Aplicações do problema de otimização de carteiras de investimento

Texto

(1)!"#$%&"'('$ )&*('+(, '$ -(./"!(& !"#$#%#& '( )*#(+,#$-*. /"#*#0"#$-* ( 1&+2%#*34& 1$(!#05-* 6 )/11. !"#$%&'() *+ !,+-"(.% *( +/#.#0%&1+ *( $%,/(#,%) *( #23()/#.(2/+ !"#$$! %# &!'(!)*+ ,)(#$ -+!'#$ !"#$$!%&!"'#()!"*#$+,-*.!(/%0-.. !""#$%&'() *# +#"%$&*). !"#$%&'(!)&*+ .'+/!0 1'!0 2345!"# 6#''#5'! %# 7#"%+"8!. ,&$#"!( '# -.// 0&12"$&3 4 56.

(2)

(3)

(4) . !" # $%&$ . &' (* #' ,#,(-# #$ , " # $%&$ ..,' !/010 002!30 4 # 56 ##( 7 (- 8, # ' 9 : . ;$ ## < #6 # ,' != 6#, !< > ,'6(->9 0 04,(- # # =$ ,0 04,(- ,, 0@0A# 6,A 0B0,'C8, :0D0 9 0=0#(!56 #0==0;$ ##< #6# ,' 0= 6#, !< > ,'6(->9 0 ===0E>60. E>6,FG &'' 9%' ,' 9', 1$ . %$,FG 8IJK # :01 9',004', 8%, :0@0L6, ,%# 0B0,'C8%, :0D0<99 9 0 0L# .#0 E6(- ,, C,# 1 9M07 M056 ##(87.;N: 1 907 056 O 1# 87&E.;1: 1 9M07 M0"A 56 # %5' 8=<.;L=&1:. 7##9 3P33 1 F ,#1Q .O #6(- #1 9 ,, . .

(5)

(6) !. !"#$%&' " $"# ("#$%&)* &)+#',)' % ("++'-%* #%,&'.) % /012 34 5'#1 6789. : ;)<=* >"'= &)# ='-" &<-" )# #',?% .'-%* %"= #)<= >%'=* #)< )=>"="* #',?% '+#@ ) #)<= ="A+"=* >"+ )=&%+)# =)#>+) %" #)< B%-"* >+">"+('",%,-" %#"+* ',(),&'."* (%+'C ,?"* +)(",?)('#),&" ) ("#>+)),=@"1 #)<= E<)+'-"= %.F= ) %#'A"=1 ;)-'("1. D %"=.

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

(8) !"#$% !"#! #$%&%'()* +$)+),)" % -!#!$,./%01) -! 2,% 3%$#!.$% -! ./4!"#.,!/#) 5#.,% 4.% 2, ,6#)-) "!, -!$.4%-%7 8%$% ."")* 2#.'.9%,)" ) ,)-!') -! ,6-.%:4%$.;/3.% +$)+)"#) +)$ <%$$= >7 >%$?)@.#9* /) A2%' ) +$)&'!,% 6 B)$,2'%-) -! ,)-) % "! ,./.,.9%$ ) $."3) -) !"#$!%&! +%$% 2, -%-) /C4!' -! $!#)$/) !"+!$%-)* )2 ,%D.,.9%$ ) /C4!' -! $!#)$/) ED%-) -) !"#$!%&! %"")3.%-) % 2, -%-) /C4!' -! $."3) ! -!#!$,./%$ #)-%" %" 3%$#!.$%" 5#.,%"* /) "!/#.-) $."3) ! $!#)$/)* B)$,%/-) % F$)/#!.$% GE3.!/#!7 )"") %'H)$.#,) 6 &%"!%-) /) >6#)-) !'-!$:>!%-* -!"#./%-) I $!")'201) -! +$)&'!,%" -! +$)H$%,%01) /1) './!%$ .$$!"#$.#)"7 J"".,* %-!A2%,)" % B)$,2'%01) -) !"#$!%&!* A2! -!+!/-! -! $!"#$.0K!"* +%$% % 2#.'.9%01) -) ,!",)7. !"!#$!%&'(!#)* L#.,.9%01) -! 3%$#!.$%" -! ./4!"#.,!/#)7 >6#)-) !'-!$:>!%-7 F$)/#!.$% GE3.!/#!7. #+.

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

(10) !"#$%& /0"+@). 12. !"#$%&#. 122. '(#$)*+,-). .. . /012"-) 32"#4$2&%. 5. 6. 9. 70)$2% 8)*0$(% *0 !"#$!%&! !". #$%&'()$* +,%-./'%).(*. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 0. !. #$%*(-'1.23'* (%(&(.(* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 4. !5. 6 /$-'7$ /8-(.9:.1(;%&(. ' . <'$1(. =$-'1%. -'. !B. C D1$%)'(1. EF&('%)' ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. "B. !G. 6)(/(@.%-$ . /8-(.9:.1(;%&(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. "H. !"#$!%&!. -' =.1>$?()@ "A. 5 8:#)*) ;0<*0$=80%*. .>. 5!". =8)$-$ I'7-'19='.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. "J. 5!. C7K$1()/$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. ". 5!5. L1$M1('-.-'* -$ =8)$-$ I'7-'19='.- ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. B. 5!B. #$%*(-'1.23'* F%.(* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. G. ? 70"#0" (+@:$2&)" B!" B!. 6A. N'*&1(2O$ -$ M1$P7'/. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. H. B!"!". 6 /$-'7$ /8-(.9:.1(;%&(. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. J. <'*)'* %,/81(&$* ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 5A. B)("2*0$%,C0" D2(%2". 5E 1222.

(11) !"!#$%&'() *'+,'-.#/0&(). 12. 3 4(5( 678'( !)9!#(8( 8! #!:-#%- ! 6(:#'; 8! &-<(#'=%&'(. 1>. * 3,.-#':6- 8- ?7:-8- @!,8!#A?!(8. 1B. C 3,.-#':6- A DE%FG- -+H!:'<- 8- 9#-+,!6( #!"-#6E,(8-. IJ. '5.

(12) !"#$ %& '$(&)$" !". #$%&'(%$ )' $*+', ',-./0()$ ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 12. !1. 3'&.%4. ',5'%$). ' '4-.4&%$).. !"#$# %(,-. $,,.-($). ! ! ! ! ! ! ! ! !. 6". !6. #$%&'(%$ 7&(8$ 5$%$ 10% )' %'&.%4. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 61. !. 3'&.%4. ',5'%$). ' '4-.4&%$).. !"#$# %(,-. $,,.-($). ! ! ! ! ! ! ! ! !. 66. !9. #$%&'(%$ 7&(8$ 5$%$ 50% )' %'&.%4. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 6. !:. #$%&'(%$ 7&(8$ 5$%$ 100% )' %'&.%4. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 69. ;!" #./<4$, " $ "= )$ 8$&%(> )' -.?$%(@4-($! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ;!1 #./<4$, "" $ 1= )$ 8$&%(> )' -.?$%(@4-($! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. 9. ;!6 #./<4$, 1" $ 6= )$ 8$&%(> )' -.?$%(@4-($! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. :. ;!. 2. #./<4$, 6" $ = )$ 8$&%(> )' -.?$%(@4-($! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !.

(13) !"#$ %& '!()*$" !" 4!". #$%&'()$* +,-)(&'( .( /*$0%1)'2. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. "3. 5(6(78%9 (7:*&;8%9 -%&'$*<8% )&'($&* ( (7'($&* .( => ;)>:?(7 ;(&.% y 2 @ % :)%$ A@$')-( ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. B. 4!. 5(.=<8% .( => ;)>:?(7. ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. ". C!". D)>:?(7 &% ℜ2 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. E. #$%&'()$* +,-)(&'( :*$* $('%$&% (;:($*.% A*$)*&.% .( 10% * 100% ! ! !. 4G. C! C!4. D)>:?(7 &% ℜ3 ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !. xi. F.

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

(15) !"#$%&'($. ). !!" #"$ %&'($)'*" &!+%,+,%'-*" " +%'.'(/" *' !&0,$-+& #"%1'2 $-$3$'(1&-+& #"$ %&'($4 )'*' ,1' %&5$!6" /$!+7%$3' !".%& " +&1' *&1"-+%'-*" ' $18"%+9-3$' *& ,1' 3'%+&$%' *& $-5&!+$1&-+"! -"! *$'! '+,'$!: ;" <'8=+,(". 2>. #"%'1 '8%&!&-+'*"! 3"-3&$+"! #,-*'1&-4. +'$! &1 ?-'-@'!> 8'%' A,& *&?-$@B&! %&(&5'-+&! '" &-+&-*$1&-+" *" +%'.'(/" &!+&C'1 *&?-$*'! -' 1"*&('0&1 *" 8%".(&1' 8%"8"!+"> '!!$1 3"1" '! #"%1,('@B&! 3(D!!$3'! *& '-D($!& *& 1E*$'45'%$9-3$' & ' F&"%$' G"*&%-' *&. !"#$!%&!':. ;" <'8=+,(" H> " 1E+"*". ;&(*&%4G&'* E '8%&!&-+'*"> '!!$1 3"1" !&, '(0"%$+1": I$-'(1&-+&> " <'8=+,(" J +&1 3"1" ".C&+$5" ' $18(&1&-+'@6" *" 1"*&("> '!!$1 3"1" ' '8%&!&-+'@6" *"! %&!,(+'4 *"! -,1E%$3"! ".+$*"! 3"1 ' '8($3'@6" *" 1&!1" '" 8%".(&1' 8%"8"!+" -" 8%&!&-+& +%'.'(/"> 5'($*'-*" " 1E+"*" *&!&-5"(5$*":.

(16) !"#$%&' (. !"#$%& '#$()*#+,. !"#$%&'($. !"#! $%&'#()*+ %&,!"!-#%.*" (.% /,!0! ,!01"2* 31"#4,1$% 5* &,*/)!.% 5! .%617 .18%92* 5* 1-0!"#1.!-#* 5! $%&1#%) %#: * 1-'$1* 5% 5:$%5% 5! ;<+ $*. * "(,=1.!-#* 5! !"#(5*" -* $%.&* 5% #!*,1% .*5!,-% 5! >-%-9%"+ &*, ?%,,@ AB A%,$*C1#8 DEEFB * &,461.* $%&'#()*+ !"#!-5!.*" % ,!01"2* 31"#4,1$% 5! .*5* % %/*,5%, %" G*,.%" $)H""1$%" 5* &,*/)!.% 5! *#1.18%92* 5! $%,#!1,%" 5! 1-0!"#1.!-#*B )*+,-($ .,-"/#,01. I 3*.!. "!.&,! #!0! 5!-#,* 5! "1 % -!$!""15%5! 5! #!, (.% .*!5% 5! #,*$% &%,% J(! $*. !)% &(5!""! "(&,1, "(%" -!$!""15%5!" /H"1$%"B K"#% /("$% 5! (.% $*-0!-92* &%,% .!51, ,1J(!8%" ! #,*$%, .!,$%5*,1%" : J(%"! #2* %-#1=% J(%-#* % 015% !. "*$1!5%5!B L* )*-=* 5% 31"#4,1%+ *" .%1" 510!,"*" %,#1=*" G*,%. ("%5*" $*. !""% >-%)15%5!B L &,1.!1,% .*!5% .!#H)1$% $*. (. 0%)*, &%5,*-18%5* &!)* K"#%5* G*1 $,1%5% &!)*" =,!=*" -* ":$()* MNN %BOB DEPFB I "%)#* $()#(,%) ! #!$-*)4=1$* 5%" .*!5%" &,1.1#10%" $*-"#1#(1( % &,1.!1,% ,!0*)(92* .*-!#H,1% 5% 31"#4,1%B QH * &%&!) .*!5% "(,=1( -% R(:$1%+ !. ESSE+ ! *" $%,#T!" 5! $,:51#* G*,%. 1-0!-#%5*" -* 1-'$1* 5* ":$()* UU &*, $*.&%-31%" 5! $*./("#'0!1"+ $*. */V!#10* 5! G%$1)1#%, % $*.&,% DEPFB.

(17) !"#$%&'(%)% *!+, -.!#/.$0%$12 .3 4.5 (67$% 89 :64#'$6! 1% ;6"/.6$%< =>?@2 !A$3! B5.2 +%3 ! 6"0%$3C#6+!2 % 16"/.6$% 4. #$!"40%$3%5 .3 63&5(4%4 .(.#$D"6+%4 6"764E7.642 (67$.4 1% .4&!F%2 1% #.3&% . 1% +%"#$%(. 1. )%7.$"%4 . +%$&%$!FG.4H I! $.!(61!1.2 % B5. 3!64 63&%$#! "J% K % 16"/.6$% .3 462 6"1.&."1."#. 1! 0%$3! .3 B5. .(. K !&$.4."#!1%2 3!4 463 !4 3.$+!1%$6!4 . 4.$76F%4 B5. .(. &%1. +%3&$!$H L 53! .4&K+6. 1. #E#5(% B5. 1C ! 4.5 &%$#!1%$ ! +!&!+61!1. 1. 4. +%"461.$!$ +$.1%$ 1! 4%+6.1!1. . 5450$56$2 !#$!7K4 1% &%1.$ 1. +%3&$!2 1. #%1%4 %4 M.".0E+6%4 1! 4%+6.1!1. 3%1.$"!H N44. 16"/.6$% "J% .4#C &$.4."#. .3 53 7C+5% +5(#5$!( %5 4%+6!(H N(. +%"4#6#56 53! 6"4#6#56FJ% 4%+6!(H O!$! #.$ 45! A"!(61!1. !44.)5$!1! ."B5!"#% 16"/.6$%2 53 3!#.$6!( "J% &%1. .P64#6$ 463&(.43."#. ."B5!"#% %MQ.#%H L &$.+64% 53 464#.3! 4%+6%+5(#5$!( .4&.+EA+%H R5!"1% .44. 464#.3! &!44! ! .P64#6$2 356#%4 %MQ.#%4 160.$."#.4 &%1.3 4.$76$ +%3% 16"/.6$%H O%$#!"#%2 &%1.S4. 1.A"6$ ! .7%(5FJ% 3%".#C$6! +%3% 53! 7%"#!1. 1% /%3.3 1. ."B5!1$!$ 45! 46#5!FJ% A"!"+.6$! T $.!(61!1. 1. 45! .+%"%36!H U$."#. ! .44. +."C$6%2 45$). ! ".+.4461!1. 1. 4. #.$ 53 +!&6#!( &!$! )6$!$ %4 $.+5$4%4H +!&6#!( &!44%5 ! 4.$ !()% 6"164&."4C7.( "% 16! ! 16! 1!4 &.44%!4 . 1!4 .3&$.4!4 .3 ).$!(H V%3 % 1.4."7%(763."#% 1% +!&6#!(643%2 % 45$)63."#% 1!4 #$!"4!FG.4 +%3.$+6!64 ."#$. !4 .3&$.4!42 &.44%!4 0E46+!4 . 6"4#6#56FG.4 A"!"+.6$!42 45$)65 ! ".+.4461!1. 1.44.4 4.)563."#%4 &%445E$.3 53 6"4#$53."#% B5. (/.4 %0.$.+.44.3 ! &%446M6(61!1. 1. 3.16$ ! +!&!+61!1. 1%4 1.7.1%$.4 .3 +53&$6$ +%3 %4 4.54 +%3&$%3644%4 A"!"+.6$%4 . )6$!$ ! .+%"%36! ! 4.5 0!7%$H 4 3.$+!1%4 A"!"+.6$%4 !#5!3 .3 53 +!3&% 6"+.$#%2 &%64 1.&."1.3 1. !)."#.4 .+%"D36+%4 4."4E7.64 ! 0!#%$.4 63&$.764E7.642 4.Q!3 3!#.$6!64 %5 &5$!3."#. &46+%(')6+%4H W 3.161! B5. ! 6"+.$#.X! 6! 4."1% 6"+%$&%$!1! "% +%"/.+63."#% +6."#EA+%2 0%$!3 4."1% &$%&%4#%4 3%1.(%4 .4#%+C4#6+%4 1.4#6"!1%4 ! 6"#$%15X6$ !()53! %$1.32 53 $!+6%+E"6% (')6+%2 !6"1! B5. &!$! 53! +%3&$.."4J% B5!(6#!#67! 1%4 0."D3."%4H I.44. +%"#.P%2 45$). % &$63.6$% #$!M!(/% "%#C7.( .3 A"!"F!42 !#$6M5E1% !% 3!#.3C#6+% 0$!"+Y4 Z%564 *.!"S[!&#64#. 9(&/%"4. [!+/.(6.$2 "!4+61% .3 Z. :!7$. "% !"% 1. >?\] =>@H ^.54 #$!M!(/%4 "J% 0%$!3 $.+%"/.+61%4 .3 761!H _3 1%4 &$63.6$%4 3!#.3C#6S +%4 ! $.+%"/.+.$ %4 $.(.7!"#.4 . &6%".6$%4 #$!M!(/%4 1. [!+/.(6.$ .3 &$%M!M6(61!1. . 3!#.3C#6+! A"!"+.6$! 0%6 [."%`# a!"1.(M$%#H :%Q.2 [!+/.(6.$ K $.+%"/.+61% 6"#.$"!S.

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`'[#'7!$#$3 ' '7($;$ C 6-'#(' *' (-J7 M-$#*'7 "#*$7 *$ G "#"&!$ &"*'-#$a $ -';"%5<=" `'[#'7!$#$3 $ -';"%5<=" *$ "&)'(!<=" &"#")"%?7(! $ "5 !&)'-6'!($ '3 )"- ,&3 $ %$-'D$ -'75%($#(' *" 57" *$ &$('&+(! $ ' G "#"&'(-!$ #$ 7"%5<=" *' )-">%'&$7 ' "#:&! "7. G%' 6"! 5& *"7 &$!"-'7 "#(-!>5!#('7 )$-$ $ 7?#('7' #'" %+77! $. ]'M5#*" " ' "#"&!7($ Z'##'(K b"7')K R--"F3 "7 )!%$-'7 *$ *"5(-!#$ #'" %+77! $ 7=" " )-!# ?)!" *$ "(!&!D$<=" )'%"7 $M'#('7 ' "#:&! "7 ' $ ""-*'#$<=" *' 75$7 $(!;!*$*'7 $(-$;H7 *" &'- $*". R &$!"- "#(-!>5!<=" #" *'7'#;"%;!&'#(" ' '#('#*!&'(" *"7 &'- $*"7 ,#$# '!-"7 ' $7 ("&$*$7 *' *' !7=" #'77' "#('O(" " "--'5 '& 01Qc3 45$#*" d$--[ U. U$- "F!(D. !& 6#-$/)( #8 9%/)/,& 7'5 )-!&'!-" '7(5*" #" $&)" *$ ('"-!$ &"*'-#$ *' ,#$#<$73 !#(!(5%$*" :#$.8#(%# ;&(&,.%#/ 8009. G& 01Q13 '7 -';'5 5& %!;-" "& " &'7&". )5>%! "5 #". #"&' *" 7'5 $-(!M" "-!M!#$% $>"-*$#*" " &'7&" ('&$. ]'57 '7(5*"7 )"77!>!%!($-$& $.

(19) !"#$"%& '! ($)$ &*$(& +,-'$" !-% ! &" #$""./!0" !"($*1&" '! 0-/!"%0)!-%$"2 3*! ",4! 0, 5,! $" 0-/!"%0'$ !" '!/!) ($-"0'! & 6,-%&)!-%! 0"($ ! !%$ -$ ! '!%! )0-& & &*$(&78$ '! +,-'$" !-% ! 0-/!"%0)!-%$" &*%! -&%0/$" 9&"!&'$" -$ !"#$%&' !-% ! !*!"2 3) :;;<= >&. ? @2 @& A$B0%C !(!9!, $ # D)0$ E$9!* !) 3($-$)0&2 @& ($B0%C )$"% $, 5,! ! & #$""./!* ($)90-& $" &%0/$" '! +$ )& 5,!= #& & ,) -./!*. ($-"0'! &'$ '!. !%$ -$= $. 0"($ ! & )0-0)0C&'$ ! $ ($-6,-%$ '! (& %!0 &" $%0)0C&'&". !",*%&/& !) ,)& F $-%!0 & 3G(0!-%!2 3""&. !',78$ '!. ($-(!0%$ '& '0/! "0G(&78$ '$" &%0/$"2 I$'!)$" '!"( !/! "!-'$ $ )!*1$. 0"($ '& (& %!0 & 9&"!0&H"! -$ & F $-%!0 & 3G(0!-%! ($)$. ($-6,-%$ #$""./!* '! (& %!0 &"= $, "!6&= $ ($-6,-%$ -$ 5,&* %$'&" &". (& %!0 &" %D) $ ).-0)$ -./!* '! 0"($ #& & ,) '&'$ -./!* '! !%$ -$2 J F $-%!0 & 3G(0!-%! ! & K!$ 0& @$'! -& '! (&! )&*+& &0-'& "8$ ),0%$ ,%0*0C&'&" -$" '0&" &%,&0"= ($)$ #$'!)$" ($-"%&%& !) LM= :<N O!""& +$ )&= & K!$ 0& @$'! -& '! (&! )&*+& !"%P %$%&*)!-%! *04&'& &$ ($-(!0%$ '! 0-/!"%0)!-%$ -$ )! (&'$ G-&-(!0 $ &%,&*= #$ 5,! 0-'0(& &$ 0-/!"%0'$ $ )$'$ ($)$ $ 0-/!"%0)!-%$ '!/! "!. !&*0C&'$2 3"%& %!$ 0& ($-"%0%,0 $ +,-'&)!-%$ # 0-(0#&* '! %$'& &. ,%0*0C&78$ '! 0-"% ,)!-%$" G-&-(!0 $"= $-'! "! !-5,&' &) $" &%0/$"= /0"&-'$ $ !%$ -$ ! & !',78$ '$ 0"($2.

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• 12-(.3. U*#$,( # +(+*.#@5( 1(& 0$/9# -(' , 3#.(&!. 1*$@5( ,! +&(D#D/./,#,! >. f (xi ) =. µ=. N X i=1. 1 ! # '>,/# > N. xi f (xi ) =. PN. i=1. N. xi. .. /"*#.'!$9! +&(343!/ ; #.

(22) ! "#$%&'()*+,(% &$&-&*&%. .. !"#$% " &%&!'"()% *%+ ,#-." / 0%#1-1.-+ /2 &+%353/-16 &%$/2%1 $/,#-+ " 3"+-7#0-" $" &%&!'"()% 0%2%. • /*)&0$-&*1. 2. σ =. PN. i=1 (xi. N. 3"'%+/1 -4!"'2/#./. − µ)2. 8 +"-9 :!"$+"$" &%1-.-3" $" 3"+-7#0-" (σ2) $/#%."+5 % $/13-%;&"$+)% $" &%&!'"()% <σ=> 8 0%3"+-7#0-" /#.+/ $!"1 3"+-53/-1 "'/".?+-"1 +/"-1 X / Y 6 0%2 3"'%+/1 /1&/+"$%1 E(X) = µX / E(Y ) = µY @ $/,#-$" 0%2% !2" 2/$-$" $/ 0%2% $!"1 3"+-53/-1 3"+-"2 0%#A!#."2/#./6 %! 1/A"6 @ !2" 2/$-$" $/ "11%0-"()% /#.+/ $!"1 3"+-53/-1 "'/".?+-"16 %B.-$" 0%2% % 3"'%+ /1&/+"$% $% &+%$!.% $/ $!"1 3"+-53/-1 "'/".?+-"1 /2 .%+#% $/ 1!" 2@$-"C %! 1/A"6. • "#2*)&0$-&*1. COV (X, Y ) = E[(X − µx )(Y − µy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n 91( %#:9;4'!( # 4 "($!<"#!% (/#(-=$!(% !4 #> ,#4 #4-#% # ! #4-!'(1#4-# !%-$!?9@ (% '+1 1A !( µ # "($!B4'!( σ 2 > 07 (1?(% &4!-(%8 C4-*+D. limn→∞ P. n 1 X √ (Xi − µ) ≤ y σ n i=1. !. 1 =√ 2π. Z. y. e. −1 2 s 2. ds.. −∞. N!*%&(%!+ ') &#!*'! #!" ! %)!*)"& &(%)*$!*+ 0&*& /" ,+$-.+/!+ ') 4 &%$.!, $(')< 0)(')(%), ) $')(%$#&")(%) '$,%*$J/;'!, #!" *)%!*(!, R1+ R2+ R3+444+Rn+ ! *)%!*(! EOI '! ,+$-.+/!+ (Rp ) : /"& .&*$D.)A &A)&%@*$& 8/) ,)P/) & '$,%*$J/$19! M&/,,$&(& 8/&('! 4 -!* ,/2#$)(%)")(%) P*&(')4 C,,$"Q n. Rp =. 1X Ri . n i=1. R) -!*"& ,$"$A&*+ 0)A! S)!*)"& '! T$"$%) )(%*&A+ & .&*$=(#$& EOI '! '&'& 0!*Q n 1 X 1 var(Rp ) = 2 var(Ri ) = 2 nσ 2 n i=1 n. var(Rp ) =. σ2 n. ,+$-.+/!+. :.

(25) !" # $%&'(% $)&*+,-+.*/01*+ ' + 2'%.*+ 3%&'.0+ &' 3+.4%5*67 !"# σ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ℜn. w = (w1 , w2 , w3 , . . . , wn )T "# 6#+ +1 !"# *%"% 6#+ wi '#6'#+#!,%. !"#$%"&'"- I ',%!,. n X i=1. '*+,'-" 6#'*#!,/%A " %,(& %++#H/'%" !. wi = 1, wi ≥ 0 para i = 1, . . . , n.. E/6 !" J+# ,%7K$7 8/# + %,(& + R1 , R2 , R3 , . . . , Rn ,#!F%7 /7 '#, '! #+6#'%". µ = (µ1 , µ2 , µ3 , . . . , µn )T # % 7%,'(L %++ *(%"% "# * &%'()!*(% "# !J$+(7% '"#7 +#>% "%"% 6 'M. . σ11 . . .  COV (R) =   σn1 . . .. !"# σij "#! ,% % * &%'()!*(% #!,'#. σij =. *. %,(& ' #.  σ1n  -  σnn. %,(& . "# 7 " 8/#. σi2 , para i = j ρij σi σj , para i 6= j.

(26) !" # $%&'(% $)&*+,-+.*/01*+ ' + 2'%.*+ 3%&'.0+ &' 3+.4%5*67 ρij ! " #$%% &"'($ )*% $ "*+,$ T. 3 2$2 w = (w1 / w2 / w3 /---/wn ). !"#$!%&!. $ "*+,$ !- .$%*")*$/ $ % *$%)$ 0$. &'. 8". "#$%&#' #. #$1 $2. ! " ,"%+4, & "& "*5%+". Rp =. n X. wi Ri. i=1. #$1 23 %")'" 0 % *$%)$. ,"%+6)#+" 0"0$2 3$%7. µp = w T µ σp2 = wT COV (R)w. 89")0$ 2. 2 & #+$)" $. 0. "#$%&#' #. 3 2$2/ $ +), 2*+0$%. 2#$&:. )*%. $2 3"% 2 0. 1!0+";,"%+6)#+" 0+23$)<, +2- ."%" #"&#9&"% $2 3 2$2 3"%" 91 3$22<, & 3"%/. 2#$&: ;2. 91" *"=" 1!0+" 23 %"0" 0 % *$%)$/ µ0 - .$% >"%?$@+*A/ $ 3%$B& 1" 0$ +), 2*+0$% ! " 1+)+1+A"'($ 0$ %+2#$/ $9 2 C"/ $ +), 2*+0$% 0 , B92#"%. min wT COV (R)w w. DE-FG. 29C +*$ H2 % 2*%+'I 2. µ0 ≤ w T µ n X. wi = 1. i=1. w ≥ 0.. J2* 3%$B& 1" #&422+#$ 0 $*+1+A"'($ 0 1!0+";,"%+6)#+" ! 0 )$*"0$ 3$% * -.% #+ &#$*/'.% #+. 'I 2 &+) "% 2/. KL/ MN- J2* ! 91 3%$B& 1" 0 $*+1+A"'($ O9"0%4*+#" #$1 % 2*%+;. " 2$&9'($ ! 0"0" 3$%. w = g + hµ0 $)0. 0. 1. $ () * + ,. 2($ 0$+2 , *$% 2. g=. 1 COV (R)−1 [cι − bµ] 2 ac − b.

(27) !" # $%&'()*%+ ,-.*)'() h=. /" 1 COV (R)−1 [aµ − bι] 2 ac − b a = ιT COV (R)−1 ι b = ιT COV (R)−1 µ c = µT COV (R)−1 µ ι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µ0 3 "'# µ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•. 1"!$."/&"0 1+,(S'&(+>. # 1+,(S'&(+ 4:'(4+ %?" &F+4+#"% #. -"!$."/&"0. A&( ') % # 48#(+$.

(28) !" # $%&'()*%+ ,-.*)'(). /0. !"#$% &'() $*+,-!$% ./0!-+,- 1- 2%$3*4!,5 •. 6 "$#7* 8*$9%1* 7*$ ,*1*: -:,-:. •. 6. ; 0<%9%1* 1- $*+,-!$% ./0!-+,-'. •. B1!0!*+%+1* * %,!>* =!>$- 1- $!:0*C % 8$*+,-!$% -/0!-+,- :- ,*$+% #9% =!+<% $-,% +* :!:,-9% 1- 0**$1-+%1%: $-,*$+* -:7-$%1*D$!:0* E1-:>!*F7%1$G*H' .:,% =!+<% ; 0<%9%1% 1- %&)*+ ,- .-"/+,! ,- /+ &#+&' '. •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

(29) !" #$%&%'()*+ ( &,*%(-.(/%0)1%(. !"! #$%! !&'()$#* +,&$)-#"!.$# ! /!")0&+)! -,. 23 !"#$!%&!. -!-! 1,"2. var(Rp ) = wT COV (R)w n 1 X 1 X = 2 var(Ri ) + 2 COV (Ri , Rj ) n i=1 n i6=j. ≤. 1 1 2 nσmax + 2 (n − 1)n × A 2 n n 2 σmax (n − 1) = + ×A n n. ,&-#2 2 2 3 ! /!")0&+)! 4'5)4! -# %,-,$ ,$ !%)/,$ )&-)/)-6!(4#&%# σmax 72 ! 43-)! #4 1!"#$ -! +,/!")0&+)! -,$ !%)/,$ A=. X 1 COV (Ri , Rj ). (n − 1)n i6=j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

(30) !" #$%&%'()*+ ( &,*%(-.(/%0)1%(. 456&. 23. !"#$%#!& '( %!))$) *(+,'('%#$-.') /#*,*0$(!) $ 1!"(/,$-2! $+"')'%#$3$ %$ )'-2!. min wT COV (R)w w. )/7'*#! 8) "')#"*-.') µ0 ≤ w T µ n X. wi = 1. i=1. w ≥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n + 1 D")#/>'3 '! %/!'?3Q'3H $3'K,H -! 3'G!'?#$ %' (/?O, 3$@)' -!, (/?O,H -! #)/R?G-($ 3$@)' -! 0(,?$H -! #'#),'%)$ '! -! '30,B$ %' #)S3 %/!'?3Q'3 ' ,33/! 3->'33/D,!'?#'5. !" #$%&'& ()*')+,#)-' &'(%')*+',% " -! !"#$%$ %' @-3>, %/)'#, J-' >$?3/3#' '! %'!,)>,) -!, )'G/C$ >$! -! 0$(/#$0$ %' n + 1 D")#/>'3H $- 3'K,H -! 3/!0('15 T$%, /#'),BC$ " @,3',%, '! -! 3/!0('1 Y = {y0, y1, . . . , yn} %' n + 1 D")#/>'3 $)%'?,%$3 0'($ ,-!'?#$ %$3 D,($)'3 %, 2-?BC$H $- 3'K,H f (y0) < f (y1) < . . . < f (yn)5 U! >,%, /#'),BC$H -! ?$D$ 3/!0('1 " $@#/%$ , 0,)#/) %$ 3/!0('1 ,?#')/$)H !$D'?%$ $ 0/$) D")#/>' %' 0$3/BC$H $- )'%-A/?%$ $ 3/!0('15 !.

(32) !" #$%&'& ()*')+,#)-'. ".. ! "#$%"&'(#! )#!!*$&%! '&!!& "+(#,# !-#. /&0&1-#2 &1)3'!-#2 4#'(/35-# %'(&/'3 & 4#'(/35-# &1(&/'36 7&!!3! #)&/358&!2 # )%#/ $+/(%4& yn *+&-$+. . #"/#0*0"12! )!, "+ )!30! 3& (*34& '"$ (*-&. !" #$%&. &'"$($ '"$ )!##"* +&*!,. yn. !" #$%&. $. yc. y = y c + δ(y c − y n ), δ ∈ ℜ, !32$. yc =. yi i=0 n . "+ 5$30,6*2$ '"$ 5!30.+ !#. Pn−1. +$(4!,$# 7.,0*5$#8. 9 7&(!, 2$. δ. *32*5& ! 0*)! 2$ !)$,&:;!8 <##! #*-3*=5& '"$ ! )*!, 7.,0*5$ . +!7*2! 3& 2*,$:;! 2! 5$30,6*2$ )&,& & ,$-*;! $+ '"$ #$ $35!30,&+ !# 2$+&*# 7.,0*5$# +*3&, '"&( 2$7$ #$, ! 7&(!, 2$. !" #$%&. &0,&12!. +$(4!,$# '"$ $($8 >$#0& 2$0$,?. δ8. @ ),*+$*,& $ ),*35*)&( !)$,&:;! $A$0"&2& $+ 5&2& *0$,&:;! . & ,$B$C;!8 D+ 7.,0*5$ ,$B$0*2! . !/0*2! ,$B$0*32! ! )*!, 7.,0*5$ 2! #*+)($C. yn. 3! #$30*2! 2! 5$30,6*2$ 2!#. n. 7.,0*5$# ,$#0&30$#8 <##! )!2$ #$, !/0*2! 2& #$-"*30$ A!,+&E. y r = y c + δ r (y c − y n ) $+ '"$. δr = 1. $. F$#0$ )!30! 0,!5&?#$. yc. ,$),$#$30& ! 5$30,6*2$ 2$#5,*0! &30$,*!,+$30$8. A&G?#$ "+& &7&(*&:;! 2$. f r = f (y r )8. H$ !5!,,$,. f 0 ≤ f r < f n−1. $30;!. y n )$(! )!30! ,$B$0*2! y r !/0$32! ! 3!7! #*+)($CE Yk+1 = {y 0 , y 1 , . . . , y n−1 , y r }8. D+ 7.,0*5$ $C)&32*2! 2$7$ #$, 5&(5"(&2! #$ !5!,,$,. f r < f 08. I30;!. $($ #$,J !/0*2!. yn. )$(! )!30! $C?. 5&(5"(&32!. y e = y c + δ e (y c − y n ) !32$. δ e > 18. @7&(*&?#$ $30;! )&32*2!. ye. f e = f (y e )8. H$. !/0$32! ! 3!7! #*+)($CE. fe ≤ fr. 0,!5&?#$ ! )*!, 7.,0*5$. Yk+1 = {y 0 , y 1 , . . . , y n−1 , y e }8. K&#! *##! 3;! !5!,,&. yn . 0,!5&2! )$(! )!30! ,$B$0*2! y r !/0$32! ! 3!7! #*+)($CE Yk+1 = {y 0 , y 1 , . . . , y n−1 , y r }8 @ 5!30,&:;! *30$,3& !5!,,$ #$. f r ≥ f n8. I(& . 5&(5"(&2& & )&,0*, 2$. y ci = y c + δ ci (y c − y n ).

(33) !" #$%&'& ()*')+,#)-'. ./. !"# −1 < δ ci < 0$. %&'( )*+),+*- * ) !.-*/0 1!.#-!*2 *!*+1(*3(# f ci = f (y ci )$ 4*( f ci < f n 2 .- )*3(#. &1 - 56-.1)# y n &#+ & !. "# ) !.-*/0 1!.#-!* y ci 7 -8*!" *((18. Yk+1 = {y 0 , y 1 , . . . , y n−1 , y ci }$. ;8 56-.1)# ) !.-*<" &*-*. ! 5 (18&+#9:. #9.#-1 - )*+),+*3(# * &*-.1- "#. y ce = y c + δ ce (y c − y n ) !"# 0 < δ ce < 1$ % ) !.-*/0 #9.#-!* 1-= ) --#- )*( f r < f n $ %&'( #7#.,=3+*2 *5*+1*3(# f ce = f (y ce )$ ># f ce ≤ f r 2 .- )*3(# y n &#+ & !. "#. ) !.-*/0 #9.#-!* y ce 2 7 -8*!" ,8 ! 5 (18&+#9: Yk+1 = {y 0 , y 1 , . . . , y n−1 , y ce }$. ?1@,-* A$B: C#D#90 2 #9&*!(0 2 ) !.-*/0 1!.#-!* # #9.#-!* "# ,8 (18&+#9 (#!" y 2 6 &1 - 56-.1)# ;8* 1.#-*/0 & "# @#-*- ,8* -#",/0 " (18&+#9 A$E2 )*( *( "#8*1( &#-*/F#( !0 .#!G*8 (1" *)#1.*($ H,*!" ,8* -#",/0 6 #7#.,*"*2 . " ( ( 56-.1)#( ) !.1" ( #8 (0 -#& (1)1 !*" (2 #9)#. ) !.-*1!". 8#+G - "#+#( y 0 $ I!.0 2 ) 8&,.*3(# ( ! 5 (. (18&+#9 *.6 y 0. yi =. yi + y0 , i = 1, . . . , n. 2. ! 56-.1)#(.

(34) !" #$%&'()*&. "+. !!" #$%&"!!% ' $"#"()*% +(' ,-" + $".)/% *"0)1)(+*+ #"0% !)1#0"2 !"3+ !-4&)"5("6 1"5(" #",-"5+7. 8).-$+ 97:; <"*-=/% *" -1 !)1#0"2. !" #$%&'()*& >%1% ?)!(% 5+ !"=/% +5("$)%$@ % ?+0%$ *" δ )5*)&+ % ()#% *" %#"$+=/% ,-" "!(A !"5*% $"+0)B+*+7 C! "!&%0D+! -()0)B+*+! 5+ )1#0"1"5(+=/% *"!(" ($+E+0D% #+$+ %! &%"4&)"5("! -!+*%! !"$/%; 1 −1 r δ ce = , δ ci = , δ = 1, δ e = 2 2 2. C !".-)$ % +0.%$)(1% *% 1'(%*% F"0*"$6G"+* )1#0"1"5(+*% 5"!(" ($+E+0D% ' +#$"6 !"5(+*%7 H"3+1 -1 !)1#0"2 )5)&)+0 5/% *"."5"$+*% *" ?'$()&"! Y0 = {y00 , y01 , . . . , y0n }@ +! &%5!6 (+5("!; 0 < γ s < 1, −1 < δ ci < 0 < δ ce < δ r < δ e. " ∆tol 7 8+=+ k = 07.

(35) !" #$%&'()*& +,--& 1.. "". !"#$# %& n + 1 '(!)*+#& "# Yk = {y0, y1, . . . , yn}, "# -%!./ 01#2 f 0 = f (y 0 ) ≤ f 1 = f (y 1 ) ≤ . . . ≤ f n = f (y n ).. 3#4#56% y 3#4*)/ % 7*%! '(!)*+# yn /)!/'(& "% +#$)!8*"# yc = Pn−1 '(!)*+#& 01# !#&)/!/.2 i=0 n "%& +,--& 2.. i. y r = y c + δ r (y c − y n ). 9'/:*# f r = f (yr ); <# f 0 ≤ f r < f n−1, #$)6% )!%01# yn 7#:% 7%$)% !#4#)*"% yr %=)#$"% % $%'% &*.7:#52 Yk+1 = {y0, y1, . . . , yn−1, yr }; >? 7/!/ % 7/&&% @; +,--& 3. A57/$&6% <# f r < f 0, #$)6% +/:+1:# % 7%$)% "# #57/$&6% y e = y c + δ e (y c − y n ). B/C/ f e = f (ye); <# f e ≤ f r , )!%01# yn 7#:% 7%$)% #57/$"*"% ye %=)#$"% % $%'% &*.7:#52 Yk+1 = {y0, y1, . . . , yn−1, ye}; D/&% +%$)!?!*%, )!%01# yn 7#:% 7%$)% !#4#)*"% y r %=)#$"% % $%'% &*.7:#52 Yk+1 = {y 0 , y 1 , . . . , y n−1 , y r }; >? 7/!/ % 7/&&% @; +,--& 4. D%$)!/C6% <# f r ≥ f n−1, #$)6% / +%$)!/C6% ( #5#+1)/"/; • /&0)',12& 34)3'0,. <# f r < f n, !#/:*E# / +%$)!/C6% #5)#!$/ y ce = y c + δ ce (y c − y n ). # -/C/ f ce = f (yce); <# f ce ≤ f r , )!%01# yn 7#:% 7%$)% "# +%$)!/C6% #5)#!$/ yce # -%!.# 1. $%'% &*.7:#52 Yk+1 = {y0, y1, . . . , yn−1, yce}, # '? 7/!/ % 7/&&% @; D/&% +%$)!?!*%, '? 7/!/ % 7/&&% F G!#/:*E# 1./ !#"1C6%H; • /&0)',12& (0)3'0,. <# f r ≥ f n, !#/:*E# / +%$)!/C6% *$)#!$/ y ci = y c + δ ci (y c − y n ).

(36) !" #$%&'()*&. ". !"#" f ci = f (y ci )$ % f ci < f n & '()*+ y n , -) ,).') / 0).'("#1) 2.' (." y ci !)(3 ) .)4) 523,- 67 Yk+1 = {y 0 , y 1 , . . . , y n−1 , y ci }&. 48 ,"(" ) ,"55) 9$ :"5). 0).'(8(2)& 48 ,"(" ) ,"55) ; <( "-2= +3" ( /+#1)>$. +,--& 5.. ? /+#1). ? ,)5202). )5 n @,2)( 5A 4B('20 5& ( 0"-0+-"./)C)57. yi = y 0 + γ s (y i − y 0 ), i = 1, . . . , n, )D' ./) ) .)4) 523,- 67 Yk+1 = {y 0 + γ s (y i − y 0 ), i = 0, . . . , n}$ E8 ,"(" ) ,"55) 9$. +,--& 6.. :(2'B(2) / ,"("/". %. v u n+1 uX (f (yi ) − f (yc ))2 < ∆tol dist = t n + 1 i=1. 2.' (()3," " 6 0+#1)& 62D2./) " 5)-+#1) 0)3). n+1. 1 X yi . x∗ = n + 1 i=0 :"5) 0).'(8(2)& !"#" k = k + 1. ( ')(. ") ,"55) F$. !"#$!"% &' ()!)&) G , 5*+25" /) ,()0 /23 .') / H -/ (CI "/ 5 )DJ '24) ,"(" ) 0).J+.') / ,).')5 ')(."3C5. .0 ((" *+"./) )5 4"-)( 5 /" !+.#1). 55 .02"-3 .' 2K+"25 <4 ( ,"55) 9 /). "-K)(2'3)>$. *+,'!-)./', L5'" / 50(2#1) /) "-K)(2'3) B ) *+ 0)5'+3"C5 0M"3"( / 2.' (,( '"#1) 3)/ (." /) "-K)(2'3) H -/ (CI "/ )(2K2."- N9O$ G 3"2)( /2! ( .#" .'( " 4 (51) )(2K2.""*+2 +'2-2="/" B *+ ." 4 (51) )(2K2."- ) ,).') /. 6,".51) y e B "0 2') 5. ". f e < f 0&.

(37) ! "#$%#&'()('* ($ +,-$($ .'/('#0+')(. 12. !"# #$%&'&(#) # *#$%# +, &,-,./# y r 0 ! ,(%#1 2' # *&# ,+(3,$%# +,"%! ",4/# *,&3(%, ! ,(%!& # 3,56#& *#$%# ,$%&, y r , y e 1 7 !58#&(%3# 9,5+,&:;,!+ &,!5(<! =3 %#%!5 +, !>!5(!4?," +! @=$4/# #AB,%(># *#& (%,&!4/#) +!+# *#&C. • n + 2 !>!5(!4?," +, @=$4/#) ", ! (%,&!4/# 0 =3! &,-,./#D • n + 3 !>!5(!4?," +, @=$4/#) ", ! (%,&!4/# 0 =3! ,.*!$"/# #= #$%&!4/#D • 2n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fkn−1 ≤ fkr < fkn 1 E8#&!) *,5! #$>,.(+!+, ,"%&(%! +,. 7. , *,5# @!%# +, ykce. ",& =3! #3A($!4/# #$>,.! +, ykc , ykr *!&! # *!&G3,%&# λ ∈ (0, 1)). f (ykce ) = f (λykc + (1 − λ)ykr ) < λf (ykc ) + (1 − λ)f (ykr ) ≤ max {fkc , fkr } ;!" max {fkc , fkr } = fkr +,"+, F=, fkn−1 ≤ fkr , fkc ≤ fkn−1 H# I5%(3# +, #&&, +!. #$>,.(+!+, ,"%&(%! +,. 7 J1. K$%/#) fkce < fkr , !. #$%&!4/# ,.%,&$! 0 !*5( !+! H, !. &,+=4/# $/# 0 ,., =%!+!J1 L,) !# ($>0" +(""#) =3! #$%&!4/# ($%,&$! 0 #$"(+,&!+!) ,$%/# =3 !&8=3,$%# "(3(5!& ",&' !*5( !+#) A!",!+# $# @!%# +, ykci ",& =3! #3A($!4/# #$>,.! +, ykn , ykc 1 9#%, F=, ! #$>,.(+!+, ,"%&(%! 0 &,F=,&(+! *!&! ,"%, !&8=3,$%# MNO1. 3'$#'4) ! !1 6'4$7*-#)89$:. ; %.3)*) !/*!+45!'* &)(($0 6)"8!+<="60' </)>'/ 2$'"*) P&#>!+# ,3 MQO1. n = 1:.

(38) !" #$%&'()*+,-)& .%+'&. /0. !"#$%&'"()* +*%* +!",!- ".! $-,*()!"/%)!!"! #$%&'! &(%)*$!*")(%)+ ! ",%!'! ') -).')*/0)&' 1!223$ #!(4)*56(#$& 5.!7&. 1&*& n = 1+ "&2 ! ",%!'! (8! 1!223$ *)23.%&'!2 ') #!(4)*56(#$& 1&*& n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

(40) !" #$%&'()*+ ,+ -'+./$01 No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−1/ L11*'? &1 <&.(&1 C4! & 2&)'#' .4."#. !1(#)%& .& '!1'& H*<!)<3#.&/ M!N#'&1? #()#,@1 #1 O*K4)#1 P/: ! P/Q? 4' 1*'<3!F .& ℜ2 ! 4' .& ℜ3 /. O*K4)# P/:R S*'<3!F .& ℜ2.

(42) !" #$%&'()*+ ,+ -'+./$01. 23. !"#$% &'() *!+,-./ 01 ℜ3 2 +13.-%".+ ,$1,145% ('6 .071-7. $.45$!89.4' 244!+: %05.4 3. $.41-7.$ 1 ,$1;-.+%: ,$.<!4%+14 $.=1$+#->?-1' @ ,$1;-.+% ,$1,1451 .45> 4#A.!51 B4 $.45$!89.4 µ0 ≤ w T µ n X. wi = 1. i=1. w≥0. 0%4 C#%!4 µ $.,$.4.05% % 5%/% +D3!% .4,.$%3% 3. $.51$01 . %4 $.-%89.4 3. <17%$!E0<!% .05$. %4 4D$!.4 F!45G$!<%4 3%4 %89.4: C#. 4H1 %"$#,%3%4 =1$+%031 % +%5$!I 3. <17%$!E0<!% COV (R)' J-%4 4H1 .0<105$%3%4 0%4 K%;.-%4 6: (: L . &: 01 %,M03!<. 2' N%$% !0<1$,1$%$ .45%4 $.45$!89.4 B +13.-%".+: =1! <$!%3% ,1$ 0G4 #+% 4#;$15!0% 01 O%5-%; <1+ #+% =#08H1 1;A.5!71 31 ,$1;-.+% $.=1$+#-%31 C#. $.<.;.# 1 01+. ! "#$%"&%'()* ' PF%+%+14 .44. ,$1<.441 3. ,.0%-!I%8H1 Q&R'.

(43) !" #$%&$% '()*+,-.%. /0. !"#$%%" &$ '$()*+,)-." !"#$% #&'&#&()! $ !&%*$ )+&*&$')+$ ) ,!-% '$.)% /)!*"0)%1. 2 f (w) = wT COV (R)w+ρ1 min 0, wT µ − µ0 +ρ2. ". 40 X i=1. wi. !. #2. − 1 +ρ3. ". 40 X i=1. (min{0, wi }). 23,"#$% )%%&# 4# /!$30"#) "54&.)0"'," ) 6178 &!!"%,!&,$8 $ 54" ,$!') /$%%9."0 ) )/0&*):;$ +$ <=,$+$ >"0+"!?<")+1 @$!)# ,"%,)+)% +&."!%)% "%*$0A)% /)!) $% /)!B#",!$% +" /"')0&():;$8 $% 54)&% *A)#)#$% +". ρ1 8 ρ 2. ". ρ3 8. /)!) C)!)',&! 54" *)+) /)!*"0) %". ,$!'" C!)'+" 54)'+$ 4%)#$% /$',$% 54" ';$ %),&%D):)# ) !"%,!&:;$1. 2% .)0$!" 54". %),&%E("!)# "%,) *$'+&:;$ D$!)#F. ρ1 = 10 ρ2 = 10 ρ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o. 123!. 456.. 7$+-! ,'8$%&,3.. D. 6\8:A. 6\80:. DJIGDDEFCGEHAGJ. F. 887<A. 8/6704<2. ?. J. <\9]J. <0\9]. DEIAH?GGBAAFGDF. A. <436J. 497 36<9236;. ?. B. 7/2]J. 7/2]649;. DIJHHHEFAHGCEA. C. 0815A. 0\8/610; 56/. HI?JEHBADFFFFH. G. 0\8/J. 0\8/60/. ?. H. 041/A. 041/0;6. FIGFJEDJGGFHJE. E. ^_0/J. ^0/ _0/93]0/. ?. D?. ]^46J. ]6^946. ?.

(45) !" #$%&$% '()*+,-.% No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`aER C RE]UKIH.

(46) !" #$%&$% '()*+,-.%. //. !"# $# !%&! '(# !"# $# #)"*(# +',# (# *%-# *"! './!% 012 034566789:82;058 035500254;:8:4:9 4870 01; 0322876052:;492; 038250;56;724787 4659 017 03;066;729:27556 038::84095778;86 4879 01: 03;5:5609265::90 0397828;47976;26 2809 016 037774475998962; 4304899;594;:29: 4862 015 03:259429088;;74 43427677:9;66;:2 287: 018 03:8697;5926;287 432490224;040555 250: 019 036::6582:802;88 43;6224799265::6 4842 4 035;8;4676906400 437540::940:8:79 ;424 "<=>?< 737@ >ABCDB >EF>C<GB > >DHBDAC<GB !"#$# CIEHB <EEBHI<GB JC>DA> < >EA> H>DKCIB1 E> B IDL>EAIGBC >EAIL>EE> IDA>C>EE<GB >M B=A>C NM ?NHCB GIKCIB G> 50%1 < >EAIM<AIL< C><? G> C>ABCDB B=AIGB E>CI< G> 37, 576092675590%1 > F<C< IEEB >?> G>L>CI< HBMFBC EN< H<CA>IC< HBM 20 <OP>E1 G>DAC> BE 70 <AILBE GIEFBDQL>IE1 G< E>RNIDA> M<D>IC<@ No. 4 2 ; 7 : 6 5 8 9 40 44 42 4; 47 4: 46 45 48 49 20 24. 012!. 'S)+7 ))(-7 -S*T; -%$'; ( #T; !)"&7 !S) ; !%" 7 JU! ; TJ%'; TT) 7 T#,,7 UT"W; U##"7 *T"'; *"'V; ,'S!; ,*T"; , !$; S'TT; S' *;. 345.. 'S)!+ ) '(!%-# -!S*T %*( $'-*#$', ( #T'%*, !S) '"!, &' !S) '! !%" !,' J! U! *$T! T'J*%' T! ('V T#, -*' U! *$T U#"!*% #"U#$ *TV'"!S* *"'V)'$-# ,#X'% 'S! *,*TU" %Y' ,#X'% !$$! S'T$!%*"' %' S' *%'. 6$+-! ,'7$%&,2.. 4;1579822708667 0 512474687:7979 0 71576508:66756 401050874672048 0 01784860999955 019694:5847587 0 0 0 01:92:54:5264; 01:07784996:04 401905;2096657; 0 0 9140928552:64: 0 0 91;99460;08900.

(47) !" #$%&$% '()*+,-.% No % 0 . 2 4 1 < %* %3 % %% %0 %. %2 %4 %1 %< 0*. 012!. !"#$% !)'+% -&+)% 6#,7% ,"+5% ,5:&% ,((&% (&-8% (#$&% ('"#0 +&!!0 +"'8% +"-#% +!&5% +6>8% +,$(% )885% 7&'"% 7$76%. / 345.. !"#$&' (&)#" !)'+$,'&-&+)5& 6#6-+6,5"7 ,"+56859( ,"5#$:&6 (;& ,65+6 (":)56 (&-+&-#"5 85 (&#$& (;& (&5&$7& '$75 +&! (;& +"'"85&( +"-#& +"'"!&5 - ' +"?+6> +5$)-@6 ,&5+ )-$8&-?6 7&'" 5 #6?" 7$76. 6$+-! ,'7$%&,2.. * ./. .01% 2234. 4/341*23132 2 */002%003.4 %. * * * *=**%4 430%04. %/*<*..*4% 1<. * * 3/40032.< %% . 2/*<* << <. < * /0* %1<*312.. * * * 4/31203* %1%0. +ABCDA 0=.E ?AFGCHFA IGHJA KAFA 50% LC FCGMFNM. ,AFA OJA PDGHJA ANQDHRC/ MBRCFSCJMR TMJM RCFHA A TMJKMRHUVM LA TAFGCHFA RC M HNSCRGHLMF WXARRC TMJM DOTFM LHQFHM JYNHJM CRKCFALM CJ 100%E. No 3 % 0 . 2 4 1 < 3* 33. 012!. &!870 88#?0 ?!$:% ?(-&% #56:% "8+,0 "!85% "(+50 @Z"5% :@(&% ::850. 345.. &!8"7 85&#"(?6 ?"!$: ($# -&?$6-&' #56:&($' "!85&+"' ,&5 "!85&"5 "(+5"'& @"5 Z"5$-:"5 :&@$(& :"5#&). 6$+-! ,'7$%&,2.. */2 20.*%242<1 * 2/00434 1%.<* * 32/<4%121 < %4* </<00223%*23*% * * * * *.