Impacto de Chuvas Extremas em
Ações da Indústria de Alimentos
Edimilson Costa Lucas
#04/2015
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!" #$%&'()%*+, &' -)+./' &) 0'1+.), 23*.)(+,
P0?!">* $?10%!- * P!Q7 /!-! !@!20!- * -0?B* >$ BIC@!? $F1-$%!?7 "* /-$?$"1$ $?1C>*7 ?$-S !/20B!>! ! ,$*-0! >$ P!2*-$? TF1-$%*? 5,PT;< K!-! %$2I*-B*%/-$$"?M* >! !/20B!E020>!>$ >! ,PT \!^U?$ "$B$??S-0! ! !/-$?$"1!NM* >! \C">!%$"1!NM* 1$Z-0B! >* %*>$2* ! ?$- C1020^!>*< T?1! ?$NM* $?1-C1C-!U?$ B*% !/*0* "*? 1-!E!2I*? >$?$"@*2@0>*? /*- T%E-$BI1? $1 !2< 5&'':;7 3$">$? 58999;7 H%01I 58998; $ ,?!_ 5899(;<
=* B*"?0>$-!- Pi,t * /-$N* >$ \$BI!%$"1* >$ C% !10@* A"!"B$0-* i 5$F<
!"#$ %& $#'&( )*+#,'-.*& /( $*)#*(&0 ,& %* t1 %#2,#3$# 4&( Ri,t1 & (#'&(,& %*5(*& %#$$# '*6& i1 ,& %* t1 $#,%& . ).7) %& .&+& Ri,t = log(Pi,t/Pi,t−1)8
9$$7+#3$# :7# &$ (#'&(,&$ %*5(*&$ %# . % 7+ %&$ '*6&$1 (#4(#$#,' %&$ 4&(
{R1, R2, ..., Rn}1 $#; + 6 (*56#*$ )# '<(* $ *,%#4#,%#,'#$ # *%#,'*. +#,'# %*$'(*/7-% $ =**%0 .&+ >7,!?& %# %*$'(*/7*!?& .7+7) % .&+7+ F8
@$$#$ (#'&(,&$ %*5(*&$ $#(?& &$ (#'&(,&$ .&((#$4&,%#,'#$ &$ 4(#!&$ % $ !"#$ %# #+4(#$ $ % *,%A$'(* /( $*)#*( %# )*+#,'&$1 :7# $#(?& #+4(#B %&$ ,& 4(#$#,'# #,$ *&8
9 4 ('*( % +&$'( %# (#'&(,&$ %*5(*&$1 %#2,#3$#Mn#Ln.&+& & +5C*+& # +-,*+& %#$' +&$'( 1 (#$4#.'*6 +#,'#8 D7 $#; E
Mn= max{R1, R2, ..., Rn}1
Ln= min{R1, R2, ..., Rn}8
@$$#$ +5C*+&$ =&7 +-,*+&$0 $?& .&)#' %&$ #+ /)&.&$ %# ' + ,F& 2C&8 9$$*+ $#,%&1 & &/;#'*6& % GH@ I #$'7% ( & .&+4&(' +#,'& $$*,'<'*.& %#
Mn # Ln8 J& 4(#$#,'# '( / )F&1 '&% $ $ 4(&4(*#% %#$ 4&,' % $ $#(?& (#3 >#(#,'#$ #C.)7$*6 +#,'# & +5C*+&8 K&,'7%&1 (#$7)' %&$ $#+#)F ,'#$ 6 )#+ 4 ( & +-,*+&1 6*$'& :7#E
max{R1, R2, ..., Rn}=−min{−R1,−R2, ...,−Rn}8
L ( %#'#(+*, !?& % %*$'(*/7*!?& .7+7) % %& +5C*+&1 .&,$*%#( ,%& :7# &$ (#'&(,&$ $?& **%1 '#+3$# $#B7*,'# %*$'(*/7*!?&E
P[Mn≤x] = P[R1 ≤x, R2 ≤x, ..., Rn ≤x] =Fn(x)8
!" #$$!% ! !&'()#*! + #,(-)#./01 0 ,#$)1#&2#34! 5#"#)( /!11($6!-,(-)( 0 ($)0 ,#$)1#&2#34! ,! "78#"! Fn(x)9 :010 #$)!% + ,(.-#,! x
0 = sup{x :
F(x)<1}9 ;&$(1*0-,! <2( Fn(x)→0,∀x < x
0 ( Fn →1,∀x≥x09 !"!
($)0 ,#$)1#&2#34! 5#"#)( 0$$2"( $!"(-)( *05!1($ $#-=2501($ >06(-0$ ? !2 @A -0 1()0 1(05% ,#BC$( <2( ($)0 ,#$)1#&2#34! + ,(=(-(10,09 D($)0 E!1"0% -4! (8#$)( 2"0 ,#$)1#&2#34! 5#"#)( -4!C,(=(-(10,0 6010 ! "78#"!% 0 "(-!$ <2( ! "($"! $('0 -!1"05#B0,!9
F(105"(-)(% ($$0 -!1"05#B034! ,!$ "78#"!$% + E(#)0 0 601)#1 ,0 "+,#0 ( ,($*#!C60,14! ,0 $(<2G-/#0 ,( ,0,!$ (" 0-75#$(9 D#0-)( ,( )05 E0)!% 6010 =010-)#1 0 /!-*(1=G-/#0 ,0 ,#$)1#&2#34! ,( 61!&0#,0,($ ,!$ "78#"!$ 6010 05=2"0 ,#$)1#&2#34! 5#"#)( <2( $('0 -4! ,(=(-(10,0% $4! -(/($$71#0$ 0$ ,(.C -#3H($ 061($(-)0,0$ (" I9@ ( I9J9
!"#$%&' ()*) K( (8#$)(" /!-$)0-)($ an >0% bn ∈R% ,( "!,! <2(
limn→∞Fn(anx+bn) = H(x)
6010 05=2"0 E2-34! ,( ,#$)1#&2#34! -4!C,(=(-(10,0 H% (-)4! H + /!-L(/#,0
/!"! 2"0 !"#$!%&!'() ) *+,)$ -.#$-/) ) /0.!/) ( F + ,#)0 6(1)(-/(1 0!
)/12!) - +#$+'() ) /0.!/)>MDNA ,( H% !2 $('0% F ∈M DA(H)9
!"#$%&' ()+) D20$ E2-3H($ ,( ,#$)1#&2#34! H(x) ( G(x) $4! ,! "($"!
)#6!% $( (8#$)(" -O"(1!$ 1(0#$ a (b% /!" b >0% )0#$ <2( G(x) =H(a+bx)%
6010 )!,! x∈R9
N 601)#1 ,#$$!% (" )(1"!$ ,( 065#/034! ("6P1#/0% ($$0$ /!-$)0-)($ a(b% ("
=(105% /!11($6!-,(" Q "+,#0 ( ,($*#!C60,14!% 1($6(/)#*0"(-)(% ,0 *01#7*(5 05(0)R1#0 (" ($)2,!9 ;2 $('0% 6!,(C$( )10&05L01 /!" 0 "+,#0 ( ! ,($*#!C 60,14! ,0$ $+1#($ ,( 1()!1-!$ .-0-/(#1!$ ,0$ ("61($0$ ,! $()!1 ,( 05#"(-)!$9 S" 0,#34!% ($$0$ ,(.-#3H($ 0$$(=210" <2( $( (8#$)( 2"0 ,#$)1#&2#34! 5#"#)( -4! ,(=(-(10,0% "($"! "(,#0-)( 2"0 -!1"05#B034! ,0 *01#7*(5 ,( #-)(1($$(%
!"#$%"&'"()* +,-*,$% ! #+%. !* /+#/* $"0* ! !"#$%"&'"()* *%"1", 2 !*# ! !*#3 4*,#+5'+,$+/+,$+6 * %+#'2$ !* -+,$% 2 ! 7+*%" !+ 8 2*%+# 9:$%+/*# ; 0%+#+,$ !* ,* $+*%+/ <=3>?3
!"#!$% &'(' !"# $Xn% &'# (!)&*+,-# .! /#0-1/!-( #2!#340-#( --.5 ! !6-(7 3!' ,8+(3#+3!( an>09 bn∈R ! #2:&'# ;&+<=8 .-(30->&-<=8 +=87.!:!+!0#.# H .! '8.8 )&!
Mn−bn an →
dH
!+3=8 H ?!03!+,! # &' .8( 30*( 3-?8( .! .-(30->&-<=8@
A0B,C!3@Dα >0E
Φα(x) = { exp(0 , x≤0
−x−α) , x > 0 F!->&22@Dα >0E
Ψα(x) ={ exp(−(−x)
α) , x≤0
1 , x >0
G&'>!2@
Λ(x) = exp(−e−x) , x∈R
@# A',(B+# !"#$%"&'"(B+# Φα6 Ψα + Λ #)* -C / ! # !+ .-(30->&-<H!( .! /#280 !630!'83
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!" #$%&'$()$*+, -./.'01$2030 3, 401,' 56&'.7,
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
!% $&'8&.*! + .)&"*.46+2=! )+ ).'*$.01.2=! 6!$$&*+ )! &J*$&%! )& 1%+ )&*&$%."+)+ +%!'*$+/ ? #&.*+ 1%+ %1)+"2+ 8+$+%?*$.6+ ,!6+2=!K&'6+,+ 8+$+ $&8$&'&"*+$ !' *$L' 6+'!' 8+)$=! )+' ).'*$.01.23&' )& A+,!$ &J*$&%! &% 1%+ M".6+ #+%9,.+ 8+$+%?*$.6+ I1& &":,!0+ +' *$L' ).'*$.01.23&' DN%.*-/ FGGFH; O &'*+ #+%9,.+ +*$.01.5'& ! "!%& )& <.'*$.01.2=! P&"&$+,.Q+)+ )! B+,!$ RJ*$&%! DPRBH/ '&")! )&4".)+ &% S;T;
Hξ(x) = { exp{−(1 +ξx)
−1
ξ} se ξ 6= 0
exp{−exp(−x)} se ξ = 0 DS;TH
!")&U 1 +ξx >0;
V 8+$W%&*$!ξ8!)& '&$ ."*&$8$&*+)! 6!%! 8+$W%&*$! )& 6+1)+ & + ,.:+2=!
&"*$& +' *$L' ).'*$.01.23&' 8+)$=! 6!% + PRB ? )+)+ )+ '&:1."*& #!$%+U
ξ=α−1 >06!$$&'8!")& + ).'*$.01.2=! )& X$?6-&* Φ
α/ ξ= 0 6!$$&'8!")& + ).'*$.01.2=! )& P1%0&, Λ/
ξ=−α−1 <06!$$&'8!")& + ).'*$.01.2=! )& Y&.01,, Ψ
α;
O *9*1,! )& .,1'*$+2=!/ + 4:1$+ Z +8$&'&"*+ + #!$%+ )+' *$L' #1"23&' )&"5
!"#"$ "$ %&'(#(!)!"#"$ # '*!#"# # +,-. /#&# 0#) !)1 0'4 5'! #0&!(16"'
ξ = −0,4 78$!(1))94 ξ = 0 7+1:($)9 $ ξ = 0,4 7;&<*=$09. >$ 0# :#?$!
@$&!A*#B $ C&#A*#:$?0$ D1$ ' %#&E:$0&' ξ "$0$&:!?# # ?#01&$F# "# *#1"# "#
"! 0&!(1!23' *'&&$ %'?"$?0$. G( $&@#B $4 #!?"#4 D1$ # "! 0&!(1!23' "$ ;&<*=$0 #%&$ $?0# *#1"# :#! %$ #"# "$?0&$ # 0&H #%&$ $?0#"# .
;!C1&# IJ ;1?2K$ >$? !"#"$ "$ /&'(#(!)!"#"$ B
!"#$ %&'()! *+,-"&#"*.! )!&&/-0!12/1"/ #! 3!&4#"! 2#- "&5- 2*-"&*6,*78/- %9:; <!* #"&*= 6,>2! ξ=−0,4?@/*6,++AB ξ= 0?%,46/+A /ξ= 0,4 ?<&C)D/"A;
, $ %#&E:$0&' "$ *#1"# ξ #%&$ $?0#B $ !:%'&0#?0$ ?# #%)!*#2K$ $:B
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
!"# $#%&!'!( ! (#"!)*+ #,-%&#.&# /012(#'3&% #& !"45 67789 #.&(# !% &(:% $-%&(-2;-)<#% !"#$%& '()*+$, # -$.#!%% $!$! =+(>
X ∼F rechet(α)⇔ln(Xα)∼Gumbel⇔ −X−1 ∼W eibull(α).
! "#$%&#'(#)*+ ,-.-&/0#1/2/ 2- 3/&-%+
?! %#)*+ !.&#(-+(5 @+- !2+($!$! ;1! 1#&+$+"+A-! =!(! !B;%&!( ! $-%&(-2;-)*+ $+% 1C,-1+% '+"#&!$+% #1 2"+'+% $# &!1!.3+% D,+%4 ?! =(#%#.&# %#)*+5 E !=(#%#.&!$! ! F-%&(-2;-)*+ G#.#(!"-H!$! $# I!(#&+ /GIF95 ';B+ #1=#.3+ #%&C .+ !B;%&# $+% #,'#%%+% $# ;1! J!(-CJ#" !"#!&K(-! X !"E1 $# ;1 '#(&+
"-1-!( u5 +; %#B!5 !B;%&!( ! $-%&(-2;-)*+ $! J!(-CJ#" X−u4
?#%&# %#.&-$+5 !"A;1!% 1#&+$+"+A-!% =!(! $#&#(1-.!( + "-1-!( u =+$#1
%#( #.'+.&(!$!% #1 L3!J#HMF#1+;"-.5 012(#'3&% # N!($O /PQ6R94 ?+ #.M &!.&+5 .*+ #,-%&# .#.3;1 1E&+$+ '+.%-$#(!$+ K&-1+ =!(! ! #%'+"3! $+ "-1-!( !=(+=(-!$+5 %#.$+ ;1 "-1-!( $# ST +; 6QT '+.%-$#(!$+ ;1! 2+! !=(+,-1!M )*+ =!(! u /F#" U(-+5 V+(!M !"#.'-! # I#(+ PQ6R94
F#J-$+ !+% &!1!.3+% $!% B!.#"!% /SQQ5 8SQ # 64QQQ9 $# +2%#(J!)<#% W;# %#(*+ ;&-"-H!$!% =!(! + !B;%&# $+% 1+$#"+%5 .+ =(#%#.&# &(!2!"3+5 %#(C !&(-M 2;X$+ ;1 "-1-!( (#@#(#.&# !+% 6QT $!% 1!-+(#% +2%#(J!)<#%5 $+ 1#%1+ 1+$+ W;# E @#-&+ #1 V'?#-" # Y(#O /PQQQ94 I!(! !=(+@;.$!1#.&+ %+2(# ! #%'+M "3! $+ "-1-!(5 N'!((+&& # V!'F+.!"$ /PQ6P9 !=(#%#.&!1 ;1! 2+! (#J-%*+ $! "-&#(!&;(! #%&!&X%&-'! '+1 !% $-J#(%!% 1#&+$+"+A-!% =!(! #%&-1!( &!" "-1-!(4
V#$-!.&# #%%# (#=#(&K(-+ -.-'-!"5 =!(! ! '!(!'&#(-H!)*+ $! $-%&(-2;-)*+ GIF5 $#.+&!M%# =+( Nu + .Z1#(+ $# #,'#$#.&#% !"E1 $+ "-1-!( u5 +; %#B!5
Nu =PNi=1I(Xi ≥u)5 +.$# I(Xi ≥u) = { 1 seXi ≥u
0 c.c.
I+( '+.%#A;-. +% J!"+(#% Xi−u≥05 W;# %*+ +% #,'#%%+% !"E1 $+ "-1-!( u5 %#(*+ (#=(#%#.&!$+% =+( Y1, Y2, ..., YNu4
!!"# $%#% &! '"!()"*+",-.! '. /&0%).! .1().#%! !2% &! '"!()"*+",-.! 0"3 #"(. 4&)& % #51"#%6 & '"!()"*+",2% 7.8.)&0"9&'& '. :&).(% ; & <%)#& 4&)&3 #;()"$& 4&)& '"!()"*+",-.! 0"#"(. '%! .1$.!!%! '.=8"'%! 4%) Y1, Y2, ..., YNu>
'"$"%8&0#.8(.6 &! '"!()"*+",-.! 7.8.)&0"9&'&! '. :&).(% 4%'.# !.) '.3 8%#"8&'&! 4.0&! '"!()"*+",-.!? @14%8.8$"&06 :&).(% . A.(&> @6 &"8'&6 !+&! 4&)&#.()"9&,-.! !2% ).4).!.8(&'&! '& !.7+"8(. <%)#&?
@14%8.8$"&0?
W0(y) ={ 0 , y <0 1−e−y , y ≥0 :&).(%6 α >0?
W1,α(y) = { 0 , y <1
1−y−α , y ≥1 A.(&6α <0?
W2,α(y) = {
0 , y <−1 1−(−y)−α ,−1≤y≤0
1 , y >0
B+#&)"9&8'%6 & '"!()"*+",2% 7.8.)&0"9&'& '. :&).(% CD:EF 4%'. ).4).!.83 (&) &! ()G! '"!()"*+",-.! $"(&'&! .# +#& H8"$& <%)#&6 !%* +#&γ34&)&#.()"9&,2%
CIJ&/.93E.#%+0"8 .( &0>6 KLMNF '&'& 4%)?
Wγ(y) = 1−(1 +γy)−1/γ
I%8<%)#. '.!(&$&'% .# B#"(J CKLLKF6 & <&#O0"& '. 0%$&,2% . .!$&0& Wγ,δ,ψ 4%'. !.) %*("'& !+*!("(+"8'%3!. % &)7+#.8(% y4%) y−δ
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