Portfolio selection problem: a comparison of fuzzy goal programming and linear physical programming

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1 A Iteratioal Joural of Optimizatio ad Cotrol: Theories & Applicatios Vol.6, No., pp.-8 (6) IJOCTA ISSN: eissn: DOI:./ijocta Portfolio selectio problem: a compariso of fuzzy goal programmig ad liear physical programmig Fusu Kucukbay a ad Ceyhu Araz b a Celal Bayar Uiversity, Faculty of Busiess Admiistratio, Ecoomy ad Fiace Departmet, Maisa, Turkey b Celal Bayar Uiversity, Faculty of Egieerig, Idustrial Egieerig Departmet, Maisa, Turkey fusu.gokalp@cbu.edu.tr, ceyhu.araz@cbu.edu.tr (Received November, 5; i fial form April, 6) Abstract. Ivestors have limited budget ad they try to maximize their retur with miimum risk. Therefore this study aims to deal with the portfolio selectio problem. I the study two criteria are cosidered which are expected retur, ad risk. I this respect, liear physical programmig (PP) techique is applied o Bist stocks to be able to fid out the optimum portfolio. The aalysis covers the period from April 9 to March 5. This period is divided ito two; April 9-March 4 ad April 4 March 5. April 9-March 4 period is used as data to fid a optimal solutio. April 4-March 5 period is used to test the real performace of portfolios. The performace of the obtaied portfolio is compared with that obtaied from fuzzy goal programmig (FGP). The the performaces of both method, PP ad FGP, are compared with BIST i terms of their Sharpe Idexes. The fidigs reveal that PP for portfolio selectio problem is a good alterative to FGP. Keywords: Portfolio selectio problem; liear physical programmig; fuzzy goal programmig. AMS Classificatio: 9C7, 9C5. Itroductio The purpose of ivestors is to maximize the total retur of their ivestmets while cosiderig the risk factor. I order to miimize the risk, a portfolio cocept has arise. Ivestig fuds ito a portfolio istead of oe asset may be less risky because poor performace of oe ivestmet istrumet ca be easily balaced with good performace of aother ivestmet istrumet. I order to maximize the retur of assets i their portfolio, ivestors eed to maage the portfolio efficietly []. Portfolio maagemet ca be defied as the allocatio of the fuds betwee the securities for esurig the maximum retur ad miimum risk []. I real world, the ambiguity exists because of ucertaity ad the lack of efficiet iformatio. Therefore, portfolio selectio problem is a challegig problem for researchers. Ad various studies have bee doe so far about portfolio selectio problem. Moder portfolio optimizatio studies bega with the work of Markowitz i the 95 s. Markowitz suggested a mea-variace model. Markowitz studied how to esure a portfolio that icludes stocks with maximum retur at a give level of risk [3]. Markowitz portfolio optimizatio model was the source of ispiratio to may studies ad was theoretically mostly kow model, but the model was criticized for the eed of gatherig accurate iformatio ad the large umber of calculatios [4, 5]. Several authors have tried to develop Markowitz moder portfolio theory. Sharpe [6] proposed to estimate the total risk of market Correspodig Author. fusu.gokalp@cbu.edu.tr

2 F. Kucukbay, C. Araz / Vol.6, No., pp.-8 (6) IJOCTA istead of estimatig the risk of each stock with simple regressio model. ater o moder portfolio theory was elaborated with Sharpe [7], itler [8], Ross [9], Huberma [] by proposig capital asset pricig ad the multifactor arbitrage pricig models. Sharpe [] ad itler [] developed Capital Asset Pricig Model (CAPM). I this model differet from moder portfolio theory ivestors have the opportuity to ivest i risk-free assets. Ad this theory was evolved with arbitrage pricig theory which was proposed by Ross [3] ad exteded by Huberma [4] I later years, there were studies that try to trasform the quadratic problems ito liear problems such as [5], [6], [7] ad [8]. Oe of the most popular of them was mea absolute deviatio model. Koo ad Yamazki [9] proposed a alterative model to quadratic models called as mea absolute deviatio model. I this model, they accepted absolute deviatio as the risk measure istead of the stadard deviatio. May researchers try to exted portfolio selectio problem by usig liear models such as maximum model [] miimax model [, ]. I real world, ucertaity exists for determiig the expected retur ad expected risk of stocks, therefore some researchers have devoted cosiderable efforts to deal with the vague aspiratios of a decisio maker usig fuzzy set theory. Whe the iformatio about the objectives is aturally vague, Fuzzy goal programmig (FGP) approach lets the ivolvemet of decisio makers (DMs) to the determiatio process of imprecise aspiratio levels for the goals. FGP have already bee applied to the portfolio selectio problem by Parra et al. [3] ad Aliezhad et al. [4]. I order to obtai the optimum stocks for portfolio, this study proposes to use liear physical programmig (PP) approach. I PP approach, DM ca take i accout differet goals ad determie these goals i differet desirability levels such as ideal desirable, tolerable, udesirable, highly desirable ad uacceptable. The major advatage of liear physical programmig is its capability of takig ito accout of umerous costraits, umerous goals ad cosiderig the preferece rage for the goals [5]. To show the effectiveess of the use of PP i portfolio selectio problem, FGP was also applied to the problem ad the results of the both methods are compared. The remaider of this paper is orgaized as follows. The secod sectio briefly explais the liear physical programmig method. Mathematical modellig of portfolio selectio problem will be preseted i the third sectio. I the fourth part, a real life portfolio selectio problem will be solved uder two coflictig objectives: maximum retur ad miimum risk possible. Fially coclusio ad further research will be discussed... iear physical programmig PP is a multi-objective optimizatio method that proposes specific algorithms for obtaiig the weights of multiple objectives ad use these weights i the optimizatio process to obtai optimal results [6]. Differet from goal programmig ad fuzzy goal programmig techiques that have already bee applied to portfolio optimizatio problem, PP uses the satisfactio levels (such as desirable, tolerable, udesirable, highly udesirable, or uacceptable) at which a particular goal (i.e. expected retur) to obtai the weights ad reach optimal results. Maily, PP distiguishes itself from the other techiques by removig the decisio maker from the weight determiatio process [7]. Weight determiatio is oe of the essetial steps of multi-objective optimizatio which has iheretly the challege i determiig the correct weights. A traditioally preprocessig costat weight determiatio may lead to bias i some cases [8]. I the PP, it is ot eeded to set the weights of objectives i priori. Differetly, PP determies the weights i a systematic approach with the itegratio of the solutio phase to fid optimal results. Weight determiatio procedure uses the oe versus others criterio rule (OVO rule) ad cotais a little complicated arithmetic. Therefore it eeds to use a computer program to obtai the weights. The details of weight determiatio procedure ca be foud i [8] I the PP, DMs use four differet classes amed as soft classes to express their prefereces accordig to each objective fuctio. The most frequetly used class fuctios, Class-S (Smaller is Better) ad Class-S (arger is Better), ca be show i figure. The decisio variables, the qth objective fuctio, the class fuctio that will be miimized for the qth objective fuctio are deoted by x ad g q(x), q respectively. I the figure, g q(x) is o the horizotal axis, q is o the vertical axis. As it ca be deduced from the figure, the smaller value of a class fuctio improves the satisfactio level of the goal. Therefore, it is desired to obtai the value of the class fuctio as zero. Besides soft classes, the

3 Portfolio selectio problem 3 costraits that must be satisfied without ay deviatio is defied as Hard Classes. Each soft class fuctio is a part of the weighted aggregated objective fuctio of PP that is wated to be miimized. The weights of Soft Class fuctios are determied by PP weight algorithm [8]. d qs g q + d qs t q(s ) ; ; g q t q5, s =,,5 q i S (3) ad hard costraits + where, s deotes a rage, d qs ad d qs are deviatioal variables, sc deotes the umber of soft classes, t qs is the limit of differet rages, ad, w qs deotes the weight of rage s i goal q. As it ca be see from Figure, there are five rages that differs six degrees of desirability form ideal to uacceptable. 3. Model costructio for portfolio selectio problem I a portfolio selectio problem, it is assumed that there are N stocks from M sectors ad K idexes to be selected for satisfyig decisio maker s objectives. The selected objectives are as follows; Expected Rate of Retur: The expected rate of retur measures the retur of each stock. The price of the stock x at time t is subtracted from the price of stock x at time (t-) the divided by the price of stock x at time (t-) Risk: The stadard deviatio of the expected rate of retur of each stock The system parameters ad assumptios are give i below. i stock type i =,,... j sector type j=,,..m. k Stock idexes k=,..,k. X i the ratio of stock i. The mathematical represetatio of the objective fuctios are show as below: Figure. Smaller is Better ad arger is Better soft class fuctios [7] Step. Selectio of the appropriate soft ad hard classes for each criterio, Step. Determiatio of the target values that ca be defied as the limits of the rages of differet degrees of desirability (i.e. t qs, t + qs ). Step 3. Determiatio of the weights for each criterio by usig the weight algorithm. Step 4. Solvig the followig P problem: Mi = sc 5 q= s=(w qs d qs + w + qs d + qs ) () g q d + + qs t q(s ) ; d + qs ; g q t + q5, s =,,5 q i S () Objective : Maximizatio of the Expected Rate of Retur = i= (r i X i ) (4) Where r i deotes the expected rate of retur of the Stock i over the plaig period. Objective : Miimizatio of the Risk = i= (σ i X i ) (5) Where σ i represets the stadard deviatio of the expected rate of retur of each stock over the plaig period. The costraits of the portfolio selectio problem are represeted below;

4 4 F. Kucukbay, C. Araz / Vol.6, No., pp.-8 (6) IJOCTA Costrait : The followig formula esures that the total weights of the stocks must equal to. i= X i = (6) Costrait : Beyod the objective of miimizig expected risk of portfolio, it is importat to avoid allocatig all resources to the small umber of stocks which operates i the same sector. I order to diversify the portfolio, at least four differet sectors must be icluded i the portfolio selected. I other words, the weights of each sector must be at most 5 %. i IE X i.5, j (7) Where SE j represets the set of stocks which belog to the j th sector. Costrait 3: I order to esure the log-term profitability ad to maximize the possibility of success i the log ru, the model proposes to ivest at least 5 % or more o the firms i Bist 5 idex. i IE X i.5( i IE X i + + i IEk X i ) (8) Where IE k represet the set of stocks which belog to the kth Bist idex. Moreover i the model lower ad upper boud for each stock was decided as xj. i order to esure diversity. X i, (9) 3.. iear physical programmig model for portfolio selectio problem To maximize the expected rate of retur of the selected portfolio, the first goal is defied as Class-S type (i.e. arger is Better). (r i X i ) i= + d S t (s ) ; d S ; i=(r i X i ) t 5 ; s =,,5 () The secod goal is for portfolio risk measuremet which is represeted by Class-S type (i.e. Smaller is Better ). (σ i X i ) d + S t (s ) ; d + S ; i= + i=(σ i X i ) t + 5 ; s =,,5 () The, the PP model ca be costructed as follows: 5 s= d qs + d + qs ) Mi = q= (w qs + w qs () Subject to (6) - (). 3.. Fuzzy goal programmig model for portfolio selectio problem Sice there are two coflictig objectives which force decisio maker to accept trade-off values i the fial decisio, the problem ca also be modelled by usig fuzzy goal programmig which ca hadle the ambiguity of the decisio makig process as follows: Objective : The expected rate of retur i=(r i X i ) (3) Where represets the desirable achievemet value for the expected rate of retur objective. The symbol deotes the statemet of approximately greater tha or equal to. The fuzzy goal ca be expressed as a triagular membership fuctio μ( ) with tolerace limits for the goal (, U ) as follows: if U μ( ) = { U if U (4) The membership fuctio for fuzzy expected rate of retur goal is show as i the Figure. Figure. Membership fuctio of fuzzy expected rate of retur goal Objective : The Risk U i= (σ i X i ) (5) Where represets the desirable achievemet value for the risk. The symbol meas that the objective fuctio should be approximately less tha or equal to the predefied limits. The fuzzy goal ca be expressed as a triagular membership fuctio μ( ) with two parameters (, U ) as follows:

5 Portfolio selectio problem 5 μ( ) = { U U if if U U (6) The membership fuctio for fuzzy risk goal is show as i the Figure 3. The lower ad upper tolerace limits (i.e., U aspiratio levels) are determied by costructig a pay-off table which cotais the solutios of two sigle objective problems. I the solutio methodology, the problem is solved separately with expected rate of retur ad risk objectives, respectively. The the best ad worst values are determied ad used as the aspiratio levels for the fuzzy goals. U Figure 3. Membership fuctio of fuzzy risk goal After costructig fuzzy membership fuctios for the goals, fuzzy goal programmig model ca be preseted as follows [9]: Max = λ (7) μ( ) λ μ( ) λ Subject to (6) - (9) Where λ deotes overall achievemet level of fuzzy goals. 4. A portfolio selectio model with the help of liear physical programmig I order to show the effectiveess of PP o portfolio optimizatio problem, a real-life experimetal study was performed by selectig stocks operatig i Borsa İstabul. The performace of the obtaied portfolio is compared with that obtaied from fuzzy goal programmig (FGP). The the performaces of both methods, PP ad FGP are compared with BIST i terms of their Sharpe Idexes. The details about umerical aalysis ca be foud i the followig subsectios. 4.. Results for liear physical programmig model I the model we cosider two criteria: the expected retur of stocks ad risk. The sample cosists of 89 compaies that traded cotiuously at BIST betwee April 9 - March 4. The observatio period is April 9- March 5. This period is divided ito two; April 9- March 4 ad April 4 March 5. April 9-March 4 period is used as data to fid a optimal portfolio. April 4-March 5 period is used to test the real performace of selected portfolios. The expected rate of retur values are calculated by usig the closig prices at the begiig of each moth for each stock. The data are gathered o mothly basis for April 9 March 4 period. The umber of observatios gathered was 6. The physical programmig represets differet desirability degrees for each criteria. These desirability degrees are expressed by usig six types of rages which are ideal, desirable, tolerable, udesirable, highly udesirable ad uacceptable [3]. Table represets the target values for expected rate of retur ad risk. Geerally, decisio makers estimate the target values based o their kowledge ad experiece. I the paper, the iterval target values are also estimated, however, a payoff table (see Table ) which cotais the solutios of -sigle objective problem is costructed to estimate the max. ad mi. limits of these target values which are also used i costructig the FGP membership fuctios. Table. Correspodig pay-off table Objectives Maximize Expected Rate of Retur Expected Rate of Retur Risk Miimize Risk Table. Target values for criteria s Expected Risk Rate Of Retur Ideal >4.59 <.34 Desirable Tolerable Udesirable Highly Udesirable Uacceptable <.4 >6.3 Oce the class fuctio is defied accordig to the target values, the PP weight algorithm was used to calculate the weights preseted below.

6 6 F. Kucukbay, C. Araz / Vol.6, No., pp.-8 (6) IJOCTA w + =.3, w + 3 =.58, w + 4 =.366 w + 5 =.584 w =.484, w 3 =.887, w 4 =.4 w 5 =.9 By solvig the P mathematical model, stocks were selected for our portfolio. Table 3 presets the stocks otatios, their expected rate of returs, the risks for April 9 March 4 period, Bist idex classificatio ad their sectors. Ad the last colum shows the proportios of each stock i the portfolio for optimal solutio. Table 3. Selected stocks for portfolio with the help of liear physical programmig Notatio Expected Retur Risk Bist Idex Proportios DOAS E NTTUR E TCE E CCOA E TTRAK E UKER E NTHO E TAVH E YAIC E ASES E OGO E NETAS E Results for fuzzy goal programmig model I order to compare the performace of PP, we have also solved the problem with FGP. The lower ad upper tolerace limits are determied as i Table by costructig a pay-off table which cotais the solutios of -sigle objective problem. These max-mi limits guaratee the feasibility of each fuzzy goal i the solutio. Figure 4 ad 5 shows the membership fuctios for satisfactio levels of the expected retur ad risk goals, respectively Figure 5. The membership fuctio of risk goal After applyig FGP, the followig results were obtaied. Table 4. Selected stocks for portfolio with the help of fuzzy goal programmig Notatio Expected Retur Risk Bist Idex Proportios DOAS E NTTUR E TCE E CCOA E TTRAK E UKER E NTHO E TAVH E VKGYO E YAIC E ASES E OGO E NETAS E Table 4 presets the stocks, their otatio, their expected rate of retur ad the risk for April 9 March 4 period whe the problem is solved with the help of fuzzy goal programmig. Ad the last colum shows the weights of each stock i the portfolio for optimal solutio. The overall results of both PP ad FGP models are provided i Table 5. The results show that the portfolio returs obtaied from PP model is fewer tha those obtaied from the FGP model. However, the risk obtaied from PP is fewer tha the oe obtaied from FGP model. Although both FGP ad PP models provide compromise solutios, the piecewise liear goal fuctios ad multiple target values of PP model allow to geerate the differet sets of Pareto optimal solutios Figure 4. The membership fuctio of expected retur goal Table 5. Results for fuzzy goal programmig ad liear physical programmig Objective FGP PP Expected Rate of Retur Risk.35.76

7 Portfolio selectio problem The performace of portfolio for cotrol period I order to test the performace of our portfolio, we use the cotrol period. The cotrol period is April 4 March 5. The performace of our portfolio performace will be compared with the preset market. We assume that the ivestor ivests his/her fud i the selected portfolio determied by liear physical programmig i April 4 ad hold this portfolio for moths till March 5. BIST idex is selected to represet the market. Firstly the retur ad risk are calculated for both BIST ad the selected portfolio o mothly basis. I order to compare the performace of selected portfolio ad the idex more vigorously, Sharpe idex [3] was used. The success of the portfolio will be evaluated by comparig the Sharpe idex of market ad Sharpe idex of selected portfolio. Higher Sharpe Idex is better. So if the Sharpe Idex value of the portfolio is higher tha the Sharpe Idex value of the market, the performace of portfolio will be better tha the market. Sharpe idex cosiders both risk ad retur at the same time. Sharpe idex is calculated as follows; S = (rp rf)/ rp (7) rp = The average retur of portfolio for a give period, rf = The average risk free iterest rate (usually state bod or treasury bod iterest rates are accepted) for a give period, rp= the stadard deviatio of portfolio for a give period (represets the risk criteria). Table 6. Sharpe idex value of Bist ad the portfolios selected with the help of PP ad FGP BIST Idex PP FGP Expected Rate of %.4 % 4.33 %3.9 Retur Risk % 6.48 % 8.4 %8.96 Sharpe Idex Table 6 represets the expected rate of retur, risk ad Sharpe Idex value of both BIST ad the portfolio obtaied by PP ad FGP. Sharpe Idex of BIST ad the portfolio are calculated for April 4 ad March 5. The risk-free iterest rate is take as the average Treasury bod iterest rate for April 4 March 5 period (%,7). The fidigs reveal that the performace of the portfolio obtaied by PP was better tha both the performace of BIST Idex ad the portfolio obtaied by FGP i terms of sharpe idex. 5. Coclusio The success of portfolio selectio problem ca oly be esured by successful selectio of stocks. I this paper, a ew techique for portfolio optimizatio problem with the aid of PP is preseted. The mai purpose of the study is to fid a optimum portfolio that maximizes the retur which at the same time miimizes the risk. We compared the performace of the portfolios obtaied by PP ad FGP approaches with the preset market (BIST Idex) for the cotrol period withi April 4 March 5 i terms of Sharpe idex. The results revealed that PP has potetial to help the ivestors to fid the efficiet portfolio as much as possible. Fially, this study is thought to make cotributio to literature by itroducig the PP for portfolio selectio problems. Further research may cosider more criteria ad more costraits. Refereces [] Ertua, I.O., Yatırım ve Portföy Aalizi (Bilgisayar Uygulama Örekleriyle), Boğaziçi Uiversitesi, (99). [] Korkmaz, T, Ceyla, A.,, Sermaye Piyasası ve Mekul Değer Aalizi, Eki Yayıevi, Bursa (7). [3] İskederoğlu, Ö., Karadeiz, E., Optimum Portföyü Seçimi: İMKB 3 Üzeride Bir Uygulama, CÜ İktisadi ve İdari Bilimler Dergisi Cilt, Sayı: (). [4] Parra, M. A., Terol, A. B., Uria, M.V. R., A Fuzzy Goal Programmig Approach to Portfolio Selectio, Europea Joural of Operatioal Research, 33, (). [5] Aliezhad, A., ohrehbadia, M., Kia, M., Ekhtiari, M., Esfadiari, N., Extesio of Portfolio Selectio Problem with Fuzzy Goal Programmig: A Fuzzy Allocated Portfolio Apprach, Joural Of Optimizatio i Idustrial Egieerig, 9, (). [6] Sharpe, W. F., A Simplified Model For Portfolio Aalysis, Maagemet Sciece, 9; (963). [7] Sharpe,, W. F., Capital Asset Prices: A Theory of Market Equilibrium uder Coditios of Risk, Joural of Fiace,, 9, (964). [8] itler, J., The Valuatio of Risky Assets ad the Selectio of Risky Ivestmets i Stock Portfolios ad Capital Budgets, Review of

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