Optimization Method for Interval Portfolio Selection Based on. Satisfaction Index of Interval inequality Relation

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1 Opmzao Meho fo Ieval Pofolo Seleco Base o Sasfaco Ie of Ieval equal Relao Yuchol Jog a a Cee of Naual Scece Uves of Sceces Pogag DPR Koea E-mal: ucholog@ahoo.com Absac: I hs pape e cose a eval pofolo seleco poblem h ucea eus a ouce a clusve cocep of sasfaco e fo eval equal elao. Base o he sasfaco e e popose a appoach o euce he eval pogammg poblem h ucea obecve a cosas o a saa lea pogammg poblem h o paamees. We shoe b smulao epeme ha ou meho s capable of helpg vesos o f effce pofolos accog o he pefeece. Keos: Ieval pofolo seleco sasfaco e sem-absolue evao s paamec lea pogammg. Iouco he heo of opmal pofolos seleco as evelope b Ha Maoz he 95's. Hs o [] fomalze he vesfcao pcple pofolo seleco a eae hm he 99 Nobel Pze fo Ecoomcs. Cose a veso ho has a cea amou of moe o be vese a umbe of ffee secues (socs bos ec.) h ucea eus. he pofolo veco mus sasf he fs cosa... hee ma o ma o be aoal feasbl cosas. A feasble pofolo s calle effce f has he mamal epece eu amog all pofolos h he same vaace o aleavel f has he mmum s (vaace) amog all pofolos ha have a leas a cea epece eu. he colleco of effce pofolos foms he effce foe of he pofolo uvese. Maoz' pofolo opmzao poblem also calle he mea-vaace opmzao (MVO) poblem ca be fomulae hee ffee bu equvale as. Oe fomulao (MVO) esuls he poblem of fg a mmum vaace pofolo of he secues o ha els a leas a age value R of epece eu. he he seco cosa caes ha he epece eu s o less ha he age value a as e scusse above he obecve fuco coespos o he oal s of he pofolo. Noegav cosas o... ae ouce o ule ou sho sales (sellg a secu ha ou o o have). As a aleave o he poblem MVO e ma a

2 cose he poblem (MVO) ha s o mamze he epece eu of a pofolo hle lmg he vaace of s eu. aoall has bee assume ha he sbuo fucos of possbl eus ae o hle solvg pofolo seleco moels. Hoeve e secues a classes of asses have emege ece mes a s o alas possble fo a veso o specf hem. I some cases fo sace hsocal aa of socs ae o avalable. I such cases he ucea eus of asses ma be eeme as eval umbes b usg epes olege. I hs pape e popose a MVO-le eval sem-absolue evao moel fo pofolo seleco hee he epece eus of secues ae eae as eval umbes. Base o he cocep of sasfaco e of eval equal elao e cove he eval sem-absolue evao pofolo seleco poblem o o paamec lea pogammg poblems. hs pape s ogaze as follos. I Seco. e gve some oaos fo eval umbes a befl ouce some eval ahmecs. A oe of elaos ove evals s ouce. he coceps of sasfaco egee of eval equal elaos ae gve. Base o hs cocep a appoach o compae eval umbes s popose. I Seco. a appoach s pesee fo esmag he evals of aes of eus of secues. I Seco.3 a eval absolue evao moel fo pofolo seleco s popose. Accog o he appoach popose Seco. hch coces abou compag eval umbes he eval pofolo seleco poblem s covee o a paamec lea pogammg poblem h o paamees. I Seco 3 a eample s gve o llusae ou appoach. A fe coclug emas ae fall gve Seco 4.. Lea Pogammg Moel h Ieval Coeffces.. Ieval umbe a eval equal Defo.([]): Le { } be a ba opeao o R. If a a b ae o close evals he a b { : a b} efes a ba opeao o he se of all he close evals. I he case of vso s alas assume ha s o b. he opeaos o evals use hs pape ae as follos: fo a o eval umbes a [ a a b [ b b ] a b [ a b a b ] a b [ a b a b] a ± [a ± a ± ]

3 [ a a [ a [ a < hee s a eal umbe. A eval umbe ca be vee as a specal fuzz umbe hose membeshp fuco aes value ove he eval a ahee else. Fo a eval umbe a [ a he mea ( a) m a h ( a) s efe b m ( a) ( a a) / a ( a) ( a a) / especvel. he hee opeaos of evals ae equvale o he opeaos of ao subaco a scala mulplcao of fuzz umbes va he eeso pcple. Ishbuch a aaa suggese a oe elao beee o evals as follos [6]. Defo.: If evals a [ a a b [ b b ] ae o evals he oe elao beee a a b s efe as a b f a ol f a b a a<b f a ol f a b a a b (.) a b (.) Fo escbg he eval equal elao eal he follog hee coceps ee ouce [3]: Defo.3: Fo a o eval umbes a [ a a b [ b b ] hee s a eval equal elao a p b beee he o eval umbes a a b f a ol f m(a) m(b). Fuhemoe f s opmsc sasfaco; f a a b s pessmsc sasfaco. a b e sa he eval equal elao a p b beee a a b a > b e sa he eval equal elao a p b beee Defo.4: Fo a o eval umbes a [ a a b [ b b ] f he eval equal elao beee hem s pessmsc sasfaco he pessmsc sasfaco e of he eval equal elao a p b ca be efe as b a PSD( a p b). (.3) ( a) ( b) Defo.5: Fo a o eval umbes a [ a a b [ b b ] f he eval equal elao beee hem s opmsc sasfaco he opmsc sasfaco e of he eval equal elao a p b ca be efe as b a OSD( a p b). (.4) ( a) ( b) Rema.: I s eas o see ha ( a b) PSD( a p b) f a ol f m (a) m (b) a ( a b) f PSD( a p b). Sce PSD p f a ol f m (a) m (b) OSD p f a ol b > a mples ha hee ma be some possbl fo b o be geae ha a ca o be sa ha he efo. a.3 coa all possbles fo eval equal o hol. heefoe e ouce a clusve cocep of eval equal elao a sasfaco e. Defo.6: Fo a o eval umbes a [ a a b [ b b ] hee s a eval 3

4 equal elao ap b beee he o eval umbes a a b f a ol f he sasfaco e of he eval equal elao ap b s efe as b a SD( ap b) ma. (.5) a a b b Fom (.5) follos ha fo eve ( ) ( a a b b) α ( a b) α b aα.e. ( α ) b αb( α) a αa b a. Bu OSD( ap b) f a ol f b a. Rema.: he possbl egee of b > a. SD p f a ol f a ( a b) ap b [7] as efe b b a PD( ap b) m ma. a a b b SD p f a ol f Accog o efos of he pessmsc he opmsc sasfaco ces a possbl egee e ca see ha he age of he pessmsc sasfaco e a possbl egee ca be [ ) a he age of he opmsc sasfaco e a he sasfaco e ca be [ ). Ou popose sasfaco e s moe geeal cocep ha he pessmsc opmsc sasfaco ces of [3] a possbl egee of [7]. hs fac s llusae b follog eample. Suppose ha possble elaos beee eval umbes a a b ae such as follog fgues: he quauple umbes cossg of coespog opmsc a pessmsc sasfaco ces possbl egee a sasfaco e ae (a) (.5) (b) ( ) (c) (.5.5) () (.5.5) (e) (.5.5) (f) ( ) a (g) ( ) especvel. Obvousl ou sasfaco e epeses moe pecsel he eval equal elaos ha ohes. he lage value of he sasfaco e s he bee s sasfe he eval equal elao.. he Epece Reu Ievals of Secues I s ell o ha fuue eus of secues cao be accuael pece a 4

5 emegg secues mae. aoall eseaches cose he ahmec mea of hsocal eus as he epece eu of he secu. So he epece eu of he secu s a csp value hs a. Hoeve fo hs echque o ma poblems ee o be solve [3]: () If he me hozo of he hsocal aa of a secu s ve log he fluece of he eale hsocal aa s he same as ha of he ece aa. Hoeve ece aa of a secu mos ofe cae ha he pefomace of a copoao s moe mpoa ece aa ha he eale hsocal aa. () If he hsocal aa of a secu ae o eough oe cao accuael esmae he sascal paamees ue o aa scac. Coseg hese o poblems pehaps s a goo ea o cose he epece eu of a secu as a eval umbe ahe ha a csp value base o he ahmec mea of hsocal aa. Ivesos ma mae use of a copoao s facal epos a he secu s hsocal aa o eeme he epece eu eval s age. o eeme he age of chage epece eus of secues e ll cose he follog hee facos as [3]: () Ahmec mea: Alhough ahmec meas of eus of secues shoul o be epesse as epece eus ecl he ae a goo appomao. Deoe he ahmec mea eu faco b a hch ca be calculae h hsocal aa. () Hsocal eu eec: If ece eus of a secu have bee ceasg e ca beleve ha he epece eu of he secu s geae ha he ahmec mea base o hsocal aa. Hoeve f ece eus of a secu have bee eclg e ca assume ha he epece eu of he secu s smalle ha he ahmec mea base o hsocal aa. Deoe he hsocal eu eec faco b h hch eflecs he eec of he eu o he secu. We ca use he ahmec mea of ece eus as h. (3) Foecas of fuue eus of a secu: he h faco fluecg he epece eu of a secu s s esmae fuue eus. Deoe he foecas eu faco b f. Esmao of f eques some foecass base o he facal epos a epes vual epeeces. Base o he above hee facos e ca eve loe a uppe lms of he epece eu of he secu. We ca pu he mmum of he hee facos a h a f as he loe lm of he epece eu hle e ca pu he mamum values of he hee facos a h a f as he uppe lm of he epece eu of he secu..3 he Ieval Pogammg Moels fo Pofolo Seleco Assume ha a veso as o allocae hs ealh amog s asses offeg aom 5

6 aes of eus a a s-fee asse offeg a fe ae of eu. We ouce some oaos as follos. ~ : he epece ae of eu eval of s asse ( ); : he ae of eu of s-fee asse ; : he popoo of he oal vesme evoe o s asse ( ) o s-fee asse ; : he popoo of he s asse ( ) o s-fee asse oe b he veso; : he hsocal ae of eu of s asse ( ) ( ); : he ae of asaco coss fo he asse ( ); u : he uppe bou of popoo of he oal vesme evoe o s asse ( ) o s-fee asse. We use a V shape fuco o epess he asaco coss so he asaco coss of he asse ( ) ca be eoe b C ( ) (.6) So he oal asaco coss of he pofolo ( ) ca be eoe b Deoe ( ) C( ) C (.7) a (.9) he ucea epece eu of he s asse ( ) ca be epesee as he follog eval umbe: ~ [ ] [m{ a h f } ma{ a h f }] (.) hee a s he ahmec mea faco of s asse h s he hsocal eu eec faco of s asse a f s he foecas eu faco of s asse. he ca be eve b usg he above meho. So he epece eu eval of pofolo ( ) he fuue ca be epesee as ˆ ~ ( ) (.) Afe emovg he asaco coss he e epece eu eval of pofolo ( ) ca be epesee as 6

7 ~ ~ (.) ( ) Because he epece eus o secues ae cosee as eval umbes e ma cose he sem-absolue evao of he aes of eu of pofolo belo he epece eu ove all he pas peos as a eval umbe oo. Sce he epece eu eval of pofolo ( ) s ( ) [ ] ˆ (.3) e ca ge he sem-absolue evao eval of eu of pofolo belo he epece eu ove he pas peo. I ca be epesee as ~ ( ) ma ( ) ma ( ). (.4) he he aveage value of he sem-absolue evao eval of eu of pofolo belo he ucea epece eu ove all he pas peos ca be epesee as ~ ( ) ~ ( ) (.5) We use ~ ( ) o measue he s of pofolo. Suppose ha he veso as o mamze he eu of a pofolo afe emovg he asaco coss h some gve level of s. If he s oleace eval ~ [ ] s gve he mahemacal fomulao of he pofolo seleco poblem s ~ ma ( ) (ILP) s.. ( ) [ ] ~ ~ p u hee epeses he pessmsc oleae s level a epeses he opmsc oleae s level. (ILP) s a opmzao poblem h eval coeffces a heefoe echques of classcal lea pogammg ca o be apple uless he above eval opmzao poblem s euce o a saa lea pogammg sucue. I he follog e pefom hs coveso. We ouce he oe elao he eval obecve fuco of (ILP). Base o he cocep of sasfaco e popose b us Seco. he eval equal elao ~ ( ) p [ ] (ILP) s epesse b a csp equal. he csp equal equvale o ~ p ca be epesee as follos: he eval cosa coo ( ) [ ] ( ( ) [ ] ) α SD ~ p (.6) 7

8 8 he he eval lea pogammg poblem (ILP) ca be epesee b eval lea pogammg poblem hch he obecve fuco s eval umbe a he cosa coos ae csp equal a equales. he eval obecve fuco lea pogammg poblem s epesee as follos: (IP) ma ( ) ~ ~ s.. ( ) [ ] ( ) α SD ~ p u hee sasfaco e α [ ) s gve b he veso. Deoe F as he feasble se of (IP). Defo.7: F s a sasfaco soluo of (IP) f a ol f hee s o ohe ' F such ha ( ) ( ) ' ~ ~ p. B Defo.7 he sasfaco soluo of (IP) s equvale o he o-feo soluo se of he follog b-obecve pogammg poblem fo gve sasfaco e α [ ): ma ( ) (BLP) ma ( ) s.. ( ) [ ] ( ) α SD ~ p u B he mul-obecve pogammg heo he o-feo soluo o (BLP) ca be geeae b solvg he follog paamec lea pogammg poblem: (PLP) ma ( ) ( ) ( ) s.. ( ) [ ] ( ) α SD ~ p u Ioucg he cocee fom of ( ) [ ] ( ) SD ~ p (PLP) ma be ee as follos: (PLP) ma ( ) ( ) ( ) ( ) ( ) ( ) ( ) s α α α α ma ma.. u

9 9 o solve (PLP) e cose he follog asfomao. Fs e ouce a e vaable such ha. (.7) Le ( ) ( ) (.8) ( ) ( ) (.9) ( ) ( ). (.) he (PLP) s equvale o he follog saa lea pogammg poblem. (PLP) ma ( ) ( ) ( ) s.. ( ) α α α α ) ( ( ) ( )... ( ) u. Oe ca use seveal algohms of lea pogammg o solve (PLP) effcel fo eample he smple meho. So e ca solve he ogal pofolo seleco poblem (ILP) b solvg (PLP). Rema.3. I [3] he eval equal as eplace b equal ( ) [ ] ( ) α SD ~ p

10 hle b equal (.6) hs pape. Fo fe sasfaco eα a paamee le F ( α) a V ( α) eoe he feasble se a opmal value of (PLP) especvel. I s eas o see fom he efo of sasfaco e ha α < α mples F( α ) F( α ) a V( α ) V( α ). A fom he cosuco of obecve fuco follos ha < mples V ( α ) V ( α ). heefoe V ( α) s o-ceasg h α a. hs meas ha he smalle sasfaco e s he lage s a u he lage eu s. Gve α [ α α] a [ ] e have V( α ) V( α ) V( α) hee α > aα < α. Fo ecso of appopae α (o ) e ma use he ge compehesve evaluao meho [8] combg AHP (Aalc Heachal Pocess)[9] a OPSIS (echque fo Oee Pefeece b Smla o Ieal Soluo)[] h abues such as eu a s fo gve (oα ). 3. Numecal Eample I hs seco e suppose ha a veso chooses 6 compoeal socs a a s-less asse fo hs vesme. he ae of eu of he s-less asse s.4 pe moh. We collece hsocal aa of he 6 socs ug 8 peos usg oe moh as a peo. Because he ahmecal mehos o o pouce goo esmaes of he acual eus ha he veso ll eceve he fuue e foecase f he eu ae of s asse accog o Wavele-Ge-SVR-Maov peco meho a oba he epece ae of eu eval of each soc. he hsocal eu eec h as obae b h m m hee m s amou of he mos ece peos (e oo m 5). he epece ae of eu evals ae gve able 3.. Suppose he veso spulaes s level eval ~ [.5.4]; b he meho popose he above seco e ca solve he pofolo seleco poblem b solvg (PLP). Fo he gve s level eval ~ moe sasfaco pofolos ca be geeae b vag he values of he paamees a α (PLP). able 3. he epece aes of eus evals Soc Soc Soc3 Soc4 Soc5 Soc6 Loe eu Uppe eu he eu evals he s evals a he values of paamees of pofolos ae lse able 3.. he coespog pofolos ae lse able 3.3 a able 3.4.

11 able 3.. he eu evals he s evals a he values of paamees of pofolos α Reu eval Rs eval Pofolo [.5.363] [ ] Pofolo. [.5.363] [ ] Pofolo 3.4 [.5.33] [ ] Pofolo 4.36 [.5.38] [.78.99] Pofolo 5.48 [.5.36] [.7.98] Pofolo 6.6 [.5.36] [.7.98] Pofolo 7.7 [.5.34] [.7.97] Pofolo 8.84 [.5.34] [.7.97] Pofolo 9.96 [.5.34] [.7.97] α.5 Reu eval Rs eval Pofolo [.3.347] [.7.975] Pofolo. [.4.336] [ ] Pofolo 3.4 [.7.33] [ ] Pofolo 4.36 [.7.33] [ ] Pofolo 5.48 [.9.3] [ ] Pofolo 6.6 [.9.3] [ ] Pofolo 7.7 [.9.3] [ ] Pofolo 8.84 [.9.3] [ ] Pofolo 9.96 [.9.3] [ ] able 3.3. he allocao of pofolo foα pofolo Soc Soc Soc Soc Soc Soc 6 Soc 7

12 able 3.4. he allocao of pofolo foα.5 pofolo Soc Soc Soc Soc Soc Soc 6 Soc 7 he veso ma choose hs o vesme saeg fom he pofolos accog o hs aue oas he secues epece eus a he egee of pofolo s h hch he s comfoable. If he veso s o sasfe h a of hese pofolos he ma oba moe b solvg he paamec lea pogammg poblems (PLP) fo ohe values of paamee aα. 4. Cocluso I [3] a appoach as pesee fo esmag evals of aes of eus of secues a he sem-absolue evao s fuco as eee o a eval case. he popose a eval sem-absolue evao moel h o sho sellg a o soc boog a fcoal mae fo pofolo seleco. I hs pape b oucg a cocep of clusve sasfaco e of he eval equal elao a appoach o compae eval umbes s gve. B usg he appoach he eval sem-absolue evao moel ca be covee o a paamec lea pogammg poblem h o paamees. We epesee he eval equal b he sasfaco e equal ule equal of [3]. Oe ca f a sasfaco soluo o he ogal poblem b solvg he coespog paamec lea pogammg poblem. A veso ma choose a sasfaco vesme saeg accog o a opmsc o pessmsc aue b choosg pope values of paamee α a. he moel ca help he veso o f a effce pofolo accog o hs/he pefeece. Refeeces. Maoz H. Pofolo seleco Joual of Face Alefel G. a Hezbege J. Iouco o Ieval Compuaos Ne Yo:

13 Acaemc Pess Fag Y. La K. K a Wag S.Y. Fuzz Pofolo Opmzao: heo a Mehos Lecue Noes Ecoomcs a Mahemacal Ssems Spge 8 Vol Fag Y. a Wag S.Y. A eval sem-absolue evao moel fo pofolo seleco Lecue Noes Compue Scece 6 Vol eo K. L. a Yag X. Q. Pofolo seleco poblem h mma pe s fuco Aals of Opeaos Reseach Ishbuch H. a aaa H. Mulobecve pogammg opmzao of he eval obecve fuco Euopea Joual of Opeaoal Reseach Yag N. a Pag Y. F. A ucea amc eghe geomec aveagg opeao a s applcao o mulple abue ecso mag Mahemacs Pacce a heo Sfeg Lu Y L Ge Ssems. heo a Applcao Bel Spge- Velag 9. Hag C.L. L M. Goup ecso mag ue mulple cea; Mehos a Applcaos Ne Yo Spge-Velag 987. Hag C.L. Yoo K. Mulple abue ecso mag Spge-Velag Bel Heelbeg Ne Yo 98 3