Data Envelopment Analysis and Fuzzy Theory: Efficiency Evaluation under uncertainty in portfolio optimization

Transcription

1 Data Envelopment Analyss and Fuzzy Theory: Effcency Evaluaton under uncertanty n portfolo optmzaton Paulo Rotela Junor * Edson de Olvera Pamplona Anerson Francsco da Slva 2 Fernando Luz Rera Salomon Vctor Eduardo de Mello Valero Leonardo Alves de Carvalho Insttute of Producton Engneerng and Management Federal Unversty of Itajuba 2 Department of Producton Sao Paulo State Unversty BPS Avenue, 303, PO Box: 50, Itajuba-MG BRAZIL * Correspondng author: paulo.rotela@gmal.com Abstract: Ths artcle ams to analyze the behavor of a portfolo selected through Data Envelopment Analyss (DEA) assocated wth fuzzy logc and optmzed usng the Sharpe approach. As a bass for comparson, two other portfolos were used, one obtaned through only the Sharpe approach. In ths research study, a fuzzy DEA model was used to evaluate effcency under uncertanty of the Brazlan Stock Exchange - Bovespa, by means of nput and output ndcators such as return, varance, earnngs per share and prce-earnngs. The study relably dentfed whch effcent stocks and whch were most senstve to the effect of uncertanty. Through the comparson of portfolos, t was observed that the resultng combnaton of the fuzzy DEA models n whch the stocks were consdered effcent n both scenaros presented the best results. Key-Words: Stock selecton, Data Envelopment Analyss, Fuzzy Logc, Effcency, Sharpe approach. Introducton Stock nvestments are a great alternatve when compared to other nvestments, especally when observed over longer tme spans. However, ths hgher return s accompaned by a certan level of rsk. In order to maxmze the return, nvestors must seek the best ways to nvest ther captal, avodng rsks larger than what they are wllng to accept []. It s noted n recent years n Brazl that fxed ncome funds and savngs accounts are becomng less attractve due to drop n real nterest rates; the nvestment optons that am for better long term returns should attract nvestors n the comng years. Stock nvestments are becomng alternatves for dversfcaton for nvestors seekng a long-term ncrease n proftablty. The frst and stll well-recognzed model for portfolo selecton was created by Markowtz [2], beng a statc model for evaluaton of a sngle perod. Ths classc mean-varance approach s stll the man model used to allocate assets and manage portfolos. It has lead to new proposals developed by academcs [3-7]. For evaluaton and comparson of organzatonal unts, the technque known n the study of Operatonal Research as Data Envelopment Analyss (DEA), a technque developed by Charnes et al. [8], can be used to evaluate and compare organzatonal unts that use many dfferent nputs to produce varous outputs over a specfed perod of tme [9-]. The concept has been amply dscussed and varatons on t contnue to be developed. Accordng to Slva et al. [2], uncertan and approxmate reasonng can be consdered n DEA models; n order to do ths, researchers use technques related to fuzzy theory. Fuzzy DEA models are based on the model proposed by E-ISSN: Volume 2, 205

2 Lertworasrkul et al. [3], a model wth constant returns of scale wth fuzzy coeffcents. To frame the research problem, ths artcle ams to evaluate the effcency of stocks of publcly traded companes through the use of fuzzy DEA models n scenaros under uncertanty. And, as specfc objectves, ths artcle seeks to: - Utlze the Fuzzy DEA model to assst n reducng the search space by dfferent crtera; - Select a portfolo wth the stocks consdered effcent by the Fuzzy DEA model; - Utlze the Sharpe approach [4-5] to determne the allocaton of captal n stocks consdered effcent by the Fuzzy DEA model n order to mnmze rsk exposure to nvestors; - Apply the model whch ncorporates the concepts of the approach proposed by Sharpe [4-5] and compare them to the other models. 2 Portfolo Selecton The basc theory of portfolo selecton began wth Markowtz [2], n whch the selecton process s based on a model of a sngle nvestment perod. The proposal of Markowtz [2] model, gven by () - (3), s operated by Quadratc Programmng technques wth the objectve of portfolo optmzaton, takng nto account the mean, varance and covarance of the expected returns of the shares, optons that are to be part of the portfolo. These parameters are estmated from nformaton of a hstorcal seres based on a mean vector and a covarance matrx of ther returns. n mnz xx. cov Subject to: j = j = x n = () = j= E( r) = E * j j (2) x = (3) Where x and x j represent the percentage share of the asset and stock j n the optmal portfolo, E ( r ) s the expected return stock, = a j, and E * s the expected return of the portfolo. Sharpe [4] extended the work of Markowtz [2]. He ntroduced a smplfed model of the relatonshps between assets that offered evdence of costs [6]. It was also convenent to use the model for practcal applcatons, as n equatons (4)-(6) [2]: N + N + 2 λ XA X Q = = maxz = (4) Whch s subjected to: X 0 for each from to N. N XB = X n + (5) = N X = (6) = Ths formula shows why parameters A n+ and Q n+ are used to descrbe varance and the future expected value of I. Ths fact also shows why ths s called the dagonal model. The varance and covarance model, whch s complete when N assets are consdered, can be expressed as a matrx wth non-zero values on the dagonal, ncludng an (n+) th asset defned as has been shown [4]. Ths drastcally reduces the number of calculatons needed to solve the portfolo analyss problem. It allows the problem to be recognzed drectly n terms of the parameters of the dagonal model. Accordng to Ben Abdelazz et al. [7], the mean-varance methodology proposed by Markowtz [2] for portfolo selecton has been essental for the actvty of research and has served as the bass for the development of modern fnancal theory. In the lterature, some algorthms, such as those proposed by Sharpe [8], were created n order to lnearze and mprove the effcency of the covarance Markowtz [, 6, 8-9]. However, researchers have developed more sophstcated models that use mult-perod or dynamc extensons, n whch the assocaton between data envelopment analyss and fuzzy logc can be an alternatve, lke the model presented below. 3 Data Envelopment Analyss assocated to Fuzzy Logc The man purpose of Data Envelopment Analyss (DEA) s to evaluate the effcency of productve unts whch perform smlar tasks, called decsonmakng unts (DMUs). These unts are compared to each other and are dfferentated by amounts of resources (nputs) they consume and the goods (outputs) they produce [5, 20-22]. DEA models allow more than smple analyss of the DMUs that can be classfed as effcent; they also allow one to measure and locate the E-ISSN: Volume 2, 205

3 neffcency and estmate a lnear producton functon that provdes the benchmark for the DMUs classfed as neffectve. Ths benchmark can be establshed by the projecton of neffcent DMUs on the effcency fronter [22]. For Kao [] and Slva et al. [2], DEA enables the dentfcaton of DMUs that are references (benchmark) for the dentfcaton of other decson makng unts under analyss. Authors as Bal et al. [23] and Rotela et al. [] show that DEA has stood out among the realm of quanttatve modelng technques whch ad decson makng, assstng managers from varous felds, such as banks, schools, and hosptals [, 2, 24-25]. Charnes, Cooper and Rhodes [8] developed the frst DEA model to evaluate the effectveness of publc programs. Cooper et al. [20] assert that the nput and output varables for each DMU should be chosen to represent the nterest of managers, and there must be postve numerc data for each nput and output, and t s preferable to use a smaller number of nputs compared to the outputs. Outlned below are the classcal DEA models: CCR and BCC. Accordng to Malekmohammad et al. [26] and Mranda et al. [27], the frst, known as the DEA CCR model, or model wth constant returns of scale, proposed by Charnes et al. [8] as (7) - (): s max w = u. y (7) Subject to: m = s o r r0 r= v = (8) xo m u y 0 v xo 0 j =,2,..., n (9) r r r= = u r 0, r =,2,..., s. (0) v 0, =,2,..., m. () In whch j represents the ndex of the DMU, j =,..., n; r s the output ndex, wth r =,..., s; s the nput ndex, =,..., m; y rj s the value of the r- th output for the j-th DMU; x j s the value of the th nput to the j th DMU; u r s the weght assocated wth the r-th output; v s the weght assocated to the th nput; w o s the relatve effcency of DMU o, whch s the DMU under evaluaton; and y r0 and x o are the technology coeffcents of data arrays for nputs and outputs, respectvely [27]. When w o =, ths ndcates that DMU 0, under analyss, shall be consdered effcent when compared to the other unts ncluded n the model. If w o <, ths should be consdered an neffcent DMU [28]. Banker et al. [29] transformed the assumpton of constant returns to scale n the CCR model through a restrcton of convexty, creatng the DEA-BCC model that admts varable returns of scale, as seen n (2) - (6): s max wo = ur. yr0 + c0 (2) Subject to: m = s r= v = (3) xo m u y 0 v xo + c0 0 j =,2,..., n (4) r r r= = u r 0, r =,2,..., s. (5) v 0, =,2,..., m. (6) It s worth notng that to obtan the formulaton of the DEA-BCC model, an unrestrcted auxlary varable c 0, known as a scale factor, was nserted addtvely n equatons (2) and (4). Kao and Lu [30] develop a method to fnd the membershp functons of the fuzzy effcency scores when some observatons are fuzzy numbers [3]. Accordng to the defnton provded by Lertworasrkul et al. [3], Fuzzy Data Envelopment Analyss (FDEA or Fuzzy DEA) s a tool to compare the performance of a set of actvtes or organzatons under uncertan envronment. Inaccurate data n FDEA models are represented by fuzzy sets and FDEA models can take the form of fuzzy lnear programmng models. The FDEA models may be more realstc than models representng real envelopment analyss of conventonal data problems. However, there are some FDEA models that do not use fuzzy lnear programmng; such models are based on the concepts of upper and lower fronters [32-33]. In ther research, Hatam-Marbn et al. [34] cte four major approaches that deal wth the fuzzy DEA models: The tolerance approach; the α-level based approach; the Fuzzy rankng approach; and the Possblty approach. Hatam-Marbn et al. [34] and other authors such as Slva et al. [2] and Wen and L [3], beleve that the α-level approach s the most commonly used FDEA model, and s also the approach proposed by Kao and Lu [30]. E-ISSN: Volume 2, 205

4 Accordng to Chen et al. [35], when assessng rsk characterstcs and estmatng effcency, t s approprate to use the FDEA model, where nputs and outputs are non-specfc values. Hatam-Marbn et al. [34] and Mranda et al. [27] state that many researchers have formulated FDEA models to deal wth stuatons that present mprecse or vague data nput and output. The FDEA BCC model, whch admts varable returns of scale, s shown n (7) - (2): max E j ur. yr0 + c0 (7) r R Subject to: o I uy r rj vx j + c0 0 j J (9) r R I vx = (8) u 0, r R (20) r v 0, I (2) Where, accordng to Slva et al. [2], DMU 0 s beng analyzed; x 0 are the fuzzy varables of the - th entry of DMU 0 ; y r0 are the fuzzy varables of the r-th output of DMU 0 ; X represents the matrx of j fuzzy varables of the -th nput of the DMU j; represents the matrx of fuzzy varables of r-th output DMU j. The objectve functon (7) may have a greater value than due to constrants (8) and (9) whch nvolve Fuzzy parameters and are solved usng probablty [7, 20]. In ths artcle, wll be adopted the Hatam- Marbn et al [34] α-level based approach. The α Є [0, ] value allows to generate dfferent scenaros,.e., dfferent values of effcency, respectng the varaton range determned by the pertnence functon [2]. Based on the α-level approach, accordng to Slva et al. [2], when the value of α = 0, the parameter values are below average; that s, they wll approach the lower lmt value of the pertnence functon. However, when the parameter value s above average, t wll be exactly the values of the upper lmt of the pertnence functon. Yet accordng to the same authors, when the value of α =, there s a scenaro n whch the values of output parameters and nput data are formed by the average of the pertnence functon, then consderng the most probable values; that s, a scenaro that gnores uncertanty. For ths, n ths research were consdered two FDEA-BCC models used by Mranda et al. [28]. These two models, resultng from the Banker et al. [29] and Kao e Lu [30] approaches, analyses pessmst (22)-(26) and optmstc scenaros, accordngly (27)-(30), where c 0 s unrestrcted: max E j ur.(y r0 ρr0 α) + c0 (22) r R Subject to: v ( xo +Ψo α) = (23) I r R I u ( Y ρ α) r rj rj v ( X +Ψ α) + c 0 j J j j 0 (24) u 0, r R (25) r v 0, I (26) j r r0 + ρr0 α + 0 (27) r R max E u.( y ) c Subject to: v (X o Ψo α) = (28) I r R I u ( y + ρ α) r rj rj v ( X Ψ α) + c 0 j J j j 0 (29) u 0, r R (30) r v 0, I (3) Where, accordng to Mranda et al. [27], α s the value chosen for the α-level based approach, wth varaton [0, ]; Ψ o s the α coeffcent n constrants lnked to the -th Fuzzy nput for DMU 0 ; ρ rj s the α coeffcent n constrants lnked to the j-th Fuzzy output for DMU 0 ; ρ j0 s the α coeffcent for constrants lnked to the r-th Fuzzy output of the j-th DMU; and Ψ j s the α coeffcent for constrants lnked to the -th Fuzzy nput of the j-th DMU. E-ISSN: Volume 2, 205

5 4 Materals and Methods Ths research uses the scentfc bass developed by Markowtz [2] and Sharpe [4] to present a method for selectng stock portfolos. To select assets consdered effcent Fuzzy DEA model wll be used, as descrbed n Fgure. Fgure : Research flowchart The ntal sample was composed of stocks of publcly traded companes negotated on the São Paulo Stock Exchange (Bovespa), obtaned by consultng the database Economátca software. Intally, the set of nputs and outputs to be used n the analyss of effcency by DEA model were determned. Authors such as Powers and McMullen [36], and Lopes et al. [37] used ndcators wth returns after one, two and three years and earnngs per share to compose the set of outputs whle Beta (60 months), prce-earnngs and volatlty (36 months) to compose the nputs of the DEA model ndcators. However, Rotela et al. [] proposed replacng the ndcators that make up the nputs for volatlty n wndows of 2 months over years, 2 and 3, along wth prce-earnngs. Ths change n the rsk measurement ndcator set can provde nformaton that wll allow the DEA model to choose assets that are comparatvely analyzed and found to be effcent []. They treated attrbutes wth advantages as outputs and those wth costs as nputs for the model. 24 companes wth the largest partcpaton n Bovespa were selected for consderaton, regardless of ther classfcaton as ether common or preferred stocks. The sample could have been greater, but the mnmum number was selected to meet the requrement of DEA wth respect to the number of DMUs, consderng usng eght ndcators, four for nput and four for output. Data were collected usng Economátca software durng the perod from Aprl 20 to March 204. For each of the nput and output varables, the mnmum, average and maxmum values obtaned n the perod were dentfed. Ths artcle ams to evaluate the effectveness of assets (DMUs), n whch the nputs were consdered ndcators amng to mnmze, as shown n Table. The outputs, ndcators to be maxmzed, are shown n Table 2. Table : Trangular pertnence functons for nputs Table 2: Trangular pertnence functons for outputs Trangular pertnence functons were used to nsert uncertanty n the nput parameters (nputs) and output (outputs) of the FDEA model. Accordng to authors such as Slva et al. [2], Lang and Wang [38] and Aoun et al. [39] they represent human expertse to correctly judge the behavor of varables common n many practcal stuatons. It s noteworthy that the negatve data were treated as proposed by Cook and Zhu [40], whch adds the value that makes the most negatve value postve, wthout changng the model analyss Fuzzy DEA. For modelng the FDEA model, the software General Algebrc Modelng (GAMS ), verson and CPLEX solver n verson 2.2. were used. Wth the purpose of comparson, three portfolos were proposed. Then, the monthly returns of the past 36 months for each of the assets were utlzed. Data were collected wth the ad of Economátca software and treated n order to avod the presence of outlers. Effcency results based on the α-level are presented n Table 3. The values n parentheses represent the pessmstc and optmstc scenaro, respectvely, for the values for each DMU. In the artcles proposed by Kao and Lu [] and Mranda et al. [27], the authors use the geometrc mean between the optmstc and pessmstc DEA approaches to establsh an ndex of overall rsk, as s also proposed by Fare et al. [4] and Chn et al. [42]. The geometrc mean of the results presented by the α-level was used, then obtanng n Table 3, where the effcency for each of the scenaros s gven. Through analyss of Table 3, t s noted that only sx stocks were effcent n both scenaros. These assets consdered effcent n both scenaros were used n assemblng the portfolo denomnated FDEA-6. However, to put together the portfolo denomnated FDEA-, all the stocks consdered effcent n at least the optmstc or pessmstc scenaro were aggregated. Ths portfolo conssts of stocks. Table 3: Effcency values n α-level functons In both portfolos, after the mplementaton of the fuzzy DEA model, the Solver add-n for Mcrosoft Excel was used to optmze the model through the Sharpe approach for the stocks consdered effcent. Then the recommended weghts for each of the stocks comprsng the portfolos were determned. E-ISSN: Volume 2, 205

6 Fnally, a thrd portfolo, called the comparatve portfolo (Cp), n whch only Solver was used n Mcrosoft Excel to optmze the 24 stocks through the Sharpe Rato approach, obtanng the recommended weght for each stocks. It s noteworthy that there was no pre-selecton through FDEA modelng. Later, has begun the process of verfyng the exstence of abnormal returns. For ths, t was used the Captal Asset Prcng Model (CAPM) methodology, proposed by Sharpe [5], n whch t was possble to compare the portfolo n relaton wth expected return. To dentfy abnormal returns for the assets of each portfolo, t was calculated the dfference between the expected return and return provded by the CAPM observed durng the analyss perod. By usng the CAPM, n addton to obtanng the rate of return, could be quantfed the relatonshp between rsk and return for whch the company s stocks are subject [5]. Already about rsk, the CAPM conssts of two pllars, the dversfable or unsystematc rsk and un-dversfable or systematc rsk. In CAPM equaton, for the rsk-free rate (Rf), t was appled the Specal System of Clearance and Custody (Selc) average rate durng the perod from Aprl 200 and March 204, for beng consdered an ndcator of basc nterest rate of the economy of the Brazlan Federal Treasury. In revewng the performance of the expected return of the market (Rm), t was observed the behavor of Bovespa Index from the begnnng of 995 to the frst quarter of 204. Expected return of the market was then calculated based on the average of the observed years. Dscounted the nflaton rate, t was used value of 5.% p.a. for Selc average rate and 0.80% p.a. for (Rm) n the defnton of the expected return of portfolos. Next, t was calculated the Beta (β) term, n whch, for each of the three portfolos formulated, was calculated usng the weghted average of the ndvdual Beta multpled by the percentage of each stock. For ths, there was obtaned the ndvdual Beta for each asset wth the ad of Economátca software, durng the same perod used for the monthly returns. Fnally, could be possble to verfy f the portfolos resulted n returns above expected. For a more complete analyss, was also used the Sharpe Rato (SR), whch, accordng Rotela et al. [], Schuhmachera and Elngb [43] and Schuster and Benjamn [44] s a measure of effcency n the use of rsk to generate returns. The SR establshes a relatonshp between the excess return of a gven nvestment portfolo relatve to the rsk-free rate and nvestment rsk. Selc average n the prevously proposed perod was assumed as rsk-free rate, and standard devaton of monthly returns of portfolos as rsk measure. 5 Results and Analyss The result was the formaton of three portfolos, one usng only the Sharpe approach called comparatve portfolo (Cp), and two resultng from a selecton of stocks consdered effcent usng the FDEA model. It s noteworthy that after selecton wth the FDEA model, the Sharpe approach was used to determne captal allocaton for each stock. The goal was to buld a portfolo wthout sgnfcant addtonal rsks, vewed as a portfolo wth superor performance; that s, wth a rato of return and rsk better than the optmzed portfolo wth only the Sharpe model (Cp). Applyng only the Sharpe approach n the lst of 24 stocks wth the largest holdngs n the Bovespa Index, the comparatve portfolo (Cp) was obtaned consstng of 3 stocks. In assemblng the portfolo FDEA-, an evaluaton of the effcency of the 24 stocks (DMUs) was performed, and eleven were consdered effcent n at least one of the scenaros. The Sharpe approach was appled and only fve such assets composng the portfolo were obtaned. Fnally, to select the FDEA-6 portfolo, an effcency analyss of the ntal sample was performed and only sx stocks (DMUs) were consdered effectve n both the optmstc scenaro and pessmstc scenaro. The Sharpe approach was agan appled to dentfy the optmal allocaton of resources, and fnally obtaned the FDEA-6 portfolo, composed of only four stocks, as shown n Table 4. Table 4: Asset allocaton It s mportant to note that all four stocks used n assemblng the portfolo FDEA-6 are part of the portfolo FDEA-, whch can be expected, snce t s composed of assets that were effcent n both scenaros. However, t s nterestng to note that the same assets are also part of the group used n the composton of the comparatve portfolo (Cp). As n the work from Rotela et al. [], as the focus of optmzaton was to obtan effcent stocks usng the rsk measured by varance n smlar perods as nput varables, prce-earnngs rato, and as output varables the monthly ndex return and earnngs per share, t s beleved that the assets consdered E-ISSN: Volume 2, 205

7 effcent showed good relatonshps between the ndcators n the perod, postvely mpactng the result. Table 5 shows the standard devaton results, return and Sharpe Rato (SR), obtaned wth the proposed portfolos. Comparng the results obtaned by the SR, t can be seen that the comparatve portfolo (Cp) showed postve SI of On the other portfolo FDEA- showed an SR of And fnally for the FDEA-6 portfolo n whch the effcency was evaluated for the 24 ntally proposed stocks and sx were consdered effcent. Then these were optmzed by the Sharpe approach, obtanng the partcpaton of each of the stock n order to maxmze the SR. The optmzed portfolo showed an SR of 3.2, showng that the rsk offerng s rewarded, even surpassng the results of other portfolos, beng an attractve nvestment. Table 5: Results Table Performng a comparson between the values obtaned n each SR portfolos, one can observe the outperformance of portfolo FDEA-6 over the others, n whch an nvestor could earn a hgher premum for the rsk assumed. It can be seen, n Table 5, that the expected return of the three portfolos are near from each other, where the portfolo (Cp) presents (0.58%), FDEA- (0.59%) and FDEA-6 (0.59%) per month. The return effectvely earned by portfolos (Cp) (.78%), FDEA- (2.93%) and FDEA-6 (.98%), per month, demonstrate the exstence of abnormal returns compared wth ther expected returns obtaned by CAPM. The observed rsk measured by the standard devaton, as can be seen n Table 5, was vrtually dentcal n both portfolos put together through the FDEA model, as n the comparatve portfolo (Cp). However, the average monthly return surpassed the comparatve proftablty of the portfolo, formed only wth the ad of the proposed Sharpe, demonstratng a better return on rsk rato for portfolos usng the DEA model assocated wth the fuzzy approach. Another beneft that can be cted s the reduced number of stocks comprsng the portfolo FDEA-6, whch may ultmately produce savngs related to the cost of re-balancng the portfolo. The data from the 36 months used n the study were obtaned from the accumulated portfolos (Cp), FDEA-, FDEA-6, and the Bovespa Index return over the same perod. Thus, t was possble to observe that the portfolos outperform the Bovespa Index, as shown n Fgure 2. Optmzed portfolo FDEA-6 gets the best performance, wth a cumulatve return of 02.4%, followed by portfolo FDEA-, whch presents a cumulatve return of 98.25%. The portfolo (Cp) has a cumulatve return of 4.32% n the perod. Fgure 2: Return accumulated per portfolo Another advantage to be mentoned s that a purely determnstc model was not used; nstead, stocks were selected accordng to ther effcency through multple crtera, not just rsk and return, and consderng the uncertanty and mprecson, thus provdng a more robust analyss. 6 Conclusons Ths artcle amed to evaluate the feasblty and vablty of usng data envelopment analyss assocated wth fuzzy logc n order to select a portfolo and then optmze ts equty. For ths purpose, three portfolos were proposed: the frst smply by usng an optmzed Sharpe approach and the other two usng fuzzy logc assocated wth the DEA. Several studes, not only wth the use of DEA as well as other mathematcal models, treat the case as determnstc, beng accurate n ts determnaton of the nput and output varables. However, t s known that n practce, real world problems are naturally uncertan and mprecse. The goal was to obtan a model that consdered uncertanty and mprecson, and stll presented a better rato of rsk/ return, however, wthout leadng to a sharp ncrease n the varance. The applcaton of fuzzy DEA models was feasble, provdng an excellent breakdown of the analyss unts. It s worth notng that the fuzzy DEA model asssts n reducng the search space, snce t determnes whch assets should be consdered effcent through varous scenaros by reducng the number of stocks that wll be submtted to the Sharpe approach, for example. The well-recognzed Sharpe approach proves to be effcent n ts objectve; however, ts practcal use can be dffcult and even engaged n some stuatons, snce one obtans a large number of assets composng the portfolo holdngs and mpossble to be performed. Another beneft to be cted s the reduced number of stocks comprsng the FDEA-6 portfolo. Such reducton assocated wth the mantenance of the level of rsk under control tends to generate savngs related to the cost of re-balancng portfolos, provdng ndrect nvestor gans. E-ISSN: Volume 2, 205

8 Data Envelopment Analyss s already consdered to be part of a set of technques that assst n selectng assets for portfolo composton. Moreover, the combnaton of the FDEA approach can further assst n portfolo composton. The assocaton of the FDEA approach wth Sharpe has been successful, and two man advantages can be cted: a larger number of varables was used n the analyss to make t more robust, whch may accommodate dfferent ndcators; As a result, a reduced number of stocks composng the portfolo was obtaned, whch makes ts use possble and gave the best SR between the portfolos analyzed. As a suggeston for further research, the use of other ndcators and a long-term analyss are suggested. It s also recommended that a more comprehensve study usng a greater number of stocks to check whether the model would replcate the results obtaned n ths study. References: [] P. Rotela Junor, E. O. Pamplona and F. R. Salomon. Portfolo optmzaton: effcency analyss, RAE, v. 54, n. 4, 204. [2] H. Markowtz. Portfolo selecton, Journal of Fnance, v. 7, n., p. 77-9, 952. [3] T. Cover and D. Julan. Performance of unversal portfolos n the stock market: nformaton theory, Proceedngs of IEEE Internatonal Symposum on Informaton Theory, Sorrento, p. 232, [4] J. Tu and G. Zhou. Markowtz meets Talmud: A combnaton of sophstcated and nave dversfcaton strateges. Omega, v. 28, p , [5] J. Brode, I. Daubeches, C. Mol, D. Gannone and I. Lors. Sparce and stable Markowtz portfolos. Proceedngs of the Natonal Academy of Scences of the Unted States of Amerca, v. 06. n. 30, [6] F. Dtragla, J. Gerlach. Portfolo selecton: An extreme value approach. Journal of Bankng & Fnance, v. 37, n. 2, p , 203. [7] C. Zopounds, M. Doumpos, F. Fabozz. Preface to the Specal Issue: 60 years followng Harry Markowtz s contrbutons n portfolo theory and operatons research. European Journal of Operatonal Research, v. 234, n. 2, p , 204. [8] A. Charnes, W. Cooper and E. Rhodes. Measurng the effcency of decson-makng unts, European Journal of Operatonal Research, v. 2, n. 6, p , 978. [9] L. Kuo, S. Huang, Y. Wu. Operatonal effcency ntegratngthe evaluaton of envronmental nvestment: the case of Japan. Management Decson, v. 48, n. 0, p , 200. [0] A. Amrtermoor. A DEA two-stage decson processes wth shared resources. Central European Journal of Operatons Research, v. 2, p. 4-5, 20. [] C. Kao and S. Lu. Mult-perod effcency measurement n data envelopment analyss: The case of Tawanese commercal banks. Omega, v. 47, p , 204. [2] A. F. Slva, F. A. Marns and M. V. Santos. Goal Programmng and Data Envelopment Analyss combned wth Fuzzy Theory to evaluate effcency under uncertanty: applcaton n mn-factores of the auto parts segment, Gestão & Produção, ahead of prnt, São Carlos, 204, [3] S. Lertworasrkul, S. Fang, J. Jones and H. Nuttle. Fuzzy Data Envelopment Analyss (DEA): A possblty approach, Fuzzy Sets and Systems, v. 39, [4] W. F. Sharpe. A smplfed model for portfolo analyss, Management Scence, n. 9, 963. [5] W. F. Sharpe. Captal Assets prces: A Theory of Market Equlbrum under condtons of Rsk, Journal of Fnance, v. 9, 964. [6] S. Darolles and C. Goureroux. Condtonally ftted Sharpe performance wth an applcaton to hedge fund ratng. Journal of Bankng & Fnance, v. 34, p , 200. [7] F. Ben Abdelazz, B. Aoun, R. El Fayedh. Mult-objectve stochastc programmng for portfolo selecton, European Journal of Operatonal Research, v. 77, n. 3, [8] D. N. Nawrock and W. L. Carter. Earnngs announcements and portfolo selecton. Do they add value?, Internatonal Revew of Fnancal Analyss, n. 7, 998. [9] C. Shng and H. Nagasawa. Interactve decson system n stochastc mult-objectve portfolo selecton, Internatonal Journal of Producton Economcs, v. 60, 999. [20] W. Cooper, L. Seford and K. Tone. Data envelopment analyss: a comprehensve text wth models, applcaton, references and DEA- Solver Software, 2nd. ed. New York: Sprnger Scence + Busness, [2] L-J. Kao, C-J. Lu and C-C Chu. Effcency measurement usng ndependent component analyss and data envelopment analyss. E-ISSN: Volume 2, 205

9 European Journal of Operatonal Research, v. 20, p , 20. [22] W. Cook and J. Zhu. Data Envelopment Analyss A Handbook on the modelng of nternal structures and networks. New York: Sprnger Scence + Busness, 204. [23] H. Bal, H. Orkcu and S. Çeleboglu. Improvng the dscrmnaton power and weghts dsperson n the data envelopment analyss, Computer & Industral Engneerng, v. 37. n., 200. [24] P. Bachller. Effect of ownershp on effcency n Spansh companes. Management Decson, v. 47. n. 2, p , [25] J. Lu, L Lu, W-M. Lu and B. Ln. A survey of DEA applcatons. Omega, v. 4, n. 5, p , 203. [26] N. Malekmohammad, F. Lotf and A. Jaafar. Data envelopment scenaro analyss wth mprecse data. Central European Journal of Operatons Research, v. 9, p , [27] R. C. Mranda, J. A. B. Montevech, A. F. Slva and F. A. Marns. A new approach to reducng search space and ncreasng effcency n smulaton optmzaton problems va the Fuzzy-DEA-BCC, Mathematcal Problems n Engneerng, Volume 204, Artcle ID , 204. [28] J. Jablonsky. Multcrtera approaches for rankng of effcent unts n DEA models. Central European Journal of Operatons Research, v. 20, p , 202. [29] R. Banker, A. Charnes and W. Cooper. Some models for estmatng techncal and scale neffcences n Data Envelopment Analyss, Management Scence, v. 30, n. 9, 984. [30] C. Kao and S. Lu. Fuzzy effcency measures n data envelopment analyss, Fuzzy Sets and Systems, v. 33, [3] M. Wen and H. L. Fuzzy data envelopment analyss (DEA): Model and rankng method, Journal of Computatonal and Appled Mathematcs, v. 223, n. 2, [32] D. Desposts and Y. Smrls. Data Envelopment Analyss wth Imprecse Data, European Journal of Operatonal Research, v. 40, n., [33] L. Bond Neto, J. Mello, L Meza, E. Gomes and N Bergante. A geometrcal approach for Fuzzy DEA fronters usng dfferent T norms, WSEAS Transactons on Systems, v. 0, n. 5, p , 20. [34] A. Hatam-Marbn, A. Emrouznejad and M. Tavana. Taxonomy and revew of the Fuzzy Data Envelopment Analyss lterature: two decades n the makng, European Journal of Operatonal Research, v. 24, 20. [35] Y-C. Chen, Y-H. Chu, C-W. Huang and C-H. Tu. The analyss of bank busness performance and market rsk Applyng Fuzzy DEA, Economc Modellng, v. 32, p , 203. [36] J. Powers and P. McMullen. Usng data envelopment analyss to select effcent large market cap securtes, Journal of Busness and Management, v. 7, n. 2, p. 3-42, [37] A. L. Lopes, M. Carnero and A. Schneder. Markowtz na otmzação de carteras seleconadas por Data Envelopment Analyss DEA, Revsta Gestão e Socedade, v. 4, n. 9, , 200. [38] G. Lang and M. Wang. Evaluatng human relablty usng Fuzzy Relaton, Mcroelectroncs Relablty, v. 33, n., 993. [39] B. Aoun, J. Martel and A. Hassane. Fuzzy Goal Programmng Model: An overvew of the current state of the art, Journal of Mult- Crtera Decson Analyss, v. 6. n. 5, [40] W. Cook and J. Zhu. Data envelopment analyss: modelng operatonal processes and measurng productvty. Worcester: Create Space Independent Publshng Platform, [4] R. Färe, S. Grosskopf, M. Norrs and Z. Zhang. Productvty growth, techncal progress and effcency change n ndustralzed countres, Amercan Economc Revew, n. 84, p , 994. [42] K-S. Chn, Y-M. Wang, G. Poon and J-B. Yang. Falure mode and effects analyss usng a group-based evdental reasonng, Computers & Operatons Research, v. 36, n. 6, p , [43] F. Schuhmacher and M. Elng. Suffcent condtons for expected utlty to mply drawdown-based performance rankngs, Journal of Bankng and Fnance, v. 35, n. 9, p , 20. [44] M. Schuster and A. Benjamn. A note on emprcal Sharpe Rato dynamcs, Economc Letters, v. 6, n., p , 202. ACKNOWLEDGMENTS The authors would lke to express ther grattude to the Brazlan agences CNPq (Natonal Counsel of Technologcal and Scentfc Development), CAPES (Post-Graduate Federal Agency), and FAPEMIG (Foundaton for the Promoton of Scence of the State of Mnas Geras), whch have been supportng the efforts for the development of ths work n dfferent ways and perods. E-ISSN: Volume 2, 205

10 Table : Trangular pertnence functons for nputs DMU s Volatlty Year 3 Volatlty Year 2 Volatlty Year Prce-Earnngs DMU (7.87; 9.69; 0.70) (6.66; 7.54; 8.3) (5.65; 6.69; 7.72) (6.0; 9.74; 5.79) DMU2 (5.42; 5.9; 6.63) (6.87; 7.38; 7.88) (6.05; 7.07; 8.) (9.6;.2; 4.02) DMU3 (4.09; 6.46; 7.40) (3.49; 4.39; 5.05) (4.02; 5.96; 8.60) (9.4; 0.85; 2.63) DMU4 (3.; 4.49; 5.74) (5.55; 6.29; 7.78) (6.8; 7.38; 8.40) (4.32; 22.44; 28.44) DMU5 (6.2; 6.95; 7.56) (5.42; 7.00; 7.83) (3.79; 4.94; 6.68) (5.42; 7.59; 59.89) DMU6 (0.78; 2.35; 3.64) (6.82; 8.45; 0.3) (6.34; 7.35; 8.67) (6.73; 9.83; 6.44) DMU7 (5.0; 5.5; 6.28) (6.03; 6.72; 7.38) (6.03; 6.97; 7.92) (7.90; 9.8;.39) DMU8 (8.24; 9.20; 9.94) (6.0; 8.62; 0.22) (3.5; 5.32; 3.5) (3.58; 5.46; 6.94) DMU9 (4.25; 5.3; 6.35) (5.40; 5.96; 6.54) (4.0; 5.77; 6.34) (5.89; 36.03; 90.8) DMU0 (3.08; 4.75; 6.40) (5.40; 6.83; 7.95) (4.42; 6.00; 6.78) (0.9; 6.64; 20.80) DMU (6.27; 8.48; 0.6) (6.26; 8.2; 9.92) (7.34; 8.25; 9.6) (0.87; 7.09; 27.55) DMU2 (3.34; 4.6; 5.65) (2.87; 3.94; 5.0) (3.09; 3.9; 4.9) (9.7; 27.48; 34.83) DMU3 (6.6; 7.07; 7.77) (7.35; 8.24; 9.5) (6.36; 7.6; 8.60) (9.76; 46.07; 86.79) DMU4 (5.60; 6.9; 7.72) (3.02; 4.87; 6.34) (4.4; 5.65; 7.60) (7.77; 0.20; 3.3) DMU5 (5.32; 8.06;.49) (6.2; 0.8; 2.94) (4.77; 5.5; 6.26) (3.76; 7.32; 2.84) DMU6 (5.02; 6.54; 9.48) (4.79; 5.62; 4.79) (4.62; 6.05; 6.7) (4.93;.27; 20.2) DMU7 (3.60; 4.8; 5.62) (4.46; 5.93; 6.34) (2.67; 3.50; 4.38) (8.99; 2.00; 4.07) DMU8 (2.89; 5.22; 7.36) (6.94; 7.88; 9.05) (6.44; 7.39; 8.3) (7.39; 23.29; 36.50) DMU9 (4.83; 6.45; 0.62) (5.89; 0.24;.56) (3.8; 5.06; 5.84) (7.44; 4.65; 20.82) DMU20 (5.9; 8.06; 0.) (5.03; 5.92; 6.30) (5.62; 6.79; 7.67) (6.26; 9.57; 4.8) DMU2 (4.70; 6.30; 6.88) (5.0; 7.8; 9.53) (8.34; 9.24; 9.99) (6.95; 22.86; 29.39) DMU22 (5.3; 6.5; 7.24) (6.05; 6.93; 7.42) (5.57; 6.5; 7.09) (8.35; 24.99; 30.68) DMU23 (5.75; 7.; 8.3) (7.99; 9.37; 0.86) (9.44; 0.2;.2) (4.3; 24.88; 36.27) DMU24 (4.59; 5.3; 6.07) (5.4; 6.85; 8.30) (6.90; 7.84; 8.48) (7.59; 23.05; 29.20) E-ISSN: Volume 2, 205

11 Table 2: Trangular pertnence functons for outputs DMU s Return Year 3 Return Year 2 Return Year Earnngs per Share DMU (-2.32; -0.76; 0.72) (-3.00; -.07;.6) (-2.60; -.26; -0.43) (.42; 2.23; 3.50) DMU2 (-0.4; 0.89; 2.39) (-.8; -0.5;.07) (-2.48; -0.73;.8) (2.4; 2.69; 3.3) DMU3 (-.74; 0.09; 2.32) (-0.8;.02;.82) (-.39; 0.07; 2.0) (2.54; 2.68; 2.86) DMU4 (-0.68;.49; 2.68) (2.58; 3.83; 5.03) (.85; 2.97; 4.46) (0.52; 0.60; 0.67) DMU5 (-2.7; -0.7; 0.64) (-.93; -0.94;.05) (-2.27; -0.72;.29) (0.02; 4.43; 8.03) DMU6 (-2.49; -.47; -0.23) (-4.55; -.76;.8) (-2.48; -.59; -0.57) (.42; 2.23; 3.50) DMU7 (-0.72; 0.63;.95) (-.73; -0.9; 0.85) (-2.22; -0.63;.2) (0.85; 0.90;.05) DMU8 (-.96; 0.9; 3.55) (-2.39; -0.49;.22) (.98; -0.45;.58) (3.93; 5.55; 5.55) DMU9 (-0.0; 2.59; 4.6) (0.08;.05; 2.05) (.2; 2.24; 3.32) (0.43;.27;.84) DMU0 (.25; 2.5; 4.48) (.56; 3.69; 5.82) (-0.55; 2.57; 5.95) (2.23; 2.74; 3.4) DMU (-2.70; -0.82; 0.30) (-0.85;.; 3.29) (-4.69; -2.96; -.0) (0.62;.02;.4) DMU2 (-.97; 0.8; 2.73) (.9; 3.97; 4.76) (0.23;.53; 2.79) (0.35; 0.57; 0.79) DMU3 (0.68; 2.23; 3.77) (.09;.87; 2.89) (-0.78; 0.49; 2.36) (0.07; 0.63;.07) DMU4 (-.29;.79; 4.3) (-0.25;.55; 3.22) (-.05; 0.44; 2.28) (2.54; 2.68; 2.86) DMU5 (-2.48; -0.79;.2) (-.62;.85; 4.98) (0.50; 2.2; 3.82) (.80; 2.54; 3.59) DMU6 (-4.5; -2.05; 0.09) (0.52; 2.53; 3.49) (0.57; 2.42; 3.94) (.09; 2.7; 3.86) DMU7 (-0. 98; 0.49;.56) (-0.5; 0.57; 2.92) (2.60; 2.94; 3.44) (3.3; 4.2; 4.70) DMU8 (-.67; -0.30;.45) (-2.3; -0.5;.60) (.08; 3.9; 6.04) (0.88;.08;.33) DMU9 (-2.43;.59; 4.24) (-2.04; -0.34; 2.5) (.58; 2.45; 3.0) (0.53; 0.70;.4) DMU20 (-3.3; -.35;.27) (2.74; 4.62; 5.84) (.58; 2.77; 4.29) (.79; 2.22; 2.80) DMU2 (-2.76; -0.54; 2.34) (-0.6;.94; 4.54) (-.75; 0.89; 3.02) (2.57; 2.77; 3.27) DMU22 (-3.33; -.09; 0.9) (0.74; 3.00; 4.80) (.42; 2.99; 4.64) (0.76;.0;.) DMU23 (2.08; 4.5; 6.4) (0.37; 4.79; 7.39) (-3.6; -0.96;.67) (0.29; 0.44; 0.78) DMU24 (-2.80; -0.97;.64) (0.32; 2.97; 5.04) (-2.72; -0.75;.87) (.74;.92; 2.09) E-ISSN: Volume 2, 205

12 E-ISSN: Volume 2, 205 Table 3: Effcency values n α-level functons Pessmstc Optmstc Alfa (α) Stocks Effcency DMU (0.0; 0.32) (0.0; 0.32) (0.6; 0.34) (0.2; 0.37) (0.26; 0.39) (0.3; 0.40) (0.36; 0.4) (0.40; 0.42) (0.44; 0.43) (0.47; 0.43) (0.5; 0.4) DMU2 (0.8; 0.39) (0.8; 0.39) (0.87; 0.36) (0.92; 0.32) (0.98; 0.28) (.00; 0.24) (.00; 0.20) (.00; 0.7) (.00; 0.5) (.00; 0.3) (.00; 0.09) DMU3 (0.77; 0.44) (0.77; 0.44) (0.80; 0.4) (0.84; 0.39) (0.87; 0.37) (0.93; 0.34) (0.98; 0.32) (.00; 0.29) (.00; 0.27) (.00; 0.24) (.00; 0.22) DMU4 (.00; 0.35) (.00; 0.35) (.00; 0.30) (.00; 0.26) (.00; 0.22) (.00; 0.8) (.00; 0.4) (.00; 0.09) (.00; 0.04) (.00; 0.0) (.00; 0.06) DMU5 (0.03;.00) (0.03;.00) (0.06;.00) (0.09;.00) (0.2;.00) (0.5;.00) (0.8;.00) (0.2;.00) (0.24;.00) (0.27;.00) (0.30;.00) DMU6 (0.08; 0.26) (0.08; 0.26) (0.4; 0.28) (0.9; 0.30) (0.25; 0.32) (0.30; 0.34) (0.36; 0.35) (0.4; 0.36) (0.45; 0.38) (0.50; 0.38) (0.54; 0.37) DMU7 (0.5; 0.4) (0.5; 0.4) (0.57; 0.0) (0.62; 0.06) (0.68; 0.03) (0.73; 0.02) (0.78; 0.05) (0.82; 0.08) (0.86; 0.09) (0.90; 0.0) (0.92; 0.2) DMU8 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU9 (0.84;.00) (0.84;.00) (0.83;.00) (0.82;.00) (0.8;.00) (0.80;.00) (0.79;.00) (0.78;.00) (0.77;.00) (0.76; 0.98) (0.75; 0.96) DMU0 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU (0.04; 0.0) (0.04; 0.0) (0.07; 0.02) (0.0; 0.03) (0.3; 0.03) (0.5; 0.04) (0.7; 0.05) (0.9; 0.06) (0.2; 0.07) (0.23; 0.08) (0.25; 0.09) DMU2 (0.49; 0.6) (0.49; 0.6) (0.49; 0.6) (0.50; 0.6) (0.50; 0.62) (0.5; 0.62) (0.5; 0.62) (0.52; 0.62) (0.53; 0.62) (0.55; 0.62) (0.57; 0.62) DMU3 (0.72; 0.55) (0.72; 0.55) (0.64; 0.53) (0.65; 0.5) (0.68; 0.48) (0.7; 0.46) (0.76; 0.44) (0.82; 0.42) (.00; 0.39) (.00; 0.35) (.00; 0.3) DMU4 (0.95;.00) (0.95;.00) (0.92;.00) (0.90;.00) (0.89;.00) (0.92;.00) (0.95;.00) (0.95;.00) (0.94;.00) (0.94;.00) (0.94;.00) DMU5 (0.25;.00) (0.25;.00) (0.30;.00) (0.36;.00) (0.4;.00) (0.46;.00) (0.50;.00) (0.54;.00) (0.57;.00) (0.60;.00) (0.63;.00) DMU6 (0.06; 0.73) (0.06; 0.73) (0.; 0.82) (0.5; 0.92) (0.8;.00) (0.20;.00) (0.23;.00) (0.25;.00) (0.27;.00) (0.29;.00) (0.3;.00) DMU7 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU8 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU9 (0.28;.00) (0.28;.00) (0.27;.00) (0.3;.00) (0.33;.00) (0.35;.00) (0.37;.00) ( ) (0.40;.00) (0.4;.00) (0.43;.00) DMU20 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00; 0.86) (.00; 0.75) (.00; 0.67) (.00;.00) DMU2 (0.48; 0.38) (0.48; 0.38) (0.50; 0.36) (0.52; 0.35) (0.54; 0.36) (0.56; 0.37) (0.59; 0.37) (0.6; 0.39) (0.63; 0.40) (0.65; 0.40) (0.67; 0.43) DMU22 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU23 (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) (.00;.00) DMU24 (0.0; 0.08) (0.0; 0.08) (0.5; 0.08) (0.20; 0.08) (0.25; 0.08) (0.29; 0.08) (0.33; 0.07) (0.37; 0.07) (0.4; 0.06) (0.44; 0.06) (0.48; 0.05)

13 Table 4: Asset allocaton (Cp) FDEA- FDEA-6 DMU 0.00% - - DMU2 0.00% - - DMU3 0.43% - - DMU4.27% 0.00% - DMU5 0.00% 0.00% - DMU6 0.00% - - DMU7 0.00% - - DMU8 0.00% 0.00% 0.00% DMU9 5.6% - - DMU0 4.63% 20.72% 25.69% DMU.99% - - DMU2 0.94% - - DMU3 7.56% - - DMU4 0.00% 0.00% - DMU5 0.76% 0.00% - DMU6 0.00% - - DMU7 29.3% 3.05% 32.62% DMU8 7.98% 2.73% 8.22% DMU9 0.64%.% - DMU % - - DMU2 0.00% - - DMU % 0.00% 0.00% DMU % 24.39% 23.48% DMU % - - TOTAL 00% 00% 00% Table 5: Results table (Cp) FDEA- FDEA-6 Beta Expected Return 0.58% 0.59% 0.59% Standard Devaton 0.39% 0.39% 0.39% Return.78%.92%.98% Sharpe Rato (SR) Number of stocks E-ISSN: Volume 2, 205

14 Fgure : Research flowchart 20.00% 00.00% 80.00% 60.00% 40.00% 20.00% 0.00% % % (Cc) FDEA-6 FDEA- Ibovespa Fgure 2: Return accumulated per portfolo E-ISSN: Volume 2, 205