ENTROPİ OPTİMİZASYON ÖLÇÜSÜ İLE OPTİMAL PORTFÖY SEÇİMİ VE BİST ULUSAL-30 ENDEKSİ ÜZERİNE BİR ÇALIŞMA

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1 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı ENTROPİ OPTİMİZASYON ÖLÇÜSÜ İLE OPTİMAL PORTFÖY SEÇİMİ VE BİST ULUSAL-30 ENDEKSİ ÜZERİNE BİR ÇALIŞMA Doktora Öğrecs Görkem SARIKAYA Başket Üverstes Prof.Dr. Hüsey TATLIDİL Hacettee Üverstes Özet Fasal karar alma yötemlerde otmzasyo modeller gü geçtkçe öem arttırmaktadır. Güümüzde ortföy yöetcler, ortföyler oluştururke e yaygı moder ortföy teorem ola Markowtz Ortalama-Varyas (MV) yötem le mmum rsk ve maksmum getr düzeyde ortföyler oluşturmaktadırlar. Bu çalışmaı amacı alışıgelmş otmal ortföy yötem ola Markowtz Ortalama-Varyas (MV) model dışıda mumum etro ve maksmum etro ölçüsü le de otmal ortföye ulaşılı ulaşılamayacağıı test etmek ve hesalaa ortföy erformas değerler karşılaştırmaktır. Çalışmada BİST Ulusal-30 Edeksde yer ala hsse seetler le hem Markowtz Ortalama-Varyas(MV) teorem hemde Mmum etro ve Maksmum etro ölçüsü kullaılarak ortföyler seçlmştr. Öcelkle eşt ağırlıklı ortföy oluşturularak ortföyü getrs ve rsk hesalamıştır. Daha sora gerekl kısıtlar ekleerek, ortföyler ağırlıkları buluu mmu rsk ve maksmum getr düzeyde üç farklı ortföy oluşturulmuştur. Oluşturula ortföyler erformasları; Shar Oraı, Treyor Edeks ve Jese Ölçüsü le hesalamıştır.portföy ağırlıkları le lg geleceğe yöelk tahmler yaılmış ve elde edle souçları edeler belrtlmş olu, e uygu otmal ortföy ve e uygu otmzasyo yötem belrlemştr. Aahtar Kelmeler: Markowtz Ortalama Varyas(MV), Mumum Etro, Maksmum Etro, Otmzasyo Jel Kodu: G, C6, C58, C52 38

2 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı A STUDY ON OPTIMAL PORTFOLIO SELECTION VIA ENTROPY OPTIMIZATION METHOD AND APPLICATION WITH BIST NATIONAL-30 INDEX DATA Abstract The mortace of otmzato models o facal decso makg rocess has bee creasg over tme. Markowtz Mea-Varace Method (MV) s the most wdely used method by ortfolo maagers to estmate the mmum rsk ad maxmum retur o ther ortfolos. The urose of ths study s to determe whether a otmal ortfolo ca be formed by aother method, amely, mmum etroy ad maxmum etroy otmzato. By usg BIST Natoal 30 Idex data, three ortfolos are geerated ths study wth mmum etroy, maxmum etroy, ad Markowtz Mea-Varace (MV) methods. Frst, rsk ad retur of the equally weghted ortfolo s calculated. Subsequetly, ecessary costrats are added, the weghts of the ortfolos are determed, ad three ortfolos are formed at the mmum rsk ad maxmum retur codto. The erformaces of the ew ortfolos are calculated ad comared usg Shar Rato, Treyor Idex ad Jese Scale. Based o the results obtaed, the most arorate otmal ortfolo ad the otmzato method s determed ad also ortfolo weghts made redctos for the future ad the results obtaed dcated reasos. Key Words: Markowtz Mea Varace (MV), Mumum Etroy, Maxmum Etroy, Otmzato, Jel Classfcato: G, C6, C58, C52. Itroducto Today, vestors are terested the stock market order to ga roft. Idvdual ad sttutoal vestors should refer vestg more tha oe stock by creatg a ortfolo, rather tha a sgle stock, order to maxmze roft ad avod ossble rsks. The ortfolos that are created should ot be chose radomly, but rather at the maxmum level of retur ad mmum level of rsk. The reaso for dversfcato ortfolo theory s reducg rsk. The most commo method used for rsk calculato s stadard devato. Stadard devato s wdely used vestmet decsos because ts calculato s smle. Whe we talk about the cocet of dversfcato, the referece s the Markowtz Mea Varace (MV) model, whch s the most revalet moder ortfolo theorem today (Brka, 2006). The Moder Portfolo Theorem s based o the eed to maxmze returs from stock vestmets at a certa level of rsk. The frst study ths feld has bee 382

3 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı coducted by Hery S. Markowtz 952. I hs 952 artcle Portfolo Selecto, Markowtz studed the exected retur ad retur varace lmtato, ad clamed that the otmal ortfolo ca be created uder these lmtatos (Markowtz, 952). Markowtz Mea-Varace (MV) ortfolo otmzato theory s based o the mea-retur, varace ad covarace of the assets. However, the crtcsms towards the model clam that the weght estmates are based o the frst two momets obtaed from the samle ad therefore statstcal estmato error s robable (Altaylgl, 2008). Whe the mea ad covarace matrx of the chose samle s used ortfolo selecto, the weghts the ortfolo ca be very dfferet, a way that does ot corresod to the samle ad ths ca have a egatve effect o erformace (Bera ad Park, 2008). The statstcal errors the lead to msevaluato of the weghts of the ortfolo ad ths tur creates a rsk of wrog ratos of vestmet o the dfferet stocks. Varous studes have bee coducted order to reduce statstcal errors that mght arse the Markowtz Mea-Varace (MV) model. Frost ad Savaro 986, Joro 986, Ledot ad Wolf 2003, roosed the use of Shrkage Predctors order to reduce statstcal errors that mght arse the Markowtz Mea-Varace (MV) model. However, sce shrkage redctors are based o emrcal Bayesa aroach, t has ot bee easy to determe sutable a ror dstrbutos ad there s ot a arorate way to select these dstrbutos. I ther 2008 artcle Dversfcato Usg Maxmum Etroy Prcle, Bera ad Park roosed the cross etroy method order to elmate the ossble statstcal error (Bera ad Park,2008). The ma urose of ths study s to test whether the otmal ortfolo ca be created wth the measure of etroy ad the valdty of the ortfolos that are created by alcatos o stocks from the BIST Natoal-30 Idex. Frstly, the study does a ortfolo otmzato wth the moder ortfolo theorem, the Markowtz Mea-Varace (MV) method. The, otmal ortfolos were created by the maxmum etroy ad mmum etroy models, wth the ecessary lmtatos cluded. Performaces of the three dfferet ortfolos were calculated ad the sueror model was foud. Fally the otmzato rocesses for each of the three models were reeated wth the Pearso s skewess lmt added. Mcrosoft Offce Excel software was used for the calculatos eeded for the mathematcal models the study. 383

4 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı 2. Geeral Cocets 2.. Portfolo Aalyss 2.. Portfolo Uder curret codtos dvdual ad sttutoal vestors vest stocks order to crease ther rofts. The level of retur from a stock vestmet deeds o the rsk level take by the vestor. Ivestors am to vest at the mmum rsk level ad maxmum retur level that they ca bear durg the vestmet erod. The dea that stock vestmets have hgher levels of retur, ad the fact that the cost are relatvely low comarso to other vestmets, led dvdual ad sttutoal vestors towards stocks (Halc, 2008). The cocet of stock vestmet gave rse to the ortfolo cocet. I order to reach ther vestmet goal, or to maxmze returs, vestors eed to vest stocks that are arorate for them ad that are exected to crease value. However, there s a rsk elemet that they have to bear order to collect the exected returs. I order to reduce ths rsk, the umber of stocks eeds to be creased. Ths the leads to the eed for vestg multle stocks ad therefore creatg ad dversfyg a ortfolo (Halc, 2008). A ortfolo s the cumulatve value that a vestor created by vestg multle strumets that have smlar or dfferet characterstcs (such as currecy, gold, foreg currecy, tme deost or facal deosts such as bod, stock or treasury bll). The ortfolo structure ad ortfolo maagemet of dvduals or sttutos deed o ther tedeces to take rsk, refereces of lqudty ad retur rates of varous deosts. I other words, ortfolos are collectos of varous facal strumets, maly stocks, bods ad ther dervatves, held by a dvdual or grou (Korkmaz ad Ceyla, 2006: 47). The cocet of ortfolo selecto was brought forward by Hery S. Markowtz s 952 artcle ttled Portfolo Selecto. Markowtz Mea-Varace (MV) theorem, whch has bee used wdely utl today, s ow crtczed ad alteratve methods are beg derved Portfolo Maagemet Portfolo maagemet s the maagemet of multle vestmet strumets oe basket at the mmum level of rsk ad maxmum level of retur that s ossble wth the lmts determed by the real vestor (customer). The erso who s maagg ths ortfolo s called the ortfolo maager. 384

5 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı Otmzato-Dversfcato The urose of ortfolo maagemet s to reach the otmal ortfolo. Otmal ortfolo the, s creatg a balace betwee the exected returs ad rsks of the selected strumets from a certa umber of strumets (Kose, 200). The urose of ths balace s to rovde hghest retur wth mmum rsk uder market codtos. The vestmets rates o the selected strumets are determed wth ths cosderato. Ths stuato that the ortfolo maager s tryg to create s called ortfolo otmzato. Portfolo maagers try to crease the umber of strumets that they hold order to lower the rsk of the ortfolo. I other words, they ca reduce rsk by dversfyg ther ortfolo. There are two aroaches ortfolo dversfcato: the tradtoal aroach ad the moder aroach Tradtoal Portfolo Maagemet Aroach The urose of tradtoal ortfolo maagemet s to maxmze the retur of the vestmet. I order to do ths, t has bee argued that the ortfolo eeds to cota a lot of strumets ad that the rsk of the ortfolo ca be reduced as the umber of strumets crease. However, ths aroach assumes that the rsk of the ortfolo ca be lowered by dversfyg the ortfolo wthout takg to cosderato the relatoshs betwee the returs of the strumets cluded the ortfolo (Cetdemr, 2006) Moder Portfolo Maagemet Aroach Hery S. Markowtz s 952 artcle ttled Portfolo Selecto has bee the bass of moder ortfolo theory. Markowtz s Mea-Varace model valdated the tradtoal aroach because Markowtz clamed that lowerg the rsk of the ortfolo s more lkely va the drecto ad degree of the relatosh betwee the strumets the ortfolo rather tha merely cosderg dversfcato. Oe of the mortat ssues dversfyg the strumets the ortfolo s choosg egatve or zero correlato strumets rather tha ostve oes. Markowtz clams that dversfcato of ooste drecto correlato strumets s more effectve lowerg the rsk of the ortfolo (Cetdemr, 2006). 385

6 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı Basc Cocets Used I Portfolo Aalyss Stadard Devato ad Varace The cocets of Stadard Devato ad Varace defe rsk ortfolo aalyss. The lower the stadard devato s the lower the rsk. Stadard devato shows how much the retur of a stock devates from the exected retur. Therefore the stadard devato of a ortfolo shows how much the retur of a stock devates from the exected retur ad the am of ortfolo maagemet s to mmze stadard devato. Mea-Retur The mea of a ortfolo defes the retur of a ortfolo. I ortfolo maagemet the am s to maxmze retur. The exected retur a vestmet s the mea of the ast returs. Correlato Coeffcet Correlato s a coeffcet that dcates the relatosh betwee varats. Correlato value vares betwee - ad +. Whe the coeffcet s close to - t meas that there s a strog but ooste relatosh betwee varats whle values closer to + dcate strog ad same drecto (lear) relatoshs. As the calculated value gets closer to 0, the stregth of the relatosh s lower ad whe the coeffcet s 0 t meas that there s o relatosh betwee the varats. Covarace Matrx The matrx calculated order to create the covarace value s called the covarace matrx. The covarace value dcated the recrocal chages betwee two varats, a sese the relatosh tself. The relatosh takes a egatve or ostve value betwee - ad +. Beta Coeffcet The beta coeffcet s calculated as the rato betwee the covarace betwee the statstcal retur of the stock ad the retur of the ortfolo ad the varace of market retur. Beta coeffcet s a rsk measure. It measures the systematc rsk, the market rsk of a strumet. I other words, t s the relatosh betwee the erformace of a strumet ad the mea erformace of the market (Halc, 2008). It s the coeffcet of the chage the strumet resose to oe ut of chage the market. The beta coeffcet of the market s always cosdered to be. The beta of the ortfolo the, s equal to the weghted mea of the betas of the strumet the ortfolo. If the beta of the ortfolo s, ths meas that the ortfolo s movg arallel to the market. Whe t s bgger tha, 386

7 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı t meas that t wll fall or rse more tha the market. Whe t s smaller tha, t meas that t wll fall or rse lower tha the market. Portfolos that have betas hgher tha are more rsky comarso to ortfolos that have betas lower tha (Halc, 2008). Pearso s Skewess Coeffcet Skewess s a measure that gves formato about the shae of the dstrbuto ad t s a dcator related to symmetry. Skewess determes whether the dstrbuto s ostvely or egatvely skew. If the gve skewess value s equal to 0 the dstrbuto s sad to be symmetrcal, f t s lower tha 0 t s ostvely skew ad f t s bgger tha 0 t s egatvely skew. I ostve skewess the rght-had tal of the chart s loger. The mass of the dstrbuto has bee cocetrated o the left-had sde. Ths kd of dstrbuto called rght skewess. I egatve skewess, the left-had tal of the chart s loger ad the mass of the dstrbuto s cocetrated o the rght-had sde of the chart. Ths kd of dstrbuto s called left skewess. Skewess s calculated wth the formulas below (Brka, 2006). Skewess= 3*(Mea-mod)/Stadard devato or Skewess =3*(Mea-meda)/ Stadard devato () (2) 2.2. Portfolo Performace Evaluato Share Rato It measures the addtoal retur acheved er to oe ut of stadard devato. Therefore we ca say that the ortfolo that has hgher addtoal retur the same total rsk level has better erformace. Ths way t s determed whether the erformace of a strumet s lower or hgher that the market retur. Share Rato (SR) value s calculated wth these equatos (Brka, 2006). ( SO m = r -r Stdv ) m f ( SO r - r = Stdv ) f m (3) (4) 387

8 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı r m =market retur r f =rskless terest rate r = ortfolo retur Stdv m = stadard devato of the market Stdv = stadard devato of the ortfolo Treyor Idex Ths s the rato that measures the addtoal retur acheved er oe ut of systematc rsk take. Sce Treyor dex or measure reresets the rsk remum er systematc rsk, we say that the strumet that brgs more retur at a certa beta level, or the same rsk level, has hgher Treyor Rato or relatvely hgher erformace. Treyor Idex (TI) value s obtaed from the equatos below (Halc, 2008). TI = r m =market retur r =ortfolo retur r f =rskless terest rate β m =market beta β =ortfolo beta ( TI m = r - ) m rf (5) β m ( r - ) rf (6) β Jese s Measure It s defed as the dfferece betwee the exected retur rate of a ortfolo ad aother ortfolo that has the same rsk level o the strumets market le. Jese s Alfa whch measures the devato of ay ortfolo from the strumet market le, ca be cosdered as the dfferece betwee the actual retur ad the exected retur. If Jese alha s ostve, the Facal Asset s above the Market Le. If t s egatve, the Facal Asset s below the Market Le. Therefore, a ortfolo that has a ostve alha ca be cosdered to show a sueror erformace. Jese s Measure (JM) value s obtaed wth the equatos below (Bekc, 200). 388

9 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı JM m = r r + β *( r r )) (7) m ( f m m f JM = r r + β *( r r )) (8) ( f f r m =market retur r =ortfolo retur r f =rskless terest rate β m =market beta β =ortfolo beta Etroy Measure Etroy s a Greek word meag ambguty. It s the measure of the dsorder that exsts statstcally a system (Grftoglu, 2005). Etroy has bee frst defed by Jayes, who s the oeer of the robablty dstrbuto dervato method. Jayes method was amed Jayes Maxmum Etroy Prcle ad later develoed by Shao as a stochastc rocess. Accordg to Shao formato ca be gathered f a heomeo volves ambguty. Accordgly, occurreces of heomea that are lke to occur do ot rovde very much formato whle less lkely occurreces carry more formato (Grftoglu, 2005). I ths study, Jayes maxmum etroy rcle (MaxEt) whch maxmzes Shao s etroy measure ad Kullback s mmum cross etroy method (MxEt) has bee used order to use otmal ortfolo. Maxmum Etroy The method that determed the dstrbuto of the system by maxmzg Shao s etroy measure s called MaxEt ad the obtaed dstrbuto s called MaxEt dstrbuto (Jayes, 957). IN the MaxEt method, the cocet of ambguty s studed tesvely. The most objectve dstrbuto obtaed for a hyscal radom varat s the MaxEt dstrbuto (Jayes, 957). Probablty dstrbuto wth MaxEt s a dstrbuto where fereces are made based o mssg formato wth exstg data (Grftoglu, 2005). Sce a etroy value that s lower tha MaxEt wll ot rovde relable addtoal formato, every other dstrbuto wll have tedeces towards certa 389

10 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı behavor. Therefore t s ecessary to have a measuremet of a robablty dstrbuto ambguty (Grftoglu, 2005). For ths ambguty measuremet Shao defed a sgle fucto. Whle for each 0 ad = = = (, 2, 3,... ) s a robablty dstrbuto, the Shao measure of etroy for ths dstrbuto s gve H ()= - = l (9) I order to coverge ortfolo weghts to a uform dstrbuto, t s ecessary to maxmze Shao s etroy measure (Kaur ad Keseva, 992). Because t s thought that the covergece of the ortfolo weghts to a uform dstrbuto ca reach a welldversfed ortfolo whe the etroy measure s maxmzed uder lmtato of a certa rate of retur (Altaylgl, 2008). Certa lmtatos should be rovded order to create a well-dversfed ortfolo. These lmtatos are; whle etroy fucto, H()= - = Ad = = g r l x ( ) = = η (0) g = r r = () ad t s thought that uder these lmtatos otmal ortfolo ca be reached by maxmzg Shao s etroy measure. Mmum Etroy Mmum cross etroy method s a otmzato method that was oeered by Kullback ad t cometes wth the MaxEt method both theory ad ractce. Ths rcle s also kow as the mmum rectfed devato rcle or mmum formato dscrmato rcle. As the maxmum etroy method, the cocet of ambguty s ot utlzed the mmum cross etroy method. However there s a relatosh betwee 390

11 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı the two methods. Mmum cross etroy s the cross etroy of robablty dstrbuto to aother robablty dstrbuto q. I ractce the dstace s the dstace betwee a determed robablty dstrbuto to the uform dstrbuto u (Grftoglu, 2005). u s the dstrbuto that has the hghest ambguty. The closer a dstrbuto s to u, the hgher the ambguty. Ths dstace gves us a dea about the sze of the ambguty ad q s a a ror dstrbuto. Sce formato s a a ror dstrbuto of momet values derved from hyscal radom varats ad these radom varats, the method that mmzes Kullback-Lebler measure accordg to the momet value lmtatos derved from radom varats ad the a ror dstrbuto s called MxEt method ad the dstrbuto obtaed here s called MxEt dstrbuto (Kullback, 959). If momet codtos were gve before the closest dstrbuto to the a ror dstrbuto s selected. If uform dstrbuto s selected as a ror dstrbuto the MaxEt method aears as a secal result of the MxEt method (Usta, 2006). It mmzes a robablty dstace from a dstrbuto gve MxEt whle t maxmzes ambguty MaxEt (Katar, 2006). Uder Kullback- Leber crtera = Ad = r = g ( x ) = η r ; r=,2,..m (2) lmtatos for each, 0, D(:q)= = l q fucto s mmzed. (3) If q s ot gve, the dstrbuto that has maxmum etroy s selected stead of q. Whe there are o lmtatos u s acceted as the a ror dstrbuto ad, D(:q)= = l (4) equato s mmzed. Wth ths ersectve Jayes MaxEt rcle s cosdered to be a secal case of Kullback s MxEt rcle (Grftloglu, 2005). 39

12 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı 3. Alcato 30 stocks from BIST-30 Idex have bee studed order to comare the three models that are created ths study. Daly data betwee February, 203 ad February 28, 203 have bee used. Daly corrected closg rces of the stocks were obtaed from Osmal Mekul Comay. The otmzato rocess was coducted wth the Solver commad Excel. MV, MaxEt ad MxEt models were created wth the assumto that there s o ossblty of sellg the stocks oely. MV Model; For =,2,3..30, : weght, x : stock retur. ad uder lmtatos 0 = = ve 0...(5) the urose fucto; M = x 2 2 { E( x)} (6) MaxEt; = ad = = ; 0...(7) r g ( x ) =η r ; r=,2,..m (8) urose fucto; Max ( - MxEt = ad = l )...(9) = ; 0...(20) = g r ( x ) = η r ; r=,2,..m (2) urose fucto; M = l (22) q 392

13 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı otmzato has bee doe. I the mmum etroy model, ortfolo weghts were selected radomly so that the weght total s as a ror dstrbuto. The summarzg statstcs of the vestmet strumets are gve o Table. AKBNK ARCLK ASELS ASYAB BIMAS DOHOL EKGYO ENKAI EREGL GARAN HALKB IHLAS IPEKE ISCTR KCHOL Mea -0, ,0009 0, , , , , , ,0037-0, , , , , Stadard 0, , , , , , , , , , , , , , , Error Meda Mode Stadard 0, , , , , , , , , , , , , , , Devato Samle 4,366E-05 5,485E-05 5,867E-05 5,25E-05 3,56E-05 6,228E-05 0, ,639E-05 2,974E-05 5,6E-05 5,797E-05 0, ,7E-05 6,555E-05 4,349E-05 Kurtoss, , , , , , , , , , , , , ,835043,85963 Skewess -,576 0, , ,6685-0, , , , , , , ,848766, , , Rage 0, , , , , , , , , , , , , , , Mmum -0, , ,002-0, ,0399-0, , ,045-0, , , , , , ,03906 Maxmum 0, , , , , , , , ,0845 0, , , , , , Sum -0,0753-0, , , ,0007-0, , , ,0307-0, , , , , Cout KOZAA KOZAL KRDMD MGROS PETKM SAHOL SISE TCELL THYAO TOASO TTKOM TTRAK TUPRS VAKBN YKBNK Mea -0, , ,0004 0, ,8E-05-0, , , ,0027 0, ,33E-05 0, , , ,00062 Stadard 0, , , , , , , , ,0088 0, , , , , , Error Meda Mode Stadard 0, , , , , , , , , , , , , , , Devato Samle 4,22E-05 4,872E-05 8,53E-05 3,292E-05 7,8E-05 2,37E-05 4,8E-05 2,434E-05 8,863E-05 7,875E-05 2,599E-05 6,909E-05 3,54E-05 6,473E-05 6,273E-05 Kurtoss 2, , ,747098, ,299 0, , ,80379, , , , , ,235304, Skewess -0, , , ,3894 0, ,6578-0, , , , , , , , ,97852 Rage 0, , , , , , , , , , , , , , ,03087 Mmum -0,02-0, , ,0542-0,0524-0,0097-0, , , , , , , ,0827-0,0289 Maxmum 0, ,059 0, , , ,0069 0,0295 0,0385 0, , , , , , , Sum -0, , ,0065 0, , , , , ,0576 0, ,0069 0, , ,0568-0,0677 Cout Table : Summarzg Statstcs of the Stocks The study frst comared ortfolo otmzato wth Mea- Varace (MV), Maxmum Etroy (MaxEt) ad Mmum Etroy (MxEt) models ad stadard devato values obtaed at a certa retur level. The erformace of each ortfolo was calculated wth the data obtaed ad erformaces were evaluated. Table 2: Equal Weght Portfolo For the equal weght ortfolo created wth stocks from BIST-30 Idex, the retur s calculated as ad rsk as The fact that the retur of the ortfolo s egatve s thought to be the result of the baks that s the BIST-30 Idex, sce there s 393

14 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı a exectato that the Cometto Isttuto wll ush bak the moth of February ad ths had a egatve mact o the stocks. Table 3: MV, MaxEt ad MxEt Stadard Devatos at Dfferet Portfolo Retur Levels Whe stadard devatos of the ortfolos created at the same retur level, t s foud that the most rsky ortfolos are created by the MxEt model whle the least rsky ortfolos are created by the MV model. The reaso for the ortfolo returs beg hgher tha the equal weghted ortfolo s the fact that the bakg sector s ot cluded the ortfolos created wth the MV model, whle bakg sector stocks have very low weght MaxEt ad MxEt models. 394

15 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı Table 4: Portfolos Created at Dfferet Retur Levels The retur of the otmal ortfolo created wth the MV model s whle the rsk s The retur of the otmal ortfolo obtaed by maxmzg the etroy measure s whle ts rsk s calculated as As t ca be see the MaxEt mode s both more rsky ad brgg lower returs tha the MV model. It s also foud that the ortfolo retur obtaed by mmzg etroy measure s the same as the MxEt model, but the ortfolo s more rsky. Lookg at the created ortfolos; the stocks that are at the retur level wth the MV model are; ASELS, EKGYO, IPEKE, KOZAL, KRDMD, PETKM, SAHOL, TCELL, THY, TTRAK ad TUPRS. I the MaxEt ad MxEt models o the other had, 29 stocks that are the BIST-30 Idex, excet IHLAS, have the same retur level. Table 5: Portfolo Weghts at the Retur Level Table 6: Portfolo Performaces at Dfferet Retur Levels 395

16 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı The ortfolo erformaces at the same retur level are calculated accordg to the Share Rato, Treyor Idex ad Jese Measure. Whe the erformace measures are comared, the erformaces of the ortfolos that are created the MV model ca sad to be hgher tha ortfolos created the MaxEt ad MxEt models. Fgure : Effcet Froter of MV, MaxEt, MxEt I addto to the lmts used aalyss, Pearso Skewess measure was added the secod art of the study ad otmzato was reeated for MV, MaxEt ad MxEt models. The urose here s to show how the skewess measure affects our ortfolo. The reaso for tryg to fd the weghts of the ortfolo retur seres that wll create egatve skewess s the thought that ostve retur vestmets ca be cluded the ortfolo (Altaylgl, 2008). Pearso s Skewess Measure: 3*( R ~ R Stdv) <0 (23) R =Portfolo Retur ~ R = Portfolo Meda 396

17 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı Table 7: MV, MaxEt ad MxEt Stadard Devatos at Dfferet Retur Levels Portfolos otmzed by added Pearso s skewess lmt are comared at the same retur level ad the lowest rsk level at each retur level s foud the MV model. Table 8: Portfolos Created at Dfferet Retur Levels The retur of the otmal ortfolo created wth the MV model s whle ts rsk s foud as Although slghtly, the skewess lmt creased the rsk. The retur of the otmal ortfolo created by maxmzg etroy s whle ts rsk s calculated as It s see that the MaxEt model s both more rsky ad brgg less retur that the MV model. Whe the etroy measure s mmzed the ortfolo retur s same as the MaxEt model but the ortfolo rsk creases. I the MV 397

18 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı model, the stocks that are cluded the ortfolo at the retur level are; ASELS, EKGYO, EREGL, IPEKE, KOZAL, KRDMD, PETKM, SAHOL, TCELL, THY, TTRAK ad TUPRS. I the MaxEt ad MxEt models o the other had, 29 stocks that are the BIST-30 Idex, excet IHLAS, have the same retur level. Table 9: Portfolo Weghts at the Retur Level Table 0: Portfolo Performaces at Dfferet Retur Levels Whe ortfolo erformaces are comared at the same retur levels wth Pearso s skewess lmt s added to otmzato, the erformaces of the ortfolos created the MV model ca sad to be hgher tha ortfolos created MaxEt ad MxEt models. 398

19 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı Fgure 2: Effcet Froter of MV, MaxEt, MxEt 4. Dscusso Ad Cocluso I ths study, ortfolos are created wth MV, MaxEt ad MxEt models by usg stocks from the BIST-30 dex. Frst of all retur ad rsk has bee calculated after equal weghted ortfolo s created. Later, lmts for each model were added order to create ortfolos at certa retur levels. The otmal ortfolo was foud ad ts retur ad rsk was calculated. Fally erformaces of the three models created at the retur level of the otmal ortfolo. As a result of the alcato, we are able to say that the MV model s more otmal tha ortfolos created wth MaxEt ad MxEt models at the same retur level. Portfolos created wth the MV model are both less rsky ad hgher erformg. MaxEt ad MxEt models gve almost equal results. However, f we eed to comare, MaxEt model ca sad to be more otmal tha the MxEt model. MV model s also sueror to the other two whe Pearso s skewess lmt s added to all three models as the otmzato lmt. However, the effect of Pearso s skewess lmt s very low o all three models. It rases the rsk of the ortfolos but ths crease s gorably low. I other words, we ca say that Pearso s skewess lmt has o effect o the models. Aother mortat fdg s that dversfcato s hgher MaxEt ad MxEt models tha the MV model, meag the umber of stocks cluded the tha the MV model. I the lght of the fdgs, assumg that egatve exectatos about the bakg 399

20 Dumluıar Üverstes Sosyal Blmler Dergs EYİ 203 Özel Sayısı sector for the moth of February wll ot be reflected the followg moths, mothly returs of baks wll crease ad ther stocks wll be cluded more ortfolos ad/or ther weghts wll crease ortfolo otmzatos that wll be coducted T erod of tme later. REFERENCES Altaylgl, B., (2008). Portföy seçm ve ortalama-varya-çarıklık model. Joural of Istabul Uversty Faculty of Ecoomcs, 2, Bekc, İ. (200). Otmal ortföy oluşturulmasıda bulaık doğrusal rogramlama model ve İMKB de br uygulama. PhD Thess, Süleyma Demrel Uversty, Isarta, Turkey. Bera, A.K. & Park, S.Y., (2008). Otmal ortfolo otmzato usg maxmum etroy. Ecoometrc Revew, 27, 4-6, Brka, E.(2006). Portföy Seçm: Uluslararası hsse seetler ortföylere uygulaması. Graduate Thess, Zoguldak Karaelmas Uversty, Zoguldak, Turkey. Ceyla, Al., & Korkmaz, T. (2006). Sermaye yasası ve mekul değerler aalz.(3rd Edto) Ek Publshg House, İstabul. Çetdemr, A. E.(2006). Otmum ortföy seçm ve İMKB-30 edeks üzere br uygulama. Graduate Thess, Marmara Uversty Isttute of Bakg ad Isurace, Istabul, Turkey. Frost, P. A., & Savaro, J. E. (986). A emrcal Bayes aroach to ortfolo selecto. J. Fac. ad Quat. Aalyss 2, Grftoglu, Ç.(2005). Kesskl rassal değşkeler ç etro otmzasyo resler ve uygulamaları. Graduate Thess, Aadolu Uversty, Eskşehr, Turkey. Halc, B.(2008). Portföy seçm roblem üzere karşılaştırmalı alteratf yaklaşımlar. Graduate Thess, Gaz Uversty, Akara, Turkey. 400

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