CONSTRAINT ROBUST PORTFOLIO SELECTION BY MULTIOBJECTIVE EVOLUTIONARY GENETIC ALGORITHM

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1 CONSRAIN ROBUS PORFOLIO SELECION BY MULIOBJECIVE EVOLUIONARY GENEIC ALGORIHM S.K. MISHRA, G.PANDA, 3 S. MEHER & 4 R.MAJHI, 3 Dept of Electrocs ad Commucato Egeerg Natoal Isttute of echology, Rourkela, 7698, INDIA School of Electrcal Sceces, Ida Isttute of echology, Bhubaeswar, INDIA 4 School of Maagemet, Natoal Isttute of echology, Waragal, INDIA Emal: Sudhasu.t@gmal.com, gaapat.pada@gmal.com, smeher@trkl.ac., rtajalmajh@gmal.com Abstract -he problem of portfolo selecto s a very challegg problem computatoal face ad has receved a lot of atteto last few decades. Selectg a asset ad optmal weghtg of t from a set of avalable assets s a crtcal ssue for whch the decso maker takes several aspects to cosderato. Dfferet costrats lke cardalty costrats, mmum buy thresholds ad maxmum lmt costrat are assocated wth assets selecto. Facal returs assocated are ofte strogly o-gaussa character, ad exhbt multvarate outlers. akg these costrats to cosderato ad wth the presece of these outlers we cosder a mult-objectve problem where the percetage of each avalable asset s so selected that the total proft of the portfolo s maxmzed whle total rsk s mmzed. Nodomated Sortg Geetc Algorthm-II s used for solvg ths multobjectve portfolo selecto problem. Performace of the proposed algorthm s carred out by performg dfferet umercal expermets usg real-world data. Keywords: Multobjectve optmzato, Pareto optmal solutos, portfolo asset selecto problem, o-domated sortg, eltsm, decso makg, costrat hadlg. I.INRODUCION A portfolo s a collecto of assets held by a prvate dvdual or a sttuto. he portfolo selecto seeks a optmal way to dstrbute a gve budget o a set of avalable assets. Massve vestmet to dfferet products lke peso fuds, bakg surace polces, stock exchage ad other seres of facal assets s oe of the complex problems facal maagemet. he choce of a approprate vestmet portfolo s a mportat task for a portfolo maager. Optmal selecto of stock exchage assets as well as the optmal vestmet for each asset s a well kow portfolo selecto problem. Portfolo selecto s a complex task as t depeds o varous factors such as assets terrelatoshps, preferece of the decso makers ad resource allocato. Whe vestg moey a set of stock exchage assets, the vestors are terested obtag the maxmum proft of a vestmet ad mmum rsk smultaeously. hs optmzato problem has may costrats lke () the umber of assets a portfolo ca cota s fxed ad fte () the mmum ad maxmum amout of possble vestmets for each chose assets. () the maxmum umber of assets that the portfolo maager ca select out of all the assets.(v) the outlers preset the data. Markowtz set up a quattatve framework for the selecto of a portfolo [,]. hs framework uses the mea varace of hstorcal returs of may assets to measure ts expected retur ad rsk. Koo ad Yamazak [3] proposed the mea absolute devato (MAD) of portfolo whch s take as the rsk measure. he possble asymmetry of retur s take to accout by Koo Shrakawa ad Yamazak [4] who exteded the MAD approach to clude skewess the objectve fucto. Negatve sem-varace proposed by Markowtz [] s oe of the several objectve fuctos that cosdered dowsde rsk. But the etre algorthms rema slet about dfferet costrat assocated wth portfolo. Oe of the costrat.e. the cardalty costrat s approached by some of the researches, Mas ad Speraza [5, 6], ad Youg[7]. I these papers the MOEAs lacked geeralty. hs multobjectve decso makg (MODM) problem wth costrat ad outlers s solved by NSGA-II algorthm. I addto to that, the preset work was dd wth depth aalyss examg how the cardalty costrats affect the evoluto search process o the measuremet of dfferet metrc ad the effcet froter attaed. he remder of the paper s orgazed as follows. Secto outles the mult-objectve optmzato formulato of portfolo selecto. I Secto 3 some of the mult-objectve evolutoary techques used ths paper are dealt. For comparg dfferet multobjectve algorthm, dfferet metrc proposed by varous authors are preseted secto 4. Secto 5 deals wth the smulato study usg reallfe data. he results terms Pareto frots betwee rsk ad retur are show Secto 6. he paper cocludes secto 7 wth a summary ad some deas for further research work drecto. II. MULIOBJECIVE OPIMIZAION: BASIC CONCEPS AND A BRIEF OVERVIEW. Most of the practcal optmzato problems requre decso by smultaeously fulfllg more Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 57

2 Costrat Robust Portfolo Selecto by Multobjectve Evolutoary Geetc Algorthm tha oe goals. hese goals are the mmzato or maxmzato of fuctos geerally cotradcts ature. It s ot possble to fd a sgle soluto for such multobjectve problems. A multobjectve optmzato problem (MOOP) s defed as the problem of computg fdg a vector of decso varables that satsfes some restrctos ad optmze a vector fucto whose elemets represet the value of the fuctos. he geeralsed multobjectve optmzato problem may be formulated as: Maxmze or mmze (x) m,,3..., M () f m Subjected ( x) g j j,,3..., J () h k ( x) k,,3..., K (3) x L x x,,3..., U (4) Where x s represets a vector of decso varables x ( x, x,..., x ) ad wll optmze the vector fucto, f x) { f ( x), f ( x),..., f m ( x)} ( Where f m (x) are the x objectve fuctos. he U values x ad x represet the mmum ad maxmum acceptable values for the varable x respectvely ad defe the boudary of the search space. he J equaltes g ad the K equaltes h k are kow as costrat fuctos. Pareto Optmalty: A pot x s Pareto optmal f for every x ad I {,,3,..., k} ether I ( f ( x) f ( x )) or, there s at least oe I such that ( ) ( f x f x ). he symbols f ad represets the objectve fucto ad the feasble rego ( S) of the whole search space S respectvely. I other words, x s Pareto optmal f there exsts o feasble vector x whch would decrease some crtera wthout causg a smultaeous crease at least oe other crtero. Pareto domace: A vector s sad to be domate v v, v,..., } j u u, u,... } { { v k u k that s v u f ad oly f u s partally less tha v.e.,,..., k, u v,,..., k : u v Pareto optmal set: For a gve MOP f (x), Pareto optmal set p s defed as, the (5) ' p : { x x, f ( x') f ( x)} (6) he soluto of a MOOP s a set of vectors whch are ot domated by ay other vector, ad whch are Pareto-equvalet to each other. hs set s kow as the Pareto-optmal set. f Pareto frot: For a gve MOOP (x) ad Pareto optmal set p,the Pareto frot pf s defed as: pf : { u f ( f ( x), f ( x),...., fk ( x)) x p } (7) he Pareto optmal set whe grouped geerates a dscotuous plot kow as the Pareto frot or Pareto border. he geeralzed cocept s gve 986 by Pareto [8]. It s dffcult to fd a aalytcal expresso of the le or surface that cotas these pots. he procedure to geerate the Pareto frots s to compute the feasble pots ad the correspodg f. Whe there are suffcet umbers of pots, t s possble to determe the odomated pots ad to produce the Pareto frot. Hece the computato of complete Pareto frot volves large computatoal complexty due to the presece of large umber of suboptmal Pareto frots. It requres the soluto to be dverse to cover maxmum possble regos. III. MULI-OBJECIVE FORMULAION OF PORFOLIO he basc mea-varace portfolo selecto problem ca be formalzed as: M V w W QW Max W (8) E (9) W e () w ad,..., N () Where N s the umber of assets avalable,q deotes the covarace matrx of all vestmet alteratves, s the expected retur of asset ad e s the ut vector. he decso varables w determes what share of the budget should be dstrbuted asset. Here W w ww3... w N. Equato ad gve the two competg objectves whch are to be optmzed. Equatos 3 ad 4 show the costrats for a feasble portfolo whch meas that frst all the avalable moey s to be vested ad secodly all vestmets must be postve.e. o short sales are allowed. he costrats gve equato 3 ca be met by ormalzg the weghts Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 58

3 Costrat Robust Portfolo Selecto by Multobjectve Evolutoary Geetc Algorthm s w () he the ew values for each elemet of weght vector are ormalzed. w ' w s () here are some real world costrat that portfolo maager must cosder whle solvg the portfolo optmzato problem. Oe example of ths costrat s cardalty costrat. Let K be the maxmum umber of assets the portfolo maager ca vest moey out of N avalable asset. he K s called as cardalty costrat. N Z K ( 3) he decso varable Z {, } he varable Z f ay asset (,,..., N) s held ad Z f t ot held. hs equato esures that exactly K asset of N avalable asset. he mult-objectve portfolo selecto problem volves two competg objectves () mmze the total varace, deotg the rsk assocated wth the portfolo expressed () () maxmze the retur of the portfolo show (). Alog wth ths the maxmum umber of assets that the portfolo maager ca select out of all the assets ad the outlers preset are to be cosdered. he problem s thus to fd portfolos amogst K asset of the N avalable assets that satsfy these two objectves smultaeously wth the presece of outler. IV. MULIOBJECIVE EVOLUIONARY ALGORIHMS he classcal optmzato techques are effectve for solvg costraed optmzato problem such as portfolo maagemet. hs shortcomg has motvated researchers to develop mult-objectve optmzato usg evolutoary techques. Based o basc cocepts from the bologcal model of evoluto, the search dyamc of mult-objectve evoluto algorthm (MOEA) s guded by bologcally spred evolutoary operators lke selecto, crossover ad mutato. he crossover ad mutato operator chage ad create potetal solutos whle the selecto operator provdes the covergece property. Whe MOEA s appled for portfolo optmzato, ssues lke represetato, varato operator ad costrat hadlg techques are cosdered. MOEA matas a populato of chromosome, where each of them represets a potetal soluto to the portfolo optmzato problem. Oe chromosome represeted by a weght vector, provdes the composto of the portfolo. he poeerg work [9] the practcal applcato of geetc algorthm to MOOP s the vector evaluated geetc algorthm VEGA. For smlar applcatos a umber of algorthms based o geetc algorthm such as NSGA[],NPGA[], PESA- II[], NSGA-II [3], RDGA [4] ad DMOEA [5] have bee proposed lterature. he NSGA-II proposed [] s a useful alteratve ad popular algorthm whch allevates varous shortcomgs of NSGA. Dev ad Pratab have proposed NSGA II where selecto crtera are based o the crowdg comparso operator. Here the pool of dvduals s splt to dfferet frots ad each frot has assged a specfc rak. All dvduals from a frot F are ordered accordg to a crowdg measure whch s equal to the sum of dstace to the two closest dvduals alog each objectve. he evrometal selecto s processed based o these raks. he archve s formed by the o domated dvduals from each frot ad t begs wth the best rakg frot. Here the ew populato obtaed after evrometal selecto s used for selecto crossover ad mutato to create a ew populato. It uses a bary touramet selecto operator. hese algorthms are dealt sequel. NSGA II Algorthm:. Italze populato. Geerate radom paret populato p of sze N 3. Evaluate objectve Values 4. Assg ftess (or rak) equal to ts o domated level 5. Geerate offsprg Populato Q of sze N wth bary touramet selecto, recombato ad mutato. 6. For t to Number of Geeratos 6.a. Combe Paret ad Offsprg Populatos 6.b. Assg Rak (level) based o Pareto Domace. 6.c. Geerate sets of o-domated frots 6.d. utl the paret populato s flled do 6.e. Determe Crowdg dstace betwee pots o each frot F 6.e. Iclude the th o domated frot the ext paret populato P t 6.e.3 check the ext frot for cluso 6.f Sort the frot descedg order usg Crowded comparso operator P 6.g Choose the frst N - card t elemets from frot ad clude them the ext paret Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 59

4 Costrat Robust Portfolo Selecto by Multobjectve Evolutoary Geetc Algorthm populato P t 6.h Usg bary touramet selecto, recombato ad mutato create ext geerato 7. Retur to 6 V. PERFORMANCE MEASURE FOR COMPARISON. S metrc. It measures the spread of caddate soluto throughout odomated vectors foud. Schott [6] troduced ths metrc, measurg the dstace eghborg vectors the odomoated vectors foud. hs metrc s defed as: d d S (4) Where j j d mj f x f x f x f x ad, j,,..., (5) d mea of all d ad s the umber of odomated vectors foud so far. A value of zero for ths metrc dcates all members of the Pareto frot curretly avalable are equdstatly spaced. he S metrc dcates the extet of objectve space domated by a gve odomated set A. If the S metrc of a o domated frot f s less tha aother frot f the f s better tha f. It has bee proposed by Ztzler.. metrc. hs metrc called as spacg metrc ( ) measures how evely the pots the approxmato set are dstrbuted the objectve space. hs formulato troduced by K. Deb[3 ] s gve by N d f dl d d d d N d f l (6) Where d be the Eucldea dstace betwee cosecutve solutos the obtaed odomated set of solutos. d s the average of these dstaces. d ad d are the Eucldea dstace betwee the f l extreme solutos ad the boudary solutos of the obtaed o domated set ad N s the umber of solutos from odomated set. he low value for dcate a better dversty ad hece better s the algorthm. 3. Geerato dstace (GD): he cocept of geerato dstace was troduced by Va Veldhuze ad Lamot [7]. It estmates the dstace of elemets of odomated vectors foud, from those effcet Pareto optmal set ad s defed as: GD d (7) Where s the umber of vectors the set of odomated soluto whch are called as caddate solutos. d s the Eucldea dstace betwee each of these ad the earest member of the global effcet Pareto frot. If GD, all the caddate solutos are global effcet Pareto frot ad ay other value of GD dcates how far are the solutos from the global effcet Pareto frot. he more value of GD meas the elemets are more away from the global effcet Pareto frot. 4. Iverted geerato dstace (IGD): hs qualty dcator s used to measure how far the elemets are the global effcet Pareto frot from those odomated vectors foud from proposed algorthm ad s troduced by Va Veldhuze [7]. If IGD, all the caddate solutos are the global effcet Pareto frot coverg all ts exteso. VI. SIMULAION SUDIES I ths secto we preset the smulato results obtaed whe searchg the geeral effcet froter that resolves the problem formulated equato ad ad wth the presece of assocated cardalty costrat. All the computatoal expermets have bee computed wth a set of bechmark data avalable ole ad obtaed from OR-Lbrary beg mataed by Prof. Beasley. Fve data sets port to port 5 represet the portfolo problem. Each data set correspods to a dfferet stock market of the world. he test data comprses of weekly prces from March 99 to September 997 from the followg dces: Hag Seg Hog Kog, DAX Germay, FSE UK, S&P USA ad Nkke Japa. For each set of test data, the umbers of dfferet assets are 3,85,89,98 ad 5. I the paper we have used the frst data set whch correspods to Hag Seg stock havg 3 assets. he data ca be foud from /~mastjjb/jeb/orlb /portfo.html. I the paper oly cardalty costrats as provded equatos 3 have bee used. Alog wth ths there are some outlers the put data.e. the weekly data of retur. I the work we have selected dfferet umber of assets form the Hag Seg stock where there are 3 assets. he NSGA II has populato sze of, Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 6

5 Costrat Robust Portfolo Selecto by Multobjectve Evolutoary Geetc Algorthm umber of geeratos, crossover rate.8 ad mutato rate.5. he umber of real-coded varables s equal to umber of assets ad VII. HE PAREO FRONS OBAINED BY NSGA-II ALGORIHM he stadard effcet froter correspodg to Hag Seg bechmark problem ad the ucostrat effcet frot geerated by four algorthms are depcted Fgs. NC S matrc.4 5 Delta matrc GD matrc able K=5 K= K= 5 K= Stadard effcet froter of Hag Seg bachmark problem IGD matrc Mea retur able demostrates the values of performace metrcs. whe cardalty costrat creases these metrcs values creases. From the graph shows the value of S metrc Varace of retr x -3 Fg. Plots of UEF for Hag Sag NSGA II..5 Mea retur k= k=5 k= k=5.. Stadard EF UEF usg NSGA II Varace of retr x -3 Fg. Plots of Pareto frots acheved by NSGA II If decso maker s restrcted to select oly fve, te, fftee or twety umber of assets hs portfolo out of all the 3 assets the the Pareto curb obtaed s show the fgure 3. M e a r e t u r K= K=5 K= K= Varace of retur x -3 Fg 3. Plots of Pareto frots acheved at dfferet cardalty costrat. Fg.4. S matrc for dfferet cardalty costrat VIII. CONCLUSION he paper makes a comparatve performace study o portfolo maagemet task employg Nodomated Sortg Geetc Algorthm-II. he data set whch correspodg to Hag-Seg stock s used for carryg out smulato based expermets. Expermetal results reveal that the NSGA-II algorthm perform satsfactorly to solve the costrat portfolo selecto problem wth the presece of outlers. Future work cludes troducto of dfferet operators for local search the exstg models whch allow better explorato ad explotato of the search space whe appled to portfolo optmzato problem. Aother possble future research drecto s to hadle dfferet real world costrats lke mmum buy thresholds or maxmum lmt costrats, whch would make the problem more complex ad the devsg mproved optmzato tools to effectvely solve t. Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 6

6 Costrat Robust Portfolo Selecto by Multobjectve Evolutoary Geetc Algorthm REFERENCES []. H.M. Markowtz. Portfolo Selecto, Joural of Face7(95)77-9 []. H.M. Markowtz, Portfolo Selecto: effcet dversfcato of vestmets. New York: Yale Uversty Press. Joh Wley & Sos, (99). [3]. H.Koo ad H.Yamazk, Mea absolute-devato portfolo optmzato model ad ts applcato tookyo Stock Market. Maagemet Scece 37 (99) [4]. H.Koo, H.Shrrakawa ad H.Yamazk, A mea-absolute devato-skewess portfolo optmzato model. Aals of Operato research 45(993)5-. [5]. R. Mas ad M.G. Speraza, Heurstc algorthms for the portfolo selecto problem wth mmum trasacto lots,workg paper (997) avalable from the secod author at Dp.d Metod Quattatv, Uversta d Bresca, C.da.S Chara 48 /b,5 Bresca, Italy. [6]. H. Kellerer,R. Mas ad M.G. Speraza, O selectg a portfolo wth fxed costs ad mmum trasacto lots. Workg paper(997) avalable from the thrd author at Dp.d Metod Quattatv, Uversta d Bresca, C.da.S Chara 48/b,5 Bresca, Italy. [7]. M.R. Youg, a mmax portfolo selecto rule wth lear programmg soluto. Maagemet Scece 44(998) [8]. Vlfredo Pareto. Cours. D, Ecoome Poltque, Volume I ad II. F. Rouge, Lausae.896. [9]. Schaffer J.D. Multple objectve optmzato wth vector evaluated geetc algorthms. I Geetc Algorthms ad ther Applcatos: Proceedgs of the teratoal coferece ogeetc algorthm, Lawrece Erlbaum, (985) 93-. []. Srvas N, Deb K, Multobjectve optmzato usg odomated sortg geetc algorthms. J Evol Comput (994) -48. []. Hor. J, Nafplots. N, Goldberg D.E, A ched pareto geetc algorthm for multobjectve optmzato, I: Proceedgs of the frst IEEE coferece o evolutoary computato, IEEE world cogress o computato tellgece, 7-9 Jue, Orlado, FL, USA,( 994) []. Core D, Jerram NR, Kowles J, Oates J. PESA-II: regobased selecto evolutoary multobjectve optmzato. I:Proceedg coferece (GECCO-), Sa Fracsco, CA,. [3]. Deb K, Pratap A, Agarwal S, Meyarva. A fast ad eltst multobjectve geetc algorthm: NSGA-II. IEEE ras Evolutoary Computg 6(), () [4]. Lu H, Ye G.G, Rak-desty-based multobjectve geetc algorthm ad bechmark test fucto study, IEEE ras Evolutoary Computg 7(4), (3), [5]. Ye G.G, Lu H, Dyamc multobjectve evolutoary algorthm: adaptve cell-based rak ad desty estmato. IEEE ras Evolutoary Computg 7(3), (3) [6]. J. R. Schott, Fault tolerat desg usg sgle ad multcrtera geetc algorthm optmzato, M.S. thess, Dept. Aeroautcs ad Astroautcs, Massachusetts Ist. echol, Cambrdge, MA, May 995. [7]. D. A. Va Veldhuze ad G. B. Lamot, Multobjectve evolutoary algorthm research: A hstory ad aalyss, Dept. Elec. Comput. Eg., Graduate School of Eg., Ar Force Ist. echol., Wrght PattersoAFB, OH, ech. Rep. R-98-3, (998). Iteratoal Joural of Electrocs Sgals ad Systems (IJESS) ISSN: , Vol- Iss-3, 6