Archive of SID. ! "# $ 1- Value-at-Risk 2- Genetic Algorithms

Transcription

1 !##$!"# / / /! "# $ // : // : &% #*+, -! ) '( & #$%! "! * 89%*:;! /! 3<9*! * #*;( =*!.! 3'4 56,! (VaR) ).$( /! #*EID =F*G! 8FHE( %!! #E 'DC( #!#@A 5A 'B (GA) )9,>.*! 3'*4 #*9K +, VaR H, ) &D $(.! 'B ) & 'J!! #*MFN /! OGA &.'*@EC*( '*(!!/* )* '*L )* * A@ & CK# $( =! 5*A;! *$BP R*(Q '*,& & C*$BP )*9,> /! #*E 3% 'L /#@A 89%:;! RV*( * %*" *A,D W* TU$,! O 89%:;! =! CMS! H(.',!#9K.*! ]*!% *Y9Z*( & C:9*% '*@,( [*" C*G \!*4#* /*,5'L & 3'X 3'*4 5*6,! I!*A -%!/ ^M9_( E4 /! RQZ9( C(A 'B! /#B4 /! #*+, *@B( *!/ ) /;'( & C!E:,Z, 3'(D # `9,.! 'B* #*@A I/& I&D # )9,> 89%:;! /#@A & & ).$(.! ) & &'( =9K +, 5A.G11 OG1:JEL $ '( O#*@A 5A* 'B O5A 'B a_9,! O(VaR) ).$( /! #+, :,- +*.)9,> /#@A 89%:;! C%P =';!d, #N!%" C9$@S 3:Z,! Oc C'@A( 3'QZ,! O9@E 3&K 9! h_khaloozadeh@kntu.ac.ir nasibeh_53@yahoo.com 'AZ( C& 3:Z,! Oc 3&K O9@E '4! C@4E %6Z,! 1- Value-at-Risk 2- Genetic Algorithms

2 &' / / %% ' /. ID * fg*( R*(!%L & C;*( *!/ /;'( $UJ5'L e%, %N& *MJ )* =*!.%4C( K8Fd '@! )!'( /& hn%(.!! 3k & #( /! OC:@', O '(D Rj( C;( $( e!%,! i E A,D #MFN /! #E 3% e%@9( C;( QMFL fg( & %N%( ) e!%,! m l C *MFL )* O *B9L! )* O!/* )* O 6 )# I!% C( q o )* O C* )* O >%*;%@Q )* O CJ%*n )* O(*QMFL) s r O%*N%( * )* ^*M9_( e!%,! =.E 34! '( ) & C:@', '*4# & I!H(.'@9 #N!%(!/ ) e%, /! ) u($( je! tb #$*% & `&* h*n%( C;(!/!', & C u($( O6 ;$ C;*( * EZ*(#* /, i!h! =@XF & 3'4 ) & ''N $;U( *"! * *.! 3'4 ) K3/!',! =%, )@Q #$% vl =*! #*E%P#! 3'4 R'B!/ ) C/! a%b( $( )# VaR!* #( I!'( & C;( E4 & g( \% C$& RQ4# #+, w )* ^M9_(!HN! VaR '(. &C( E# 5A 'B ) 9@E )* *$( =!# #N% CMS! R;.'@EC( ]FN CFE $( )! FJ VaR.'*@EC*( I* 'L ) A@! 5A 'B RE ) VaR #E! =! 3/* =9Z* x@9() C,(/ Y! 9(! & /! C$ O5A 'B ) +, yu* & (%*4C( 56,! 9(! =!# #N% B( #ME &! 3'@D C,(/ 1- Business risk. 2- Market risk. 3- Credit risk. 4- Oerational risk. 5- Legal risk. 6- Technological risk. 7- Political risk. 8- Liquidity risk. 9- Model risk. 1- Gregory, P. C. (ED.) (1959). 11- Time horizon.

3 % 3&D F9n! :,Z, 9(! =! #;z( G&<(# #N% )! *G $( =! #Su" %P# & (% '!%" #;z( ID #9!%" I'4.' C( IZ, CS" yu +, %( C,(/ Y! &! +9,! %( & O {,!&%E {,!& & :/! '@ BL VaR#B( & = 8A( m l Rj( : )@Q #9B;! O %;E,%( & & C_ 3! /#B4 o & {,*!&%E {,*!& &. '*,&C( E# H, (K 9; VaR & CU" VaR.!!%9! 3/ (, ]/%. # %;E,%( & IE 8FHE( I!!K #( T' O5A 'B a_9,! )ue '( O%* ]* & 5A 'B 3/ ]/% K!.! 5A 'B +9,! %( % I!H(!*! /*8FHE*( =*! I!%* C( '4 CS" S%d"!! & }_Z(!'*( #*E=*! =FG! 56,! O5A 'B ) &D $( ) IE #@FE q & H9%E*( * =*;&!.*! 3'4 =$ 3/ =:,(! C@$( & }_Z( r!* & '*,! A@Z* )* =:,( '( e%, =! =% & -*! * ID /! {*.'*, E a*_9,!! {,*!& O)* *K3/!'*,!.[q& Oo] '*4 5*6,! */ & E {,!& =:,( /#@A s RV*( '* /! )* *K3/!'*,! *: 8*A( $( '@~ VaR O[r] ]N( #*69,.*! 3'*4 #( C B( & )% S%d" ;# /#@A /! * RQZ*( t! VaR /#@A #E=! %N& '@EC( I #( =! ) VaR =:,( 5A 'B =9 C;&! ) /#@A : e!%,! w O)* K3/!',! $( : OCP /!.! =QF( & Y RJ #;z( 1- Confidence level. 2- Variance-Covariance. 3- Historical simulation. 4- Mont Carlo. 5- Varcholova, T., Rimarcik. M., (1952). 6- Return. 7- Markowitz, H. M. 8- Tobin, J., (1958). 9- Gaivoronski, A., Pflug, G., (21). 1- Feasible.

4 &' / / %( * #*@A 5A* 'B C# /, =!@.'@9, VaR! CB@( =Z,N -%( tu(e OY%( /#B4 ) h;k CMFL Rn 3! =9 &VaR $(.! 5A* 'B* & 3'*4 h*e 5A 'B /#@A VaR 3'! #;( =!!! =F*G #E%P#! 3'(D # )9,> 89%:;! /! 3<9! #@A.%*4C*( 3&D* H, VaR & 'J O#MSn 5A 'B 3/ 8FHE( I% { *.%4C( #V!! & =B #;z( /, %( I%W%( i_!'9! #(!! RV*( /*#*@A %*J C4& I!%@L# )9,> 89%:;! l i_ /! RSn `9, O"D i_.%4c( 3! ƒ4 t!d9_( O T$9(k & T$9( ]@*S#* Y*M$9( E*4 3/!&#* %( C,(/ /! 3<9! /#B4 C:,%*:~ #;z*(#* %( I%W%( /! 3<9! I!! -%!/ ^M9_( '*S.*! 3'4 3&D )9,> /#@A 89%:;! & #@A 5A 'B a_9,! ) & 'J #E %4C( =$!#,%K# 5A 'B 8A 3/!& /! ) 3/*!! '* 3'*(D *# 5A 'B 'J =! %N& t@fg O%4 3&D.'4 8FHE( VaR 2( $ / * 5A* 'B /#@A OY =! CMS! T' '4 #9<K #E %PIF VaR =:,( #@A 5A 'B #E%P#! VaR ) &D $( %N& 'B O) I!H(! %BJ RJ W 'n =9K +, C@$.'D #.%4 =$ 8FHE( 3/ +, %( 5A *+,! i =1,...,n #(* &'*( #*L%F6( )* c% #;z( =B! c!&! & 5A* OC;*( #(* e%*, R(4 ',!% C( j( I!%@L# #E 8KC(.'@EC( }_Z(! x 5A 'B #( =! $G&.'4 #GJ a'* & * Y9Z*( OC:9% \!4 # /,! '@F,%,J Cd %69N # #E 89%:;! =!., YMU( #@A #U, # I'!

5 % x = (x 1,...,x n ) %*<!HN! %N%( $UJ5'L & ) e%, v ) %9E.'@@EC( ^S%!... & 5A FJ O%, $UJ5'L Rj( C;(!/ v = (v1,...,vn) * v )* * %9E* 3'4 3! (! x 5A 'B /! &!'( & C*U"k C$* *! =*QF( CME ;n #E %4C( 3! IZ, ( x,v ) 'B $G&# B, tw%f$( C;&.'4 v & x /! #9K C$ C9n 3'X '@! & CME f! I!% C( & 3% )Q< RJ #E C@$( =!#!!'N 5A :&D #!HN! ) ) f! /!! ( x,v ) = ( x,v ) = n i= 1 n i= 1 x i ( xi,v ) i i( v ) ().'4 CU" x # B,! =QF( C9n & ( ) O'4 CU" ) %9E H, & 5A 'B $G&# B, ',!% C( C K & C,*Q $*G& & *$J%( * C 5A R(4 5A 'B #E!%( Rj( '*@!%" 5A* *FJ IF ) %9E %S=!! #$;U( e&4 #*9K *+, *;n =*! H, 3'4 56,! #;( =! #E!/#B4.% :C@$ O! 3'4 ( x,v ) = n i= 1 x i v i IZ*, v * t = I*(/ 5%M$( & }_Z( ) %9E ( K!.*! t = 1 I*(/ 5A* 'B! =$ T' #Mn( =! %4 3! ]*.%*4 #9K +, H, #9< '@~ /& '@~ ',!% C( C,(/ Y! =! #9B;! (l) 1- Risk factor. 2- Searable.

6 &' / / %) E. 5%M$( I!% C(! t = 1 I(/ ) %9E F9n! C;:~ & /!! *A,D * & *K *+, ID!*! CS" C;:~ ] I!% C( ]J!& C;:~ ] t = 1 I(/ v 9 #Q %S.&D # #94K 3! t = 1 I*(/ (x, v) 5A 'B!'( 9.%4 I f ( v ) 5%M$( F9n! :%4C( I / %S# (x, v) 9Z( F9n! C;:~ ] O P { ( x,v ) } = f ( v ) ( x, y ) = ( x, y )dy )* * %9E* ]/% & ( x,v ) = y f ( v )dv ( x,v ) 5A 'B!'( ] )FE# #E :! I RJ / %S# #E C9J& ts%d_(.! CMQZ( E ]/% ] =! IE!' RFL O ( x,v ) 5A* 'B*!'*( ] &! I%K,%K 5A 'S &n 5A 'B =*! R*n!* C*<M9_( * /#B4 & 9(! &.! CU"k ]*!% I!%* C( H, 9(!, & /! 3<9!.! 3'4 A@Z #;z( #*E %*PI*F i &ˆ* =! O/ =F_! 8A # %( F9n! C;:~ l R,E /!%F 89%:;! /! #94K 3! =94! O'4 '!%" 34!!'6(.! 3'4 3<9!! (x).*e R*FL */ RQ*4#* I!% C( O x 5A 'B VaR ^$! O#*9K *+, t = 1 I(/ (5A 'B =:,() x 5A 'B '(!!'( :4%, I!% C( %S=! ( x ) = Ev ( x,v ) = ( x,v ) f ( v ) dv #9K +,! %4C(. w/ss w/so tw%f$( #E c <1I@FP! yu 1- Probability density function. 2- Gaivoronski, A., Pflug, G., (21). 3- Kernel smoothing algorithm. (m) (o)

7 % P ( * x, y )dy = c #*U, /! (x, y).8@ec( Rn # B,! / #;$( & C*@@( */ yu* #E '@EC( I () #U! */ RQ*4 3%*_, %4 yu #E!!#U, =!@.%4C( c! */ yu* O*! F9n! C;:~ ]/% ] # %( C@@( I%~.! c!.! 1 c! 3%" %4 yu!; & 3% )! C@@( 8 /. 7&6 &*( c F9n!# t = 1I(/ x 5A 'B /! &!'( : BL# ( x ) =* Tu9*"! VaR h* =*!#*.% '!%" = ( x ) /! 9Z.! ( x )& ( x 5A 'B /! =:,() VaR = ( x ) ( x ) (q) x 5A* 'B*!'( Oc F9n!# #E! C,!H( =9Z VaR n 5A* 'B* *N!'*( *J&! hmk!.%4 9FE +9,! %(!'( /! ',!% C( :C@$.&C( E# VaR ^$ ID '(! I!%@L# VaR = ( x,v ) ( x ) (r)! VaR '*J %*N& * #@A 5A 'B a_9,! #;z( %S I!% C( I%@E! *+9,! %*( %* #*E '*!#*,%K#*! x 5A 'B :E I / RQ4# ()

8 &' / / % max m i= 1 P ( x,v ) f ( v ) dv ( x,v ) f ( v )dv x i = 1, ( x,v ) Cd* x f ( v )dv = y i ( x ) V dy = c.! :C@$.%4 8FHE( (s) :%4 3&D / VaR & 'J #E%P# (w) # B, / #;$( Rn ( x ) () ID #E /*#*@A RV*( C(%FL #9# YM$9( n #;z( =! () & Ow]]N!(# RV( /! #9 =! S%d" & 9Z #$;U(!.! {,!& =:,( 5A 'B a_9,! #;z( Tu" #,<z9(.'@e #$N!( [ #*L%F6( =9* CM*S! RQZ*(.! RQZ( #;z( =! Rn OH9&%E( =QF( #L%F6( =!!%( = 3 C9n I%~! () (w) IQ(! m l )*FE#*! #;z*( =*! O#;( =!.'4 #9_K C9n a'( k! C*U"k & a'*( *k RV*( Rn '@F,!%!H! #E )9,> 89%:;!.8@EC( Rn! 5, $<:5; /9 o.','*4 ƒu( '@;% IN \% sw # =;&! )9,> 89%:;! & C*$BP )*9,> /! #*E '@9 'L /#@A 89%:;! O 89%:;! =! TU$,! O 89%:;! =! CMS! H(. ',!#9K 5A;! $BP R(Q ',& 1- Stochastic otimization. 2- Feasible set. 3- Non-convex. 4- Disjoint. 5- John Holland. 6- Coley, D. A., (2).

9 %* [*" C*G \!*4# /,5'L & 3'X RV( %" A,D W * #E #;&! $FN ) 89%:;! =!.! Y9Z( & C:9% '@,( /*#*@A #;z*( R*n!* A@Z a!%n ) I!%@L# $FN =! % L Cd %S# tw%f$( -'n =!.',%4C( e&4 -'n /! C &K! *:MFL #* /! { *.'*,!3'*@E! %69N & ',%4C( a_9,! l *, R*(Q C:,%*:~ * B*4 -*! #E) ian & O a_9,! F*#* C;!%*9( R, %P $FN!'! (',!3'4 a_9,! C$J!&.%4C( 3<9! O#@A #U, C!:F CQ>%*;% * 89* C*$BP a*_9,!#z(! Z Oa_9,! :MFL & 3'*4 T*n '*,! * ^$*G *QMFL #E!!.'@EC( FL! $FN#! 9Z {,4 O! 9Z A,D CK',/! & ',! 9A QMFL #E!! * #dd" & LuP! #E ' C(! IQ(! =! O :MFL.4! '@!%" *:MFL & ',%*4 #*;B( $FN <N = ORj( ';% #Z( C4&# C9f!&.%4 Š<n $FN e%@ #E! ID! H, ian \*% '* #*E! %*N& C &<9( 9(! )9,> 89%:;! FL!! m Oa_9,! :MFL e%, O CK',/! ] O$FN L!!'$.%4 =$ ƒ!p * 9(!* =*! h*@( a*_9,!.'@9*!%*( =*! #MFN /! & ian o #@A, IQ, K &! #@A #U, F# 89%:;! =! C!:F '* #;z*( [*" e%,# #N% %(.! F! HVn O C$G%(.%F, =$ & 8+@! 9(! =! 1- Selection. 2- Crossover. 3- Mutation. 4- Fitness function. 5- Global otimal oint. 6- Local otimal oint.

10 &' / / %%+ > = /9/. =*! *+, %*( 5A* 'B* /*#*@A #;z( O'4 34! #E %PIF =*!.%4C( I &'( %N& /#@A #;z( ) %S# i &ˆ %4 8FHE( (s) #U! #E! %,# { ( I&D # T' #;z( I!%*9(! T'* ]* =!*@.'@*4!*J H*, () & (w) %*J #EC;n % *L *.%*F, ^$ O%J a%mu( ( Tu9"! /! C,!% e%f6( %S# '*!%" 9Z* CK'*,/! O'*4 )~%E ID T' ]!'( #E $FN.% '!%" 9Z C;!%9( R, %P % L ID {,4 =!@ & 4! * =QF( & #n, &'( O'J %N& /#@A #;z( ) %*N& * /*#@A RV( E! )9,> 89%:;!.'@@EC( ^$! '*J *, 5/W C9@* & Tu" #E!~ O'@9 h@( 'J. '@4!!! Y9Z( & C:9% '@,( CS" CG \!4 #*FN ]*!% /! 3<9*! O'*J * /*#*@A RV( & = %F$( C9;*@ ]*!% )*FE#* '*J * /*#@A #;z( & =!.! (C9;@) I!%* C*(! CK',/! ] =!@.%4C( R'B 'J I&' /#@A #;z(# :4%, / RQ4# f (x); xfeasible region F(x) = (l) f (x) + enalty(x); xfeasible region * & =*! #*ME.K +, I!% C(! #FN ]!% /! C<M9_( e!%,! #FN ]!% O#;( =!.%4C( #9K +,!#FN O'J L5'L! * #MS /! CN&/ I!% O'J L5'L! #E ',%4C( 8+@ h =!# #*FN I!H*( * 8*@EC*( ag CBG & #9K +,! &'( /( CK'*,/! I!H( O'@Q, L! &'( #E % L =!@.'D # * #E %,# #9K +, 9(! &'( hg. '!%" i E! 1- Coello, C., (21).

11 %% '*J Gn #;z(.%4 FL! _& YJ &'( OAM, =9 () & (w) \!& IF. /9/ /! '*S) '@EC( }_Z(! x #E!! %S# $FN % L a*_9,! Cd* %*S#*! &! R*, * #*;&! *$FN.(C_9,! 5A.! F! HVn )9,> 89%:;! $FN L!!'$ =$.8@EC( & % '!%" ]& %69N# J 89%:;! O'4 9Z L!!'$ #~!*N! '*( O *L!!'*$ i!h*! CP /! (!.'4 '!%" RSn 9A `9, 5*6,! * & * 9(!'*$ # #N% #;z( =!./C( C,W%P! 89%:;!.! 3'4 #9K +, <, ow L!!'$ U"& C$ ='@~ 345 ;+"B /9/9 ='*;!& I!%*@L#* #$(N L! /! ) 5!'E #E '@EC( =$ a_9,! :MFL I!%*@L#* *;& *~ *:MFL /! #*;( =!.',%4 #9K +, '$ R, CK'*,/! #*E % *L * #*E 5%*A<( =*!#.! 3'4 3<9! a_9,! :MFL a*_9,! F9n! :MFL =!.4! '!%" 9Z jq {,4! 9Z & =*9A =@~ 8.! % L ID CK',/! h@9( O';!& I!%@L# % L ).'@@EC(!' 3! '$ R,# I&' R, (#B_, L!) #$(N % L ;+"B /9/C *LuP! #*E '* C*(! I*Q(! =*! O* *:MFL '4 34! #E %PIF /! #*;( =*!.',%*4 #;B( $FN <N = ORj( ';% #Z( C4&# l.! 3'4 3<9! I!',/ 6! & % L & he! Q9! :MFL '*;!& #94 { & %4C( a_9,! Cd %P# #U, ) & =! 1- Initial oulation. 2- Roulette. 3- Heuristic.

12 &' / / %%% '*'N * * #*E I!'*,/ #94 ',%4C( he 8 #U, =!.',%4 ';% O'@9 #;z( EA ;+"B /9/D i*, *:MFL =*!.&C*( *E# $FN e%@ 6!! ian :MFL C*M" i*an *F9n! #*E C %*S.! %69N 9K C! /!.'*@EC*( R( Œ( Cd %69N F# 89%:;! O%4 a_9,! H CM( #@A, #M 89%:;! O9(! =! I% )~%E %S : TP 3<9*! %*( i*an :MFL.'C( -&/ %M# tnuus! & 39! 5!# & *U" & C$ ID F9n! #E! "!%@Q ]/% ian :MFL #;( =! 3'4 #9K +, w/wrlso OC;!%9( R, %P $FN 9 3' Z(.! $6 /C *+, ^*M9_( 8A* * C(A* 'B Y =! /#B4 56,! I!H*( ',%4 a_9,! ^M9_( ]@S /! A9E4 =! 3'4 C$.! 3'4 #9K l /! *! *BL A9E*4 =!# %( LuP!. ' i E A,D = C:9BF * rw * /! #94K I(/ (closed rice) FJ C,(/ 3! m.'@9 %N%( [] ]N( 3! =!.rl 1- Uniform distribution. /! 3<9*! * RV( Rn [%d_( #E GA!H! #B$N /! 3<9! & Matlab 7!H! 5, \% A/ #B4.! 3'4 56,!! )9,> 89%:;! tu(*e RQ*4 #* GA /*#*@A 89%:;! & VaR #+, YBP 3'4 E & 56,! AN E4 =! l I!*! -%!/ 3'4 #9 A9E4 5F 3/!',!# A9E4!'$ CME ;n!; & ',!3'4 a_9,! 3!%_; /! 8A* O'@*4 #9*4!',! #*UQ C:9*BF 3'*4 a*_9,! A(A #Q,D P_ H, &! i!h! RJ )* * 5A* & A9E*4 /! *! =*QF( #*@A a_9,! C$J!& ;n #9B;!..',!3'4 a_9,! C<M9_( ]@S 3'*4 #*V!! & y*g% *AN *S &! CME tu(e #;( 3'4 E & & %S [" $@S.',!3'4 a_9,! E;! c% 5A Y 4- htt://

13 %%! E*4 *A~ C,*(/ * * 3! RQ*4 & A9E4 =! 5, &'N.' C( IZ, #,%F, I!%@L# $6 i C!k W%d(!K#(!K#( CF4 W%d( I%H%M & %! " C!k W%d( C%~ W%d( K & ) D OIF C9$@S!#94 '@~ E4 $UJ " & &%" C9<, 3&D )9u & )9W AZA #$% - %E%4!K#( /B;!!K#( I!D CF4&9 )9Q;! - =~ =~ $@S & ZE I!! B I IF CM( ),!K#( I!A <, /B;! )9W l m o q r s w

14 &' / / %%( $O> M% M N "6-5L $<@*6 $ / 7&6 P - I 5A* 'B* %N%( ^M9_( 5A# %( x A,/& IE!' T' #@Z* t = 1 I*(/ 5A* 'B* /! RSn % #E!!#,%K# O.&<( 5*K =*;&!.%4 3&D H, VaR $( +, %( ) 'n #E%P# %4 *9 =*$ O'*4 ƒ*u( i*_ ID '*@%*( #*E #;z( =! Rn!.! t = 1 I(/ ('@9 5A FJ 6@! #E) ) %9E * 3! =9*4! * t = 1 I*(/ f ( v )*F9n! C;*:~ ] I&D # C;*:~ ]*!% l RQ4.! 3'4 3<9! R,E /!%F 89%:;! /! #94K 3*k & 3'*@D #9< ) O3'@D /& ) T$( ',!% C( ) 'L &! C D I(/ C@ i T$( 1=t.'4 2- Kernel smoothing algorithm

15 %%. ' C( IZ,! RQ4 E4 A~! 3'(D # $<@*6 $ 7* $"< :5; F"5# ;% = /9 7&6 P - I $O> M% M N "6-5L %*( /! t = 1 I*(/!'*(! 3'4. (x) = x 5A* 'B* CG '(! ( x,v ) = n i= 1 x i v i f ( v ) I'(D # #E #Q6,D /!.%4C( =$.'DC( # / %S# 12 1 i= 1 j= 1 x v f (v) v i i j j (o ) ( x ) E*4 I*( /! j( I!%@$ A,D F9n! C;:~ ] & C,(/ # %( RQ4 & E4 A~ =!.',!3'4 8 & a_9,! 'B %N%(

16 &' / / %%).! /, ( x, y )!'(# ID#B(! #E! ( x ) '$ %A6( x i *( /! #;&!. ) #E 3'4 RFL h =!# ( x, y ) =$! 5A* 'B* *F9n! C;*:~ ] n ( x, y ) I%~.8@EC( e&4 =F_ C;:~ ] =! R,E /!%F & )FE 8 /! t = 1 I(/ *+, *$FN * * & )*9,> 89%:;!!Q E =!.%4C( 3/!* C! B,# ' ( x, y ) I'4 5%M$( n.%4c( 56,! O3'4 #9K =9*K +,.%4 'L#B( RJ (w ) 'J 84 ( x ) =$ #*U, /! #E E B$ %S =!#! ( ) I!% C( w/so I@FP! yu.%4 w/so &( ( x, y ) C@@( / yu '$# I!%* C*( W* B$ { O'DC( # ( x, y )!Q '4 #9<K #E %PIF C;*:~ ]*!* Cd*_Z( %( I%~.&D #!Q H,! *, '*L *( H*, 89%:;!!Q &, 5A 'B =$ &#B(!!:9,! =! I!% CF, CK 'DC( # ID! ^M9_( /! R*; =*F #* O*! w/so *! ID!'( 5A 'B /!!'( #~ #E E.! 3'4 3<9!!#,/& 'L :;!:9,! *FL! & * x i!* Cd ( =$ VaR! h@( 'L!'( (q) #U! /! O ( x ) & ( x )#B(! 3'4 3! yg% Q9! & * 5A* 'B ) %o j( I!%@L# #E! 3'4..%4C(#B( VaR '*n I!%@L# ( =! =:,( A,.'4 CB@(!'( Cd =!# '4!J 'J =! K! ]J!&.! loow 'L #E! 3'4 #9K +, 3& I* * C*ME *;n ) '*$ /& I w/so F9n!# #E! C@$(.E 8!%_, G I(% loow /! 9Z ('$ C,(/ #*Mn I'*4 R*FQ!* /*, %*( %*A6( *( 5*F h* =!# /*, %*( * :MFL & CK',/! ].'@DC( # x i C,/& h!g#b( * /*#B*4 /! R*Sn `*9,.','*4 I RBJ i_ H, )9,> 89%:;! ]*!* m RQ*4.*! 3'*4 3&D */ * RQ*4 )*9,> 89%:;!

17 %% %*P! *$FN *!*!!* ]* =! =:,( & % L =9A CK',/! RQ4.! 3'4!:F <S# C%" L #E.' C( IZ, C;!%9( R, /! )* * 8A & #@A I/& 'S :, #E 3'(D # x i ( =9A o.' C( IZ,!! v %( 8A 3/!& < > = ; - RB 5 > = GQ /C 7&6 (7.TTT) 5 $<7 >. K $"B RB 5 /D7&6 (< I & a*_9,! Cd* %*S#* C(A* 'B 3'(D # #69, I%(/D! #*E %*PI*F.*! 3'*4#B( ID! VaR!'( & 5A 'B /! t = 1 I*(/ 5A* 'B* /! &! *;n %*4C( 3' Z( / &'N

18 &' / / %% *! 9Z //! &!'( #@A 5A 'B /! #E! /!'( &! 3'4 Œ, )# %( 'J ;n =! #E 4! #N% C9 C;& *+, %*(!'*( /! #*E 8*! ( /) 9Z* CM" ) ;n =! %S Cd RQ4# 'B a_9,! #E 5& %@.! 9Z ( ) /! C*9n ( / ) 5A* 'B /! C;&!!J ) 'J O! #9 *F9n! C@$! 9FE H, / C@$ %@ =! )!'( &*<9( 5A* 'S : 'B a_9,!.e 8!%" G / *j( I!%*@L#*.%4C( 'L 9FE 5A 'B /! H, RQ4 #@A 'B uj(!'(# (/ ) ID #@A!'( /!! ( ) 3F4 5A 'S K! /!! (/*B;! )9*W) 5A* '*S.%*L & 3! i *E (/ ) 'B* /!!* ( / ) *( 8* i!h*! (/ )# (/ ) x*@9(!'( /! 5A 'B /! #E 'DC( # VaR! ( ) & 5A h@( #E! +, %( 'n /! 9Z VaR!'( C;&! 9Z #@A.,

19 %%* K - $6 GH4I /F-A <@ # < :< K ) (5P Q / * / / / / / / / / * / / * / :< K (PH ) / / * / / / / / / * / / / / :< K (.PH ) / / * / * / * / / / / * / * /* / * / * : < K ( ) / * / / / / / / / / * / / * C!k W%d(!K#(!K#( W%d( CF4 & %! " I%H%M C!k W%d( C%~ W%d( & ) D OIF K '@~ E4 - %E%4!K#( AZA #$%!K#( /B;! CF4&9 I!D )9Q;! - $@S & ZE =~ =~ I!! B I IF!K#( CM( ), " & &%" $UJ C9<, 3&D & )9W )9u I!A <, /B;! # - $ $<- $ - VaR ' /WF-A 5P Q - PH 345 A,/& a_9,! & )9,> 89%:;! ( ;n)cd 'B ( ;n)cd 'B #@A 'B VaR!'( / / 5A 'B /! / / / / i l m o q r s w

20 &' / / %+ $>Y5 /X #* * #*E 5A 'B /#@A )ue '( Tu" #;( =! 'B* &D ) $( I!%@L# (VaR) )!'( /! '@9 {,!& ID Rn &! a'( k!%( = 3 C9n #;z( =!.'4 3<9! 5A #E )9,> 89%:;! -! ''N &.! RQZ( CMM & * & '*4 #*V!! #;z( =! Rn! '@9,!% a'( k RV( Rn IZ*, `*9, #*E %PIF.'4 3! IZ, ID C!E 3'4 56,! /#B4 *!*K#(* #@A I!H( I!% C( =$( ) 'n )! '@ C(.&D #! 5A 'B /! 8A 1- Gregory, P. C. (ED.) (1959), Proceeding of the self adative flight control systems symosium,wright-patterson AFB. OH. WADC Technical Reort Varcholova, T., Rimarcik. M., (1952), Value at Risk Methods and Models and their Alication, 3- Markowitz, H. M, Portfolio selection, Journal of Finance, 7(1), Tobin, J., (1958), Liquidity reference as behavior towards risk, Review of Economic Studies, 25, Alexander, C. O, Leigh, C. T., (1997), "On the covariance matrices used in Value at Risk models," The Journal of Derivatives, Pholie, J., (1996), Risk 2: Measuring the risk in value at risk, Financial Analysts Journal, Singh, M. K., (1997), Value at risk using rincial comonents analysis, The Journal of Portfolio Management, Gaivoronski, A., Pflug, G., (25), Value at Risk in ortfolio otimization: roerties and comutational aroach, Journal of Risk, Vol. 7, No. 2,. 1-31, Winter Gaivoronski, A., Pflug, G., (21), Finding otimal ortfolios with constraints on Value at Risk, Proceedings of the III Stockholm Seminar on Risk Behavior and Risk Management. 1- Demster, M., (198), Stochastic Programming, Academic Press. 11- Ermoliev, Y., Wets, R. G., (1988), Numerical Techniques for Stochastic Otimization, Sringer Verlag, Berlin. 12- Gaivoronski, A. A., Messina, E., Sciomachen, A., (1994), A statistical

21 % generalized rogramming algorithm for stochastic otimization roblems, Annals of Oerations Research. 13- Coley, D. A., (2), An Introduction to Genetic Algorithms for Scientists and Engineers, World Scientific. 14- Coello, C., (21), Theoretical and numerical constraint handling techniques used with evolutionary algorithms: A survey of the state of the art, Comuter Methods in Alied Mechanics and Engineering. 15- Homaifar, A., Qi, C., Lai, S., (1994), Constrained Otimization via Genetic Algorithm, Simulation,62:4, htt://