The Impact of Transaction Costs on Rebalancing an Investment Portfolio in Portfolio Optimization

Transcription

1 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 he Impact of ransacton Costs on Rebalancng an Investment Portfolo n Portfolo Optmzaton B. Marasovć, S. Pvac, S. V. Vukasovć Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 Abstract Constructng a portfolo of nvestments s one of the most sgnfcant fnancal decsons facng ndvduals and nsttutons. In accordance wth the modern portfolo theory mamzaton of return at mnmal rsk should be the nvestment goal of any successful nvestor. In addton, the costs ncurred when settng up a new portfolo or rebalancng an estng portfolo must be ncluded n any realstc analyss. In ths paper rebalancng an nvestment portfolo n the presence of transacton costs on the Croatan captal market s analyzed. he model appled n the paper s an etenson of the standard portfolo mean-varance optmzaton model n whch transacton costs are ncurred to rebalance an nvestment portfolo. hs model allows dfferent costs for dfferent securtes, and dfferent costs for buyng and sellng. In order to fnd effcent portfolo, usng ths model, frst, the soluton of quadratc programmng problem of smlar sze to the Markowtz model, and then the soluton of a lnear programmng problem have to be found. Furthermore, n the paper the mpact of transacton costs on the effcent fronter s nvestgated. Moreover, t s shown that global mnmum varance portfolo on the effcent fronter always has the same level of the rsk regardless of the amount of transacton costs. Although effcent fronter poston depends of both transacton costs amount and ntal portfolo t can be concluded that etreme rght portfolo on the effcent fronter always contans only one stock wth the hghest epected return and the hghest rsk. Keywords Croatan captal market, Fractonal quadratc programmng, Markowtz model, Portfolo optmzaton, ransacton costs. I. IRODUCIO 1952 H. M. Markowtz [12] developed the frst model for I portfolo optmzaton and wth that model he lad the foundaton of the modern portfolo theory. Hs model s based upon only two crtera: return and rsk. he rsk s measured by the varance of returns dstrbuton. Markowtz shows how to calculate portfolo whch has the hghest epected return for a gven level of rsk, or the lowest rsk for a gven level of epected return (the so-called effcent portfolo). he problem of portfolo selecton, accordng to ths theory, s a problem of quadratc programmng whch conssts of mnmzng rsk whle keepng n mnd an epected return whch should be guaranteed. he mportance of Markowtz's work s affrmed by the obel Prze for Economcs he won n 199. However, B. Marasovć s wth the Faculty of Economcs, Department of Quanttatve methods, Unversty of Splt, 21 Splt, Croata (phone: ; fa: ; e-mal: branka.marasovc@ efst.hr). S. Pvac s wth the Faculty of Economcs, Department of Quanttatve methods, Unversty of Splt, 21 Splt, Croata (e-mal: snjezana.pvac@ efst.hr). S. V. Vukasovć s graduated student wth the Faculty of Scence, Department of Mathematcs, Unversty of Splt, 21 Splt, Croata (e-mal: stjvuk@pmfst.hr). parallel to ntroducng the Markowtz model n the common usage ts lmtatons and drawbacks were beng notced. One of dsadvantages of Markowtz model s the fact that t doesn t take nto consderaton transacton cost although costs ncurred when settng up a new portfolo or rebalancng an estng portfolo must be ncluded n any realstc analyss. In ths paper, we apply a method for fndng an optmal portfolo wth proportonal transacton costs on the Croatan captal market and analyze the same. hese costs vary lnearly wth the amount of a securty bought or sold. hs method allows dfferent costs for dfferent securtes, and dfferent costs for buyng and sellng. hs model captures the feature that transacton costs are pad when a securty s bought or sold and the transacton cost reduces the amount of that partcular securty that s avalable. In partcular, both the rsk and the return n our model are measured usng the portfolo arsng after payng the transacton costs. he portfolo rebalancng problem has smlartes to the nde trackng problem [1], [5]. See [2] for a dscusson of portfolo optmzaton models. Portfolo optmzaton models wth alternatve rsk measure have been nvestgated n [8], [1], [16]. Contrary to the epectatons of the modern portfolo theory, the tests carred out on a number of fnancal markets (AMEX, YSE, SE, Pars Stock Echange, etc.) have revealed the estence of other ndcators, besdes return and rsk, mportant n portfolo selecton. he most mportant anomales dscovered to date are the sze measured by stock market captalzaton and the Prce Earnng Rato (PER) [4]. Consderng the mportance of varables other than return and rsk, selecton of the optmal portfolo becomes a mult-crtera problem whch should be solved by usng the approprate technques. he mult-crtera nature of the portfolo selecton was well presented n the paper of Khoury et al. [9] and today an arsenal of multdmensonal and multcrtera methods such as factor analyss, goal programmng, AHP, ELECRE, MIORA, ADELAIS, etc. have been already appled n portfolo selecton [4], [7], [11], [15], [21]. An applcaton of the portfolo optmzaton model wth transacton costs on the Croatan captal market s presented n [17]. In that paper authors used the model n whch measure of rsk (varance) of effcent portfolo wasn t calculated n an approprate manner. In order to properly represent the varance of the resultng portfolo, t s necessary to rescale by the funds avalable after payng the transacton costs [14]. In the model appled n ths paper the varance s calculated on proposed way. he paper s organzed n fve sectons. After ths ntroductory secton, n the second secton the elements of the Internatonal Scholarly and Scentfc Research & Innovaton 9(3)

2 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 Modern (Markowtz's) Portfolo heory s presented. Portfolo Rebalancng Problem s eposed n the thrd secton. In the fourth secton portfolo model wth transacton cost s appled on the Croatan captal market and t s nvestgated the propertes of the obtaned effectve portfolos. Fnally, some concludng remarks are gven. II. HE ELEMES OF HE MODER (MARKOWIZ'S) PORFOLIO HEORY Between two or more portfolos of rsky assets, the nvestors wll choose the one that gves the lowest varance of return of all portfolos havng the same epected return, or the one that has the hghest epected return of all portfolos havng the same varance,.e. the nvestors wll choose an effcent portfolo [2]. he effcent fronter s the set of all effcent portfolos. ow we show how to calculate effcent portfolos and the effcent fronter [12]. We use the followng notaton: here are rsky assets, each of whch has epected return Er ( ). he varable R s the column vector of epected returns of these assets: Er ( 1) r1 Er ( 2) r 2., R.. Er ( ) r and S s the varance-covarance matr: S A portfolo of rsky assets s a column vector whose coordnates sum to 1: 1 2., Each coordnate represent the proporton of the portfolo nvested n rsky asset. he epected portfolo return E( r ) of a portfolo s gven by the product of and R: Er ( ) R Er ( ) (1) 1 2 he varance of portfolo s return, s gven by the product: 2 S j j 1 j1 (2) he covarance between the return of two portfolos and y, Cov( r, r ), s defned by the product: y y Sy y j j 1 j1. (3) Mathematcally, we may defne an effcent portfolo as 2 follows [13]. For a gven portfolo varance (or standard devaton), an effcent portfolo s one that solves: 1 ma E( r ) R E r (4) 2 S (5) 1 1 (6), 1, 2,...,. (7) Last condtons mean that short sales of assets are restrcted [3]. III. PORFOLIO REBALACIG PROBLEM Portfolo rebalancng problem n the presence of transacton costs was nvestgated by [14]. What we consder s an etenson of the basc portfolo optmzaton problem n whch transacton costs are ncurred to rebalance a portfolo? hat s, transactons are made to change an already estng portfolo,, nto a new and effcent portfolo,. A portfolo may need to be rebalanced perodcally smply as updated rsk and return nformaton s generated wth the passage of tme. Further, any alteraton to the set of nvestment choces would necesstate a rebalancng decson of ths type. In addton to the obvous cost of brokerage fees/commssons, here are two eamples of other transacton costs that can be modeled n ths way [14]: 1. Captal gans taes are a securty-specfc sellng cost that can be a major consderaton for the rebalancng a portfolo. 2. Another possblty would be to ncorporate an nvestor's confdence n the rsk/return forecast as a subjectve cost". Placng hgh buyng and sellng costs on a securty would favor mantanng the current allocaton. Placng a hgh sellng cost and low buyng cost could be used to epress optmsm that a securty may outperform ts forecast. Let u and v represent the amount bought and sold (respectvely) of securty. he amount nvested n each of the securtes wll be: Internatonal Scholarly and Scentfc Research & Innovaton 9(3)

3 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 u v. (8) S 1 2 (18) Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 We assume proportonal transacton costs. Let C B and C S denote the transacton cost of buyng and sellng one unt of securty, respectvely. We assume, 1, CS 1 and for whch s CS. We let denote the total amount spent on transacton costs, so C u C v. (9) B he total amount nvested n the securtes, after payng transacton costs, wll be 1. We obtan the constrant S e 1C u C v. (1) Eplotng the fact that, e 1, (9) mmedately gves he resultng equaton s: B S B S e e e u e v C u C v (11) e u CS e v. (12) hs equaton can be used to gve a model for mnmzng the varance of the resultng portfolo subject to meetng an epected return of E n the presence of proportonal transacton costs. he resultng model s: Mn S (13) R E (14) uv (15) e u CS e v (16) uv,,. (17) o ths pont, we have been optmzng the standard rsk measure for effcent fronters, that s: S. j j 1 j1 When there are no transacton costs to be pad, one dollar s always avalable for nvestment,.e. ( 1). hs 1 assumpton s mplct n the standard rsk measure. However, for nonzero transacton costs that mplct assumpton s no longer vald. One dollar s not avalable for nvestment; costs wll be pad to rebalance. he approprate objectve s therefore Here s agan the amount pad n transacton costs. herefore, (1 ) s the actual amount avalable for nvestment, so we are choosng to scale the standard rsk measurement by the square of the dollar amount actually nvested. hs gves the fractonal quadratc programmng problem (FQP) whch we wll solve to fnd the optmal portfolo for a gven epected return. Mn S 1 2 (19) R E (2) uv (21) e u CS e v (22) uv,, (23) he fractonal objectve f() can be made quadratc usng the technque of replacng the denomnator by the square of the recprocal of a varable. hs s a straghtforward etenson of the technque of [6] for fractonal programs where the objectve s a rato of lnear functons and the constrants are lnear. Let and then defne 1 t : (24) 1 Cu Cv B S u : tu, v : tv, : t. (25) ote that snce u and v are constraned to be nonnegatve, we must have t 1. ote that we now have tucsv 1. he constrants (2)-(22) can be multpled through by t. hus, the fractonal quadratc program (FQP) s equvalent to the quadratc programmng problem (QP) Mn S (26) REt (27) u v t (28) e u CS e v (29) t u CSv 1 (3) uvt,,,. (31) * * * * Once we fnd a soluton u, v,, t to (QP), we can obtan a * * * soluton u, v, to the orgnal problem (FQP) by rescalng * *, u and v, so * *, * u * u and * v * v. he effcent fronter * t t t Internatonal Scholarly and Scentfc Research & Innovaton 9(3)

4 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 s found by optmzng (QP) for dfferent values of E. If we do not take nto the consderaton (27) for soluton of the problem we get global mnmum varance portfolo. Etreme rght portfolo on the effcent fronter can be obtaned as the soluton of lnear programmng problem: Ma R (32) uv (33) e u CS e v (34) uv,,. (35) In [14] authors have ntroduced varables to both buy u and sell v each securty. We have not mposed an eplct constrant requrng that f a certan securty s bought then t cannot also be sold. Both buyng and sellng a securty would not be a desrable strategy n practce, but t mght decrease the rsk measure S. Soluton u, v s called complementary f t satsfes uv, that s, f no stock s both bought and sold. In the paper [14] authors shown that f the return constrant REt s actve at the optmal soluton to (QP) then the optmal soluton must be complementary. If the return constrant s not actve at the optmal soluton, then t s possble that an optmal soluton wll not be complementary. However, authors also shown that a complementary soluton can always be found effcently even n ths stuaton. IV. AALYSIS OF MEA-VARIACE PORFOLIO OPIMIZAIO MODEL WIH ICLUDED RASACIO COSS HROUGH HE APPLICAIO O HE CROAIA CAPIAL MARKE hrough the applcaton of the presented mean-varance portfolo optmzaton model wth ncluded transacton costs on the Croatan captal market we conduct analyss of effcent portfolos obtaned by presented model. From the total number of securtes quoted on the Zagreb stock echange n 213 and 214 a sample of ten stocks from CROBEX1 nde has been separated. Stocks ncluded n that nde are ten the most lqud stocks wth the hghest free float, turnover and market captalzaton on Zagreb Stock Echange. Companes ncluded n CROBEX1 nde are: AD Plastk (ADPL-R-A), Adrs grupa (ADRS-P-A), Atlantc grupa (AGR-R-A), Ercsson kola esla (ER-R-A), H (H- R-A), IA (IA-R-A), Končar-elektrondustrja (KOEI-R-A), Ledo (LEDO-R-A), Podravka (PODR-R-A), Petrokemja (PKM-R-A) [19]. For each securty from the sample we take the closng prce at the end of each two-week perod from January 1 st 213 to ovember 5 th 214. Frst we calculate the two-week returns for each securty. We choose two-week returns because the most of the stocks from the sample have normal dstrbuton of two-week returns and n ths case varance s adequate measure of rsk. For perod t and securty A, two-week return r At s defned as: P A,. t rat ln [18]. Frst we calculate effcent fronter usng P At, 1 Markowtz model based on two-week returns durng perod from January 1 st 213 to October 22 nd 214. Obtaned effcent fronter and ten effcent portfolos are shown on Fg. 1 and able I Rsk (Standard Devaton) Fg. 1 Effcent fronter on the date October 22 nd 214 ABLE I EFFICIE PORFOLIOS O OCOBER 22D 214 ADPL-R-A ADRS-P-A AGR-R-A ER-R-A H-R-A IA-R-A KOEI-R-A LEDO-R-A PODR-R-A PKM-R-A E(R) σ 3.17% 4.42%.% 1.26% 7.5% 25.32% 11.6% 37.92%.%.26%.8% 1.79% 2.75% 1.3% 3.52% 9.6%.11% 23.3% 12.2% 39.47%.%.%.21% 1.81% 1.25% 12.36% 13.4% 7.42%.% 17.% 9.7% 39.51%.%.%.34% 1.91%.% 14.59% 23.36% 5.74%.% 1.85% 6.% 39.46%.%.%.47% 2.8%.% 16.52% 33.32% 4.%.% 4.41% 2.63% 39.12%.%.%.59% 2.3%.% 17.81% 43.91% 2.6%.%.%.% 36.22%.%.%.72% 2.57%.% 17.38% 56.4%.%.%.%.% 26.58%.%.%.85% 2.91%.% 16.29% 68.94%.%.%.%.% 14.77%.%.%.98% 3.29%.% 15.2% 81.85%.%.%.%.% 2.95%.%.% 1.11% 3.71%.%.% 1.%.%.%.%.%.%.%.% 1.23% 4.19% Epected Return Mean-Varance-Effcent Fronter Portfolos from able I are effcent on October 22 nd 214. However when we nclude n analyss stocks return on ovember 5 th 214 (net two-week returns) those portfolos are no more effcent and t s necessary to conduct portfolo rebalance. Durng portfolo rebalance we assume that an Internatonal Scholarly and Scentfc Research & Innovaton 9(3)

5 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 nvestor wants to keep the same return as n ntal portfolo n prevous perod wth rsk mnmzaton. Most of Croatan brokerages charge all-n type of transacton fees. It means that Zagreb Stock Echange and SKDD (Central Depostory and Clearng Company) fees are ncluded n brokerage fees. Also most of Croatan brokerages charge the same transacton fees both for sellng and buyng orders. Durng year 213 those fees was between.35% and 1.25% [22]. herefore, n ths paper we conduct portfolo rebalance for the hghest and the lowest value of brokerage fees on the Croatan Captal Market. Results of portfolo rebalance wth 1.25% brokerage fees are gven n able II and wth.35% brokerage fees are gven n able III. he frst row n able II represent new effcent portfolo whch we obtaned by rebalancng the frst portfolo from able I wth transacton costs of 1.25% whle the frst row n able III represent new effcent portfolo whch we obtaned by rebalancng frst portfolo from able I wth transacton costs of.35%. From ables II and III we can notce that rebalanced portfolos wth the same return have hgher rsk f transacton costs are hgher. So, we can conclude that f nvestor wants to acheve gven rate of return he has to accept hgher rate of rsk for hgher transacton costs. Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 ABLE II EFFICIE PORFOLIOS O OVEMBER 5 H 214 OBAIED BY REBALACIG OF PORFOLIOS FROM ABLE 1 WIH BROKERAGE FEES 1.25% ADPL-R-A ADRS-P-A AGR-R-A ER-R-A H-R-A IA-R-A KOEI-R-A LEDO-R-A PODR-R-A PKM-R-A E(R) σ 2.51% 6.84%.% 1.% 5.89% 24.12% 11.92% 38.63%.%.%.8% % 1.64% 11.33% 5.44% 9.12%.% 21.11% 11.81% 39.47%.%.%.21% %.4% 13.7% 14.68% 7.66%.% 15.28% 9.7% 39.51%.%.%.34% %.% 15.62% 23.94% 6.9%.% 8.95% 6.% 39.36%.%.%.47% %.% 17.47% 33.32% 4.57%.% 2.82% 2.66% 39.12%.%.%.59% %.% 18.31% 43.91% 2.59%.%.%.% 35.17%.%.%.72% %.% 17.9% 55.6%.41%.%.%.% 26.6%.%.%.85% %.% 17.54% 67.66%.%.%.%.% 14.77%.%.%.98% %.% 17.19% 79.8%.%.%.%.% 2.96%.%.% 1.11% %.% 2.7% 97.88%.%.%.%.%.%.%.% 1.23% 4.959% ABLE III EFFICIE PORFOLIOS O OVEMBER 5 H 214 OBAIED BY REBALACIG OF PORFOLIOS FROM ABLE 1 WIH BROKERAGE FEES.35% ADPL-R-A ADRS-P-A AGR-R-A ER-R-A H-R-A IA-R-A KOEI-R-A LEDO-R-A PODR-R-A PKM-R-A E(R) σ 2.51% 6.84%.% 1.1% 5.9% 24.13% 11.93% 38.66%.%.%.8% % 1.6% 11.4% 5.42% 9.17%.% 21.16% 11.84% 39.37%.%.%.21% %.% 13.73% 14.66% 7.71%.% 15.26% 9.11% 39.51%.%.%.34% %.% 15.72% 23.92% 6.19%.% 8.97% 6.% 39.19%.%.%.47% %.% 17.63% 33.23% 4.68%.% 2.71% 2.87% 38.86%.%.%.59% %.% 18.85% 43.73% 2.79%.%.%.% 34.62%.%.%.72% %.% 18.78% 55.31%.72%.%.%.% 25.17%.%.%.85% %.% 18.27% 67.37%.%.%.%.% 14.35%.%.%.98% %.% 17.5% 79.67%.%.%.%.% 2.81%.%.% 1.11% %.% 2.15% 97.84%.%.%.%.%.%.%.% 1.23% 4.941% ABLE IV GLOBAL MIIMUM VARIACE PORFOLIOS O OVEMBER 5 H 214 OBAIED BY REBALACIG PORFOLIOS FROM ABLE 1 WIH BROKERAGE FEES 1.25% ADPL-R-A ADRS-P-A AGR-R-A ER-R-A H-R-A IA-R-A KOEI-R-A LEDO-R-A PODR-R-A PKM-R-A E(R) σ 2.54% 5.11%.% 1.45% 7.42% 24.58% 11.69% 37.81%.%.35%.466% 1.782% 2.53% 5.1%.% 1.43% 7.4% 24.53% 11.67% 37.72%.%.35%.465% 1.782% 2.53% 5.9%.% 1.4% 7.38% 24.46% 11.63% 37.61%.%.35%.464% 1.782% 2.52% 5.7%.% 1.37% 7.36% 24.38% 11.6% 37.5%.%.35%.462% 1.782% 2.51% 5.6%.% 1.34% 7.33% 24.31% 11.56% 37.39%.%.35%.461% 1.782% 2.5% 5.4%.% 1.31% 7.32% 24.25% 11.54% 37.3%.%.35%.46% % 2.5% 5.3%.% 1.28% 7.29% 24.18% 11.5% 37.19%.%.35%.459% 1.782% 2.49% 5.2%.% 1.25% 7.27% 24.11% 11.47% 37.8%.%.34%.457% 1.782% 2.48% 5.%.% 1.22% 7.25% 24.3% 11.43% 36.97%.%.34%.456% 1.782% 2.48% 4.99%.% 1.2% 7.24% 23.98% 11.41% 36.89%.%.34%.455% 1.782% Internatonal Scholarly and Scentfc Research & Innovaton 9(3)

6 World Academy of Scence, Engneerng and echnology Internatonal Journal of Economcs and Management Engneerng Vol:9, o:3, 215 ABLE V EXREME RIGH PORFOLIOS O OVEMBER 5 H 214 OBAIED BY REBALACIG PORFOLIOS FROM ABLE 1 WIH BROKERAGE FEES 1.25% ADPL-R-A ADRS-P-A AGR-R-A ER-R-A H-R-A IA-R-A KOEI-R-A LEDO-R-A PODR-R-A PKM-R-A E(R) σ.%.% 97.53%.%.%.%.%.%.%.% % 4.155%.%.% 97.62%.%.%.%.%.%.%.% % 4.155%.%.% 97.86%.%.%.%.%.%.%.% 1.227% 4.155%.%.% 98.11%.%.%.%.%.%.%.% % 4.155%.%.% 98.35%.%.%.%.%.%.%.% % 4.155%.%.% 98.61%.%.%.%.%.%.%.% 1.231% 4.155%.%.% 98.91%.%.%.%.%.%.%.% % 4.155%.%.% 99.23%.%.%.%.%.%.%.% % 4.155%,%,% 99,55%,%,%,%,%,%,%,% 1,2418% 4,155%.%.% 1.%.%.%.%.%.%.%.% % 4.155% Dgtal Open Scence Inde, Economcs and Management Engneerng Vol:9, o:3, 215 waset.org/publcaton/183 Furthermore, n ths paper we analyze global mnmum varance portfolo. We conduct rebalance of portfolos from able I wth transacton costs of 1.25 % wth am to fnd global mnmum varance portfolo. From able IV we can observe that global mnmum varance portfolo has always the same level of rsk (varance) regardless of ntal portfolo. However, rebalancng wth hgher volume and number of transactons (hgher costs) cause lower return. Fnally, solvng problem of lnear programmng (32)-(35) we get portfolos wth the hghest return but wth the hghest rsk. From able V we can observe that all portfolos consst of only one stock and also have the same varance. Agan, return depends of volume and number of transactons durng rebalancng and can be calculated from: 1 CuCvma Er ( ) : 1,2,...,. B S V. COCLUSIO he results show that effcent fronter s always postoned n the same rsk nterval regardless both of the amount of transacton costs and ntal portfolo. Effcent portfolo return s negatve correlated wth number and volume of transactons. Fnally, t can be concluded that effcent fronter obtaned by presented model s always postoned below effcent fronter obtaned by Markowtz model.e. Markowtz effcent portfolo always have hgher or equal return than return of effcent portfolo obtaned by presented model. REFERECES [1] C. J. Adcock, and. Meade, A smple algorthm to ncorporate transactons costs n quadratc optmzaton, European Journal of Operatonal Research, vol. 79, no. 1, 1994, pp [2] Z. Aljnovć, B. Marasovć, and. omć-plazbat, he selecton of the optmal portfolo on the Croatan captal market, n Proceedngs of Sth Internatonal Conference on Enterprse n ranston, May 25. pp [3] Z. Aljnovć, B. Marasovć, B. Šego, Fnancjsko modelranje. Ekonomsk fakultet u Spltu, Splt, 211. [4] A. Bour, J. M. Martel and H. Chabchoub, A Mult-crteron approach for selectng attractve portfolo, Journal of Mult-Crtera Decson Analyss, vol. 11, 22, pp [5]. J. Chang,. Meade, J. E. Beasley and Y. M. Sharaha, Heurstcs for cardnalty constraned portfolo optmzaton, Computers and Operatons Research, vol. 27, 2, pp [6] A. Charnes and W. W. 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