ON MULTIPERIOD PORTFOLIO SELECTION WITH DIFFERENT BORROWING AND LENDING RATES

Transcription

1 Haed DAVARI, PhD Candidae E-ail: ajid AINNAYERI, PhD (corresponding auhor) E-ail: Airkabir Universiy of Technology ON ULTIPERIOD PORTFOLIO SELECTION WITH DIFFERENT BORROWING AND LENDING RATES Absrac. This paper deals wih he proble of uliperiod porfolio selecion, where borrowing and lending are allowed wih differen raes. Indeed, his work is ainly based on a recenly published paper wih he sae subjec. In his paper he underlying proble of uliperiod porfolio selecion wih differen borrowing and lending raes is reforulaed. Afer a horough discussion abou boh concepual and aheaical poins, soe new noaions, copared o previous sudies, are inroduced. Aferward, a fuzzy represenaion of he reforulaed odel is proposed and a nuerical exaple is used for ipleening he presened odel. Finally, he copuaional resuls are discussed. Keywords: uliperiod porfolio selecion, aheaical prograing, Fuzzy se heory, Borrowing, Lending. JEL Classificaion: C8, G Inroducion uli-period porfolio selecion is raising he aenion of various researchers and praciioners. Due o he forhcoing siuaions in he arke, in a long-er horizon, an invesor odifies his/her porfolio coposiion. Hence, he uli-period porfolio selecion proble capures a ore realisic iage of he capial arke condiions han he single-period one. ulvey e al. () indicae ha exploiing he uli-period paradig in he porfolio anageen proble is indispensable, in paricular when here are ransacion coss, when reurns exhibi eporal dependence, and when he invesor is able o borrow for invesen. Hence, developen of uliperiod aheaical prograing odels in he area of porfolio anageen is a aer of paricular iporance.

2 Haed Davari, ajid Ainnayeri Here, we review soe of he os iporan works ainly focusing on developing aheaical prograing odels o deal wih porfolio anageen probles. Topaloglou e al. (8) developed a ulisage sochasic prograing odel o dynaically deal wih he inernaional porfolio anageen proble. The proposed fraework was able o joinly deerine he capial allocaed o each inernaional arke, he asses seleced in each arke and he appropriae currency hedging levels. Pınar (7) developed ulisage porfolio selecion odels o axiize he expeced erinal wealh. Also, he presened odels sough o iniize one-sided deviaion fro a arge wealh level o ensure sabiliy of he invesen policies in he face of arke risk. Edirisinghe & Paerson (7) developed a uliperiod aheaical odel for sock porfolio opiizaion. Their proposed odel incorporaed various risk and policy consrains leading o significan period-by-period linkage in he odel. Zenios e al. (998) used ulisage sochasic prograing wih recourse o develop uliperiod fixed-incoe porfolio anageen odels under uncerainy in a dynaic seing. Their presened odels inegraed he prescripive sochasic progras wih descripive one Carlo siulaion odels of he er srucure of ineres raes. Escudero e al. (9) presened a ulisage sochasic ixed - odel wih coplee recourse o opiize a ean risk porfolio anageen proble. Their proposed odel deal wih a fixed incoe asse porfolio resrucuring in which he ineres raes and he liabiliies considered o be uncerain along a given ie horizon. Lacagnina & Pecorella (6) inegraed sochasic and possibilisic prograing o develop a ulisage sochasic sof consrains fuzzy progra wih recourse for capuring boh uncerainy and iprecision in porfolio anageen proble. Lucka e al. (8) proposed a ulisage odel o allocae financial resources o bond indices denoinaed in differen currencies. Their sudy uilized hisorical daa of ineres and exchange raes o copare a wo-sage and a hree-sage sochasic prograing odel fro a financial perforance viewpoin. Consiglio & Saino () presened a ulisage sochasic prograing odel o selec bond porfolios aiing o iniize he cos of he decisions ha us be aken based on he key sochasic econoic facors underneah he odel. Raubenheier & Kruger () forulaed a ulisage dynaic sochasic prograing odel o deal wih a liquid asse porfolio anageen proble. The ai of he proposed odel was o shape an opial liquid asse porfolio for a financial insiuion wihou violaing he andaory regulaions, abou he iniu required liquid asses, i has o coply wih. Fersl & Weissenseiner () forulaed a uli-sage sochasic linear progra o deal wih a cash anageen proble in which a copany wih a given financial endowen and fuure cash flows

3 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes is o iniize he Condiional Value a Risk of he erinal wealh. In he proposed odel, ineres raes and equiy reurns were considered o be uncerain. Osorio e al. (8a) developed a ulisage ean-variance porfolio allocaion odel o invesigae he role of decisions ha affec he way axes are paid in a general porfolio invesen. To aain his goal, heir proposed ulisage porfolio opiizaion odel inegraed a nuber of risky asses grouped in wrappers wih special axaion rules. Osorio e al. (8b) developed a ixed ineger sochasic prograing approach o deal wih he ean-variance pos-ax porfolio anageen. Their presened sochasic prograing approach considered risk in a ulisage seing and allowed general wihdrawals fro original capial. Dae e al. () presened a sochasic opiizaion-based approach o build a porfolio issued over a series of governen aucions for he fixed incoe deb. Their proposed ixed ineger linear prograing odel ha uses a receding horizon, sough o iniize he cos of servicing deb while conrolling risk and ainaining arke liquidiy. Rasussen & Clausen (7) forulaed ulisage sochasic ineger progras o deal wih he orgagor s choices in he Danish orgage loan syse and also his/her aiude owards risk in a dynaic seing. Barro & Canesrelli (9) uilized sochasic prograing fraework o develop a ulisage sochasic racking error odel. Their sudy invesigaed differen racking error easures which are coon in saic odels and also a nuber of probles arising in dynaic seings. Bersias & Pachaanova (8) developed robus opiizaion forulaions o deal wih uliperiod porfolio selecion in he presence of ransacion coss. They copared he perforance of he presened robus forulaions o he perforance of he radiional single period ean-variance forulaions. As enioned above, he focus of his paper is on sudies in which he invesor(s) can borrow and lend oney wih differen raes o inves in a uliperiod porfolio anageen seing. Thus, his paper ries o focus on he sudies conduced in his conex. Seyedhosseini e al. () presened a aheaical prograing odel o deal wih he uliperiod porfolio selecion proble where he borrowing rae is greaer han he lending rae. They considered a nuerical exaple o illusrae heir presened aheaical forulaion. Sadjadi e al. () presened a fuzzy linear prograing odel o address he uliperiod porfolio selecion proble where he borrowing rae is greaer han he lending rae. Due o he inrinsic uncerainy of raes of reurn for risky asses and raes of borrowing and lending, hey considered hese paraeers as riangular fuzzy nubers raher han crisp nubers. Finally, hey

4 Haed Davari, ajid Ainnayeri presened a nuerical exaple and discussed abou he oupu resuls. Seyedhosseini e al. () presened a sochasic prograing odel o address he uliperiod porfolio selecion proble where he borrowing rae is greaer han he lending rae. To deal wih he inrinsic uncerainy of he proble, chance consrained prograing was uilized. Finally, geneic algorih was used o solve he forulaed proble. In boh of he above-enioned sudies, ransacion coss were ignored. Hassanlou () copared he above-enioned approaches for solving he uli-period porfolio selecion proble wih differen borrowing and lending raes and concluded ha he resuls peraining o he fuzzy aheaical prograing approach ouperfor hose peraining o he sochasic prograing approach. Albei he auhors inenion in he above-enioned works is abiious, here are soe concepual and aheaical poins ha islead heir works. These poins will go hrough in he following secion. Afer a horough discussion abou hese iperaive poins, he uli-period porfolio selecion proble wih differen borrowing and lending raes is forulaed. Then, a fuzzy represenaion of he proposed odel is presened. Aferward, a nuerical exaple is used o ipleen he proposed fuzzy uli-period porfolio selecion odel wih differen borrowing and lending raes. The reainder of his paper is organized as follows: In he nex secion, soe iporan, concepual and aheaical poins abou uli-period porfolio selecion odels wih differen borrowing and lending raes are horoughly discussed. In he hird secion, regarding forer discussions, he uli-period porfolio selecion proble wih differen borrowing and lending raes is forulaed. A fuzzy varian of he proposed odel is presened in he fourh secion. In he fifh secion, he proposed fuzzy linear prograing odel is ipleened using a nuerical exaple, provided fro he lieraure. Finally, he las secion concludes he paper. A ore deailed discussion abou soe poins of relaed sudies in he lieraure As enioned above, here are a nuber of iporan, concepual and aheaical poins abou uli-period porfolio selecion odels wih differen borrowing and lending raes ha islead he works conduced in his area. Here, we ry o elaborae hese poins. Sadjadi e al. () and Seyedhosseini e al. () enioned ha he ransacion cos does no play an iporan role in he opiizaion resuls since any

5 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes brokerage houses are planning o reove ransacion coss in order o creae a oivaion o absorb ore invesen. This saeen does no ake sense in he real word. We know ha brokerage houses are paries in he capial arke ha faciliae he ransacions beween buyers and sellers. In reurn, hey receive a coission fee for each ransacion. These coission fees, referred o as ransacion coss, are he ain sources of incoe for brokerage firs. Hence, reoving ransacion coss is an unrealisic assupion ha cass doub on he oivaion of esablishing hese iporan eniies of capial arke. oreover, as ulvey e al. () indicaed, ransacion coss are one of iporan reasons ha necessiae exploiing uli-period porfolio selecion odels raher han ieraively solving single-period ones. In oher words, in he absence of ransacion coss, as well as soe oher condiions, one can consider he long-er invesen process as a nuber of ieraive single-period invesen decisions. In fac, reoving ransacion coss no only is a pracically unrealisic assupion, bu also cas doub on he necessiy of uilizing uli-period paradig for invesen decisions. Thus, ransacion coss are incorporaed o ake he presened odel ore confored o he real world applicaions. Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou () ake soe assupions ha do no see o be reasonable in he real world condiions. Their assupion on selling risky asses and invesing he provided proceeds in he risk free asse wih he lending rae akes sense for proceeds fro selling only hose ones held using he invesor s own capial. In oher words, invesing he cash provided fro borrowing and also proceeds fro selling risky asses purchased using loans in he risk free asse is no affordable. This is due o he fac ha he borrowing rae is assued o be greaer han he lending rae. Hence, i is no reasonable o borrow for invesen wih he lending rae. Consequenly, defining a variable for invesen in risk free asse using cash provided fro loans as well as selling he risky asses purchased by loans and a corresponding balance equaion does no see reasonable in real world siuaions. We eliinae his variable and is corresponding balance consrain. Insead, we assue ha he loan can erely be used o purchase risky asses. Also, we assue ha he proceeds fro selling risky asses purchased wih loans are uilized o repay he principal of loans, raher han invesen in risk free asses. This avoids paying addiional ineress o crediors. The ineres payens o crediors are furher considered in balance equaions. Even hough, using loans for purchasing risk free asse does no sound reasonable, however, if one borrows in order o purchase he risk free asse, he/she should receive soe proceeds wih he lending rae. However, in spie of allowing such an acion in Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou

6 Haed Davari, ajid Ainnayeri () auhors ignore hese proceeds in heir odels. Anoher iporan poin is ha Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou () only discuss abou he ineress of loans and do no ake he repayen of he principal of loans ino consideraion. To deal wih, we define addiional variables as well as consrains peraining o he ne liabiliies in differen periods. Ne erinal liabiliies are assued o diinish he oal uiliy of invesor as well. A odified odel is suggesed o cope wih he above-enioned poins. In his regard, we reforulae he uli-period porfolio selecion proble wih differen borrowing and lending raes. Also, as in Sadjadi e al. (), we uilize fuzzy se heory o presen a fuzzy varian of his proble in which he raes of reurns and raes of borrowing and lending are considered o be riangular fuzzy nubers raher han crisp nubers. The forulaed uli-period porfolio selecion odel To forulae he uli-period porfolio selecion odel wih differen borrowing and lending raes, in addiion o he noaions uilized in Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou (), soe new variables and paraeers us be inroduced. This is due o he fac ha, as enioned above, he aheaical forulaions presened in previous sudies us be necessarily odified fro boh concepual and aheaical poins of view. N r he nuber of risky asses (socks); he nuber of rading periods; he invesor s dollar holdings in asse a he beginning of period (funded wih his/her own capial), ( =,,,), ( =,,,N), where, = denoes he risk free asse; he invesor s dollar holdings in risky asse a he beginning of period (funded wih borrowing), ( =,,), ( =,,,N); he rae of reurn for risky asse over ie period (, + ), ( =,,,),

7 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes ( =,,,N-); b r l r u v he riskless borrowing rae over ie period (, + ), ( =,,,N-); he riskless lending rae over ie period (, + ), ( =,,,N-); he aoun of risky asse funded wih he invesor s own capial which is sold in period, ( =,,), ( =,,N-); he aoun of risky asse funded wih he invesor s own capial which is purchased in period, ( =,,), ( =,,N-); u he aoun of which is sold in period, ( =,,), ( =,,N-); v V U() L he aoun of risky asse which is purchased using loan in period, ( =,,), ( =,,N-); he axiu peried aoun of purchasing each risky asse in each period he invesor s uiliy funcion he proporional ransacion cos for selling risky asses he proporional ransacion cos for purchasing risky asses he ne borrowed capial invesed in risky asse up o he beginning of period, ( =,,), ( =,,,N), Noe ha his noaion is differen fro ; Noe ha, in addiion o inroducing soe new paraeers and variables, soe unnecessary variables used in Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou () have been reoved and ranges of soe indices have been odified. The invesor can inves in risky asses, i.e. socks, and one risk free asse. Recall ha b l he borrowing rae is greaer han he lending rae, r r, he proposed odel would be as follows:

8 Haed Davari, ajid Ainnayeri N N N N N N ax U ( L ) L () ( r )( u v ), (,..., N), (,..., ) () l ( r )( ( u )( ) v ( )), (,..., N) () L, (,..., ) () L L u ( ) v ( ), (,..., N), (,..., ) (5) ( u v )( r ) L ( r ), (,..., N), (,..., ) (6) b ( ), (,..., N), [,] (7) v V, (,,..., N ), (,..., ) (8) v, u, v, u, (,,..., N ), (,..., N) (9), (,,..., N), (,,..., N) (), L, (,,..., N), (,..., N) () where, eq. (), he objecive funcion, copues he erinal value of oal risky and risk free asse holdings inus he erinal liabiliy ha us be repaid o crediors. The par peraining o liabiliy has no been considered in Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou (). Noe ha he invesor is no allowed o use borrowing for invesing in risk free asses. The balance of invesor s dollar holdings, funded wih his/her own capial, in risky asses a each period is considered in eq. (). Also, eq. () considers his balance for invesen in he risk free asse. Noe ha eq. (), addiionally, considers purchasing and selling ransacion coss whose iporance were forerly discussed. Eq. s () and (5) denoe he balance of ne borrowed capial invesed in risky asses up o differen ie periods. Noe ha eq. (5) assues ha he proceeds fro selling risky asses purchased wih borrowing are uilized o repay he principals of

9 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes liabiliies. Besides, eq. (5) akes he purchasing and selling ransacion coss ino accoun. Eq. (6) denoes he balance of invesor s dollar holdings, funded wih borrowing, in risky asses a each period. Reurns of socks and payens of ineress peraining o he borrowed capial over ie periods are also considered in eq. (6). The balance beween oal invesen using he invesor s own capial and borrowing in each period is considered in eq. (7). Eq. (8) defines an upper bound for invesor o use his/her own capial for purchasing each risky asse in each period. In addiion, eq. (9) ensures ha he aouns of purchase and sale for risky asses are nonnegaive. oreover, eq. s () and () guaranee ha he invesor s dollar holdings in various asses and he aouns of his/her liabiliies in differen periods are nonnegaive. Noe ha hese non-negaiviy consrains have been ignored in Sadjadi e al. (), Seyedhosseini e al. () and Hassanlou (). The presened fuzzy uli-period porfolio selecion odel The raes of reurn for risky asses as well as borrowing and lending raes are considered riangular fuzzy nubers, r ( l,, n), whose ebership funcion is illusraed in fig.. This helps ake a fair coparison beween he provided resuls wih hose provided in Sadjadi e al. (). Figure The ebership funcion of r

10 Haed Davari, ajid Ainnayeri Siilarly, he α-cu on ebership funcions is ipleened o provide he α- level confidence of r in ers of inerval values corresponding o he riangular fuzzy nuber r ( l,, n) as follows: r [ r, r ] [( l) l, n ( n ) ], [,] () Thus, lower and upper bounds for α-level confidence can be siply provided. The fuzzy varian of he proposed uli-period porfolio selecion odel wih differen borrowing and lending raes is as follows: N N N N N N ax U ( L ) L ( ) ( r )( u v ), (,..., n), (,..., ) ( ) l ( r )( ( u )( ) v ( )), (,..., N) ( ) L, (,..., ) ( ) L L u ( ) v ( ), (,..., N), (,..., ) (5 ) ( u v )( r ) L ( r ), (,..., N), (,..., ) (6 ) b ( ), (,..., N), [,] (7 ) v V, (,,..., N ), (,..., ) (8 ) v, u, v, u, (,,..., N ), (,..., N) (9 )

11 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes, (,,..., N), (,,..., N) ( ), L, (,,..., N), (,..., N) ( ) Now, he α-level confidence of fuzzy nubers can be used o reforulae he fuzzy linear prograing odel as follows: ax U ( N N LN ) N N LN ( ),,,, ( [ r, r ])( u v ), (,..., n), (,..., ) ( ), l, l,, ( [ r, r ])( ( u )( ) v ( )), (,..., N) ( ) L, (,..., ) ( ) L L u ( ) v ( ), (,..., N), (,..., ) (5 ) ( u v )( [ r, r ]) L ([ r, r ]), (,..., N), (,..., ) (6 ),,, b, b,,,, ( ), (,..., N), [,] (7 ) v V, (,,..., N ), (,..., ) (8 ) v, u, v, u, (,,..., N ), (,..., N) (9 ), (,,..., N), (,,..., N) ( ), L, (,,..., N), (,..., N) ( )

12 Haed Davari, ajid Ainnayeri Using he lower bound for he rae of borrowing and he upper bound for he rae of lending as well as raes of reurn peraining o he risky asses, he upper bound of invesor s uiliy funcion can be provided. Siilarly, using he upper bound for he rae of borrowing and he lower bound for he rae of lending as well as raes of reurn peraining o he risky asses, we can provide he lower bound of invesor s uiliy funcion. Hence, o achieve an inerval associaed wih he invesor s uiliy for any α, i is enough o solve he crisp odel wice wih appropriae bounds of inervals. 5 Copuaional resuls and discussion Here, we consider he nuerical exaple used in Sadjadi e al. (), o ipleen he proposed odel. This exaple considers one risk free and four risky asses ( = ). Also, he proble has four periods (T = ). The borrowing and lending raes in hese four periods are respecively as follows: r [.8,.7,.8,.9] b r [.6,.7,.5,.7] l Also, raes of reurn for risky asses in hese four periods are as follows: r where, r ij denoes he fuzzy rae of reurn for risky asse i in period j. To ipleen he proposed odel using he daa, he paraeer β has been se o. Furherore, he iniial values of invesor s holdings have been subsiued wih he values used in Sadjadi e al. (). Now, eq. () can be uilized o deerine he confidence inerval of each riangular fuzzy nuber for [,]. In all riangular fuzzy nubers, we have n l.. The confidence inervals for all riangular fuzzy nubers have been illusraed in able. Noe ha able is a odified version of ha in Sadjadi e al. ().

13 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes Table α-level confidence of fuzzy nubers in each period N r [..8,..] [..9,..] [..7,..9] [..8,..] r [..8,..] [..8,..] [..9,..] [..7,..9] r [..7,..9] [..8,..] [..8,..] [..9,..] r [..9,..] [..7,..9] [..8,..] [..7,..9] l r [..5,..7] [..6,..8] [..,..6] [..6,..8] b r [..7,..9] [..6,..8] [..7,..9] [..8,..] Source: Auhors copuaions The reforulaed odel has been ipleened using GAS. considering hree differen confidence levels naely,.7 and. Table illusraes he copuaional resuls for hese hree confidence levels. Since raes of reurn for risky asses as well as borrowing and lending raes have been considered o be riangular fuzzy nubers, a confidence inerval of he objecive funcion for each α can be provided. Table shows ha he opial uiliy funcion of he invesor for α = is Also, i provides he confidence inervals of invesor s uiliy for α = as [979.76, 77.] and for α =.7 as [6.58,.98]. As α increases, he associaed confidence inerval becoes igher. Even hough he rend of changing objecive values is he sae as ha in Sadjadi e al. (), he objecive values obained fro solving he reforulaed odel are less han hose in Sadjadi e al. (). For, he fac ha he reforulaed odel conains soe odificaions copared o ha in Sadjadi e al. (); as adding ransacion coss, adding and reoving soe variables and consrains. Regarding all hese odificaions, of course,

14 Haed Davari, ajid Ainnayeri he ain reason for his reducion is adding he realisic assupion ha ne erinal liabiliies us be repaid o he crediors. I is obvious ha his assupion solely enails a decrease in he invesor s uiliy. Table α-level confidence inervals of objecive funcion and variables for differen values of α confidence level α = α =.7 α = U [979.76,77.] [6.58,.98] [7.95,7.95] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] [,] = = = [5,5] [5,5] [5,5] [,] [,] [,] [,] [,] [,] [,] [,] [,] [5,5] [5,5] [5,5] [,] [,] [,] [8.5,67.96] [7,7.65] [8.5,8.5] [,] [6.,79] [7,7] [8,6] [8,] [,] [55,555] [585,555] [55,55] [98,6] [8,] [,] [9.5,57.96] [,587.65] [58.5,58.5] [8.85,8] [976,] [,] [5,5] [57,5] [5,5] [,] [,] [,]

15 On uliperiod Porfolio Selecion wih Differen Borrowing and Lending Raes = = [97.,67.7] [8.878,58.6] [56.78,56.78] [99.,6] [69.86,58.97] [56.,56.] [6.,796] [68.796,7.876] [78.8,78.8] [58.5,69.5] [597.5,597.75] [59,59] [998.,66.6] [56.776,7.96] [8,8] [.55,58.69] [55.,5.9] [5.8,5.8] [8.,59.78] [9.9,5.6] [5.5,5.5] [.9,568] [595.9,5.79] [558,558] [,] [,] [,] [7.5,95.96] [568.5, ] [87.,87.] [8.8,9.] [56.97,95.9] [9.7,9.7] [99.9,575.6] [59.99,575.9] [5.59,5.59] [698.,665.5] [6.8,658.] [67.6,67.6] [6.85,.59] [9.8,8.] [88.56,88.56] [.8,558.8] [.8,57.88] [5.6,5.6] [5.56,655.55] [8.5,55.5] [58.6,58.6] [8.,555.8] [5.689,5.] [5.,5.] [,] [,] [,] [.9,9.97] [79.97,9.65] [9.78,9.78] [8.7,9.97] [55.89,8.] [.88,.88] [5.89, ] [558.99,578.67] [565.85,565.85] [678.88,75.5] [695.7,77.5] [ , ] [,8.75] [.,5.] [96.5,96.5]

16 Haed Davari, ajid Ainnayeri [, ] [6.67,576.98] [587.6,587.6] [,678.6] [69.,59.68] [55.57,55.57] [,565.7] [55.,57.99] [ , ] Source: Auhors copuaions using GAS. 6 Concluding Rearks This paper akes a closer look o he uliperiod porfolio selecion wih differen borrowing and lending raes. Firs of all, soe concepual and aheaical poins abou relaed sudies in he lieraure have been addressed. Aferward, noaions have been inroduced and he uli-period porfolio selecion proble has been forulaed. Furherore, we considered he raes of reurn for risky asses as well as borrowing and lending raes o be riangular fuzzy nubers, and used α-cu on ebership funcions o yield α-level confidence inervals for hese raes. Finally, he nuerical exaple used in Sadjadi e al. () was uilized o ipleen he forulaed odel and he copuaional resuls were presened. The copuaional resuls confired ha when he confidence level increases, he inerval peraining o he invesor s uiliy becoes igher. When he invesor seeks a confidence level equal o, his/her opial uiliy has a single value. Taking advanage of sochasic prograing odels, o deal wih underlying proble, is a proising research direcion in his area. In addiion, ransacion coss can be assued o change beween disinc periods. Furherore, o deal wih raes of reurn for risky asses and borrowing and lending raes, oher definiions of fuzzy nubers such as rapezoidal fuzzy nubers can be used. REFERENCES [] Barro, D. and Canesrelli, E. (9), Tracking Error: A ulisage Porfolio odel; Annals of Operaions Research, 65(), 7-66; [] Bersias, D. and Pachaanova, D. (8), Robus uliperiod Porfolio anageen in he Presence of Transacion Coss; Copuers & Operaions Research, 5(), -7;

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