;#/r(er/o#7=s/o5r$#:s/orres*o#1e#(>(&"os/o# 1es5$/$o#es es(8#1res *e4&eDs o /o# 7= /orre"/$0# e#"os/'7$ose#s&s*re/$os2 *or "o4&e1e7e#7&s/rse(>(&"os/o#7=s/o5r$#:s*r$#/or*orr"s/r(er1e$#5ers$0#o*or(fo"$o 6 re1&/$r e" r$es%o (o(".S&*o#%'os 1os (>(&"os rr$es%1os /&6os re#1$'$e#(os %&r1# /orre"/$0# se%M# " e9*res$0# ),5+!0+( cov(1, 2)(1) ( 2)-5Evaluacin AlternativaPr($e#1o1e4&e"1es5$/$0#(>*$/1e" re#1$'$e#(o1e/&"4&$er(>(&"o s$e'*rees *os$($52 e" s$%#o 1e" /oe(&"os. E" /oe(&"os es!"#$(1, 2)=(1) (2) (2()E# es(e s&*&es(o e" re#1$'$e#(o 1e " /r(er$4&e1r8 1e "s$%&$e#(e for'!2(p)=w122(1)+w222(2)+2w1w2"#$(1,2)=w122(1)+w222(2)+2w1w2*12 (1)(2)= w12 21+ w22 22+2w1w21 12= (w1 1+w2 2)2(25)Por "o (#(o "s e9*res$o#es 4&e #os '$1e# e" re#1$'$e#(o es*er1o 6r$es%o e# es(e s&*&es(o ser8#!E(p)=w1 E(1)+ w2 E(2) (25)p= w1 1+ w2 2(2+)E# es(e s&*&es(o #o 37r8 5e#(=s e# " 1$5ers$#esre/(s! /o# *e#1$e#(e *os$($5 6 o(r /o# *e#1$e#(e #e%($5!E9*res$0# ., L$#er )E9*res$0# .,+ E9*res$0# .-L$#er )E9*res$0# .-+?)E(R))%- '1(&"os2 s$#o('7$J#1e"s/o5r$#:s*or "/orre"/$0#4&e'#($e#e# e#(re s> "os re#1$'$e#(os 1e "os (>(&"os!2(p)= i=1N wi2 2(i)+ i=1Nj=1Nwiwj (#$(i, j)L*r$'er*r(e1e"e9*res$0#se"e""'rie5#"pr"pi"-el"5t6tul"5 " rie5#" -iver5i7ca8le 6 " se%es e" rie5#" 5i5te9:tic" "n" -iver5i7ca8le. ;#$#5ersor $#%e#&o4&e1es/o#o:/e" 'er/1o(&"os #o se *&e1e# 7s(rer 1e "os 'o5$'$e#(os%e#er"es" ":o"7=1e" 'er/1o4&efe/(#/&"4&$er(>(&"o 4&e %&r1e 1e(er'$#1 /orre"/$0# /o# e" '$s'o.L s$(&/$0# se *&e1e res&'$r 1e" s$%&$e#(e 'o1o!,. Los $#5ers$o#$s(s 1e7e# *reo/&*rse 1e " re#(7$"$11es*er1 6 1e" r$es%o 1e s& /r(er. E" r$es%o se $#1$/ *or 'e1$o1e " 1es5$/$0# es(8#1r.-. E" r$es%o 1e //$0# 1e*e#1e /0'o Js( fe/("s1e'8s //$o#es 1e"/r(er. L5o"($"$111e"s //$o#es$#1$5$1&"es ($e#e *o/ $'*or(#/$.]Z&J *s s$ &'e#(o e" #M'ero 1e //$o#es 1$s($#(s e# '$ /r(er^S$ e# 5e: 1e 1os //$o#es2 ]4&J *s s$ /o'*o#%o '$ /r(er /o# (res2.-Evaluacin Alternativa/&(ro2... //$o#es1ee'*ress1$s($#(s^E#es(e/sose*&e1ee"$'$#r e" r$es%o M#$/o2 *ero #o e" r$es%o 1e 'er/1o.E" r$es%o M#$/o "o /o#s($(&6e# "os $##&'er7"es f/(ores es*e/>(&"os+e# /o#=(o 11o 1e /($5os *$/+..EEvaluacin Alternativa/. E" $#5ersor es 5erso" r$es%o*or "o4&es&s f/$o#es 1e$so&($"$11 o /&r5s 1e $#1$fere#/$ )5er (e' #(er$or+ 3# 1e ser/re/$e#(es 6 /o#5e9s!dE( p)d p2 > 0d2E( p)d 2p2 > 01. E" 3or$:o#(e(e'*or" 1e(o1os"os$#5ersores$#/"&6eM#$/o*er$o1o.- Re#)ect% a la# caracter6#tica# !e l%# activ%# " merca!%#.nancier%#:. E# e" 'er/1o e9$s(e# N /($5os s($/s re"e5#(es 1e "os /($5os so# s& re#(7$"$11es*er1 6 s& r$es%o./. E" re#1$'$e#(o 1e "os /($5os *r *er$o1o 1e ($e'*o 11o es 5r$7"e "e(or$ /&6 1$s(r$7&/$0# 1e *ro77$"$11 e"$#5ersor /o#o/e. Es(o *er'$(e es($'r (#(o re#1$'$e#(os /o'o" '(r$: 1e 5r$#:s-/o5r$#:s. Se s&*o#e 4&e e"re#1$'$e#(o 1e /1 (>(&"o s$%&e 1$s(r$7&/$0# #or'".1. Es('os #(e 'er/1os 1e /*$("es *erfe/(os! $#for'/$0#se4&$7"e *r (o1os "os %e#(esY #$#%M# $#5ersor *&e1e $#F&$re# "os *re/$osY se *&e1e *res(r 6 *e1$r *res(1o s$# "$'$(/$o#es" ($*o1e$#(erJs "$7re1er$es%oY #o36$'*&es(osY #o36$#F/$0# #$ /os(es (r#s//$o#"esY se *er'$(e# 5e#(s e#1es/&7$er(o 6 (o1s "s $#5ers$o#es so# 1$5$s$7"es.Se%M# Mr?o@$(: /r(er es e(&"os.Co'o se *&e1e *re/$r e# " $"&s(r/$0# B (e#e'os "os (res /soses(&1$1os!*ij=+1, *ij='1 C '1#$'o2 o#8"o%'e#(e2 e" 'e#or r$es%o *r re(or#o '89$'o. S$ '$r'os"/r(erD#os 1re'os /&e#(e#se%&$11e4&ees(/r(ere#(re%2 *r #$5e" 1e r$es%o +2 re(or#o es*er1o E(Ri)+ 'e#or4&e e"e#(re%1o *or " /r(er B2" /&"*osee e"'$s'o #$5e"1er$es%o *ero e#(re% re(or#o es*er1o E(Ri)('6or. Por "o (#(o ":o# s&*er$or 1e" ($/ *r o7(e#er " fro#(er e 5e#( e# 1es/&7$er(o 1e" (>(&"o i s0"os$ se 1'$(e#2 (e#$e#1o 4&e (o'r *os$/$o#es/or(o *":o.Por o(r *r(e "$%&"4&e " fro#(er e(&"o e#" /r(er (*i)Se"e//$o#re'os e" 5"or 1e"/e"1o7=e($5o'89$'o*r*o1er/"/&"r e" re#1$'$e#(oes*er1o'89$'os&=e(o"sres(r$//$o#es/o'e#(1s.T6tul%# De#% (Bi)Dr%me!i%Rentaili!a!KumaDr%!uct%#CLlcul%!elRei#$%EEEvaluacin AlternativaU, -,H EH I2IIII- I2IIII,U- 5H EH I2IIII- I2IIIIIU. IH -H I2III,6 I2IIIIIUE BH .H -I2III,6 -I2IIII,U5 ,H .H -I2IIIIA I2IIIIIU6 ,AH 6H I2III-, I2IIIIEUC .IH CH I2III.. I2III,IUB ,5H BH I2IIIEB I2IIIICUA IH 6H I2III65 I2IIIIIU,I IH .H -I2IIII- I2IIIII,IIH I2III-,Rentaili!a! !e la Cartera (R)): +>00G Cel!a P-etiv%M>#$' re#(7$"$11 1e " Cr(er)R*+! 62IIHRes(r$//$0# )1(o '#&"+R$es%o 1e " Cr(er! ,2EEHR$es%o(o"err! I25IHRes(r$//$0# )1(o '#&"+2'- 0inimi3acin !e varian3a# #%re ren!imeint%# e#)era!%#:Ls res(r$//$o#es so# s$'$"res "o M#$/o 4&e 36 4&e /'7$r e"*r8'e(ro M9$'$:r *or '$#$'$:r 6 " /e"1 o7=e($5o es e" r$es%o 1e" /r(er!E5Evaluacin AlternativaT6tul%# De#% (Bi)Dr%me!i%Rentaili!a!KumaDr%!uct%#CLlcul%!elRie#$%U, ..H EH I2IIII.I2IIII,I6BU- AH EH I2IIII.I2IIIII.I-U. IH -H I2IIIIEI2IIIIIIIIUE ,EH .H I2IIII.I2IIIIIEECU5 ,CH .H I2IIII.I2IIIII5E.U6 ,BH 6H I2IIII.I2IIIII5AEUC AH CH I2IIII.I2IIIII-B-UB IH BH I2IIIIAI2IIIIIIIIUA IH 6H I2III--I2IIIIIIIIU,I IH .H I2IIII6I2IIIIIIII,IIHI2IIII.-.CE6Evaluacin AlternativaRe#(7$"$11 1e " Cr(er )R*+! E25EHM>#$' re#(7$"$11 1e " Cr(er)R*+! .2IIHRes(r$//$0# )1(o '#&"+Rie#$% !e la Cartera: 0>5;G Cel!a P-etiv%R$es%o(o"err! I25IHRes(r$//$0# )1(o '#&"+Re*$($e#1o 5r$s 5e/es /'7$#1o "os *r8'e(ros 1e res(r$//$0#re#(7$"$11 6 r$es%o *o1re'os o7(e#er (7" 1e 1(os *r *o1erre"$:r e" %r8(&"os2 es(e /r$(er$o #o #os *er'$(ees(7"e/er /o'*r/$o#es e#(re 1os /r(ers e*$/ 6 " /orre"/$0# 1e" /($5e "$7re1e r$es%o so# #&"os!p=[(1'w)292+2w(1'w)*B9B9]1+2(52)Des*e=#1o(+.*)1e )E-+ 6 s&s($(&6J#1o"o e# )EI+o7(e#1re'os e" re#1$'$e#(o es*er1o 1e " /r(er!E(p)= (1pg) AB F pg E(9) (5()Reor1e##1o "os (Jr'$#os o7(e#1re'os e/&/$0# "$#e"4&e re"/$o# e"re#1$'$e#(o es*er1o 6 e"r$es%o 1e (o1s"s /r(ers '$9(s!5-Evaluacin AlternativaE(p)= AB F ( E( g)Rfg )p(55)D0#1e"or1e#1e#e" or$%e#esR=C"*e#1$e#(e"e9*res$0# e#(re *rJ#(es$s. Los 5"ores *os$7"es 1e * ser8#!,al%r!e2Ki$ni.ca!%1 EL $#5ersor se s$(M e# e" *(oR=$#5$r($J#1o"o (o1o e# /($5os "$7res 1er$es%o.0 I#5ers$0# /o'*"e( e# /($5osrr$es%1os (#)0:2:1 D$s(r$7&/$0# 1e" *res&*&es(o e#(re '7os/($5os. Se s$(&r8 e# "%M# *(o 1e "re/( R=.#:0 E#1e&1'$e#(o" (s "$7re 1e r$es%oR=e $#5ers$0# 1e (o1os "os re/&rsos)*res&*&es(o 6 *rJs('o+ e# " /r(er #1Se s$(M" 1ere/3 1e" *(o # (carteraen prH5ta9"% recur5"5 I+EEJ)1 O7(e#$1 " fro#(er e*$/ "o/"$:1o e# e" *(o 1e5.Evaluacin Alternativa(#%e#/$ e#(re " fro#(er e