Research Article A New Decision-Making Method for Stock Portfolio Selection Based on Computing with Linguistic Assessment

Transcription

1 Journal of Appled Mathematcs and Decson Scences Volume 2009, Artcle ID , 20 pages do: /2009/ Research Artcle A New Decson-Makng Method for Stock Portfolo Selecton Based on Computng wth Lngustc Assessment Chen-Tung Chen 1 and We-Zhan Hung 2 1 Department of Informaton Management, Natonal Unted Unversty, Mao-L 36003, Tawan 2 Graduate Insttute of Management, Natonal Unted Unversty, Mao-L 36003, Tawan Correspondence should be addressed to Chen-Tung Chen, ctchen@nuu.edu.tw Receved 30 November 2008; Revsed 18 March 2009; Accepted 13 May 2009 Recommended by Lean Yu The purpose of stock portfolo selecton s how to allocate the captal to a large number of stocks n order to brng a most proftable return for nvestors. In most of past lteratures, experts consdered the portfolo of selecton problem only based on past crsp or quanttatve data. However, many qualtatve and quanttatve factors wll nfluence the stock portfolo selecton n real nvestment stuaton. It s very mportant for experts or decson-makers to use ther experence or knowledge to predct the performance of each stock and make a stock portfolo. Because of the knowledge, experence, and background of each expert are dfferent and vague, dfferent types of 2-tuple lngustc varable are sutable used to express experts opnons for the performance evaluaton of each stock wth respect to crtera. Accordng to the lngustc evaluatons of experts, the lngustc TOPSIS and lngustc ELECTRE methods are combned to present a new decsonmakng method for dealng wth stock selecton problems n ths paper. Once the nvestment set has been determned, the rsk preferences of nvestor are consdered to calculate the nvestment rato of each stock n the nvestment set. Fnally, an example s mplemented to demonstrate the practcablty of the proposed method. Copyrght q 2009 C.-T. Chen and W.-Z. Hung. Ths s an open access artcle dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrcted use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. 1. Introducton The purpose of stock portfolo selecton s how to allocate the captal to a large number of stocks n order to brng a most proftable return for nvestors 1. For ths pont of vew, stock portfolo decson problem can be dvded nto two questons. 1 Whch stock do you choose? 2 Whch nvestment rato do you allocate your captal to ths stock? There are some lteratures to handle the stock portfolo decson problem. Markowtz proposed the mean-varance method for the stock portfolo decson problem n

2 2 Journal of Appled Mathematcs and Decson Scences In hs method, an expected return rate of a bond s treated as a random varable. Stochastc programmng s appled to solve the problem. The basc concept of hs method can be expressed as follows. 1 When the rsk of stock portfolo s constant, we should pursue to maxmze the return rate of stock portfolo. 2 When the return rate of stock portfolo s constant, we should pursue to mnmze the rsk of stock portfolo. The captal asset prcng model CAPM, Sharpe-Lntner model, Black model, and two-factor model are derved from the mean-varance method 3, 4. The captal asset prcng model CAPM was developed n 1960s. The concept of the CAPM s that the excepted return rate of the captal wth rsk s equal to the nterest rate of the captal wthout rsk and market rsk premum 4. The methods and theory of the fnancal decson makng can be found n 5 7. In 1980, Saaty proposed Analytc Herarchy Process AHP to deal wth the stock portfolo decson problem by evaluatng the performance of each company n dfferent level of crtera 8. Edrsnghe and Zhang 9 selected the securtes by usng Data Envelopment Analyss DEA. Huang 1 defned a new defnton of rsk and use genetc algorthm to cope wth stock portfolo decson problem. Generally, n the portfolo selecton problem the decson maker consders smultaneously conflctng objectves such as rate of return, lqudty, and rsk. Multobjectve programmng technques such as goal programmng GP and compromse programmng CP are used to choose the portfolo Consderng the uncertanty of nvestment envronment, Tryak transferred experts lngustc value nto trangle fuzzy number and used a new fuzzy rankng and weghtng algorthm to obtan the nvestment rato of each stock 4. In fact, the stock portfolo decson problem can be descrbed as multple crtera decson makng MCDM problem. Technque for Order Preference by Smlarty to Ideal Soluton TOPSIS method s developed by Hwang and Yoon 13, whch s one of the well-known MCDM methods. The basc prncple of the TOPSIS method s that the chosen alternatve should have the shortest dstance from the postve deal soluton PIS and the farthest dstance from the negatve deal soluton NIS.Itsaneffectve method to determne the total rankng order of decson alternatves. The Elmnaton et choce n Translatng to Realty ELECTRE method s a hghly developed multcrtera analyss model whch takes nto account the uncertanty and vagueness n the decson process 14. It s based on the axom of partal comparablty and t can smplfy the evaluaton procedure of alternatve selecton. The ELECTRE method can easly compare the degree of dfference among all of alternatves. In MCDM method, experts can express ther opnons by usng crsp value, trangle fuzzy numbers, trapezodal fuzzy numbers, nterval numbers, and lngustc varables. Due to mprecse nformaton and experts subjectve opnon that often appear n stock portfolo decson process, crsp values are nadequate for solvng the problems. A more realstc approach may be to use lngustc assessments nstead of numercal values 15, 16. The2- tuple lngustc representaton model s based on the concept of symbolc translaton 17, 18. Experts can apply 2-tuple lngustc varables to express ther opnons and obtan the fnal evaluaton result wth approprate lngustc varable. It s an effectve method to reduce the mstakes of nformaton translaton and avod nformaton loss through computng wth words 19. In general, decson makers would use the dfferent 2-tuple lngustc varables based on ther knowledge or experences to express ther opnons 20. In ths paper, we use dfferent type of 2-tuple lngustc varable to express experts opnons and combne

3 Journal of Appled Mathematcs and Decson Scences 3 μ T x 1 0 l m u Fgure 1: Trangular fuzzy number T. lngustc ELECTRE method wth TOPSIS method to obtan the fnal nvestment rato whch s reasonable n real decson envronment. Ths paper s organzed as follows. In Secton 2, we present the context of fuzzy set and the defnton and operaton of 2-tuple lngustc varable. In Secton 3, we descrbe the detal of the proposed method. In Secton 4, an example s mplemented to demonstrate the procedure for the proposed method. Fnally, the concluson s dscussed at the end of ths paper. 2. The 2-Tuple Lngustc Representaton 2.1. Fuzzy Set and Trangular Fuzzy Number Fuzzy set theory s frst ntroduced by Zadeh n Fuzzy set theory s a very feasble method to handle the mprecse and uncertan nformaton n a real world 22. Especally, t s more sutable for subjectve judgment and qualtatve assessment n the evaluaton processes of decson makng than other classcal evaluaton methods applyng crsp values 23, 24. A postve trangular fuzzy number PTFN T can be defned as T l, m, u, where l m u and l>0, shown n Fgure 1. The membershp functon μ T x of postve trangular fuzzy number PTFN T s defned as 15 μ T x x l m l, l<x<m, u x u m, m < x < u, 0, otherwse. 2.1 A lngustc varable s a varable whose values are expressed n lngustc terms. In other words, varable whose values are not numbers but words or sentences n a nature or artfcal language For example, weght s a lngustc varable whose values are very low, low, medum, hgh, very hgh, and so forth. These lngustc values can also be represented by fuzzy numbers. There are two advantages for usng trangular fuzzy number to express lngustc varable 28. Frst, t s a ratonal and smple method to use trangular

4 4 Journal of Appled Mathematcs and Decson Scences fuzzy number to express experts opnons. Second, t s easy to do fuzzy arthmetc when usng trangular fuzzy number to express the lngustc varable. It s sutable to represent the degree of subjectve judgment n qualtatve aspect than crsp value The 2-Tuple Lngustc Varable Let S {s 0,s 1,s 2,...,s g } be a fnte and totally ordered lngustc term set. The number of lngustc term s g 1nsetS. A 2-tuple lngustc varable can be expressed as s,α, where s s the central value of th lngustc term n S and α s a numercal value representng the dfference between calculated lngustc term and the closest ndex label n the ntal lngustc term set. The symbolc translaton functon Δ s presented n 29 to translate crsp value β nto a 2-tuple lngustc varable. Then, the symbolc translaton process s appled to translate β β 0, 1 nto a 2-tuple lngustc varable. The generalzed translaton functon can be represented as 30 : [ Δ : 0, 1 S 1 ) 2g, 1 2g Δ ( β ) s,α, 2.2 where round β g, α β /g and α 1/2g,1/2g. A reverse functon Δ 1 s defned to return an equvalent numercal value β from 2-tuple lngustc nformaton s,α. Accordng to the symbolc translaton, an equvalent numercal value β s obtaned as follow 30 Δ 1 s,α g α β. 2.3 Let x { r 1,α 1,..., r n,α n } be a 2-tuple lngustc varable set. The arthmetc mean X s computed as 31 ( ) 1 n X Δ Δ 1 r,α s m,α m, n where n s the amount of 2-tuple lngustc varable. The s m,α m s a 2-tuple lngustc varable whch s represented as the arthmetc mean. In general, decson makers would use the dfferent 2-tuple lngustc varables based on ther knowledge or experences to express ther opnons 20. For example, the dfferent types of lngustc varables show as Table 1. Each 2-tuple lngustc varable can be represented as a trangle fuzzy number. A transformaton functon s needed to transfer these 2-tuple lngustc varables from dfferent lngustc sets to a standard lngustc set at unque doman. In the method of Herrera and Martnez 29, the doman of the lngustc varables wll ncrease as the number of lngustc varable s ncreased. To overcome ths drawback, a new translaton functon s appled to transfer a crsp number or 2-tuple lngustc varable to a standard lngustc term at the unque doman 30. Suppose that the nterval 0, 1 s the unque doman. The lngustc varable sets wth dfferent semantcs or types wll be

5 Journal of Appled Mathematcs and Decson Scences 5 defned by parttonng the nterval 0, 1. Transformng a crsp number β β 0, 1 nto th lngustc term s n t,α n t of type t as Δ t ( β ) ( s n t,α n t ), 2.5 where round β g t, α n t β /g t,g t n t 1, and n t s the number of lngustc varable of type t. Transformng th lngustc term of type t nto a crsp number β β 0, 1 as ( Δ 1 t s n t ),α n t α n t β, 2.6 g t where g t n t 1andα n t 1/2g t, 1/2g t. Therefore, the transformaton from th lngustc term s n t,α n t of type t 1 at nterval 0, 1 can be expressed as lngustc term s n t 1 k,α n t 1 k Δ t 1 ( Δ 1 t ( s n t )) (,α n t s n t 1 k of type t to kth ),α n t 1, 2.7 k where g t 1 n t 1 1andα n t 1 k 1/2g t 1, 1/2g t Proposed Method Because of the knowledge, experence and background of each expert s dfferent and experts opnons are usually uncertan and mprecse, t s dffcult to use crsp value to express experts opnons n the process of evaluatng the performance of stock. Instead of crsp value, the 2-Tuple lngustc valuable whch s an effectve method to reduce the mstakes of nformaton translaton and avod nformaton loss through computng wth words to express experts opnons 19. In ths paper, dfferent types of 2-tuple lngustc varables are used to express experts opnons. The TOPSIS method s one of the well-known MCDM methods. It s an effectve method to determne the rankng order of decson alternatves. However, ths method cannot dstngush the dfference degree between two decson alternatves easly. Based on the axom of partal comparablty, the ELECTRE method can easly compare the degree of dfference among of all alternatves. Ths method always cannot provde the total orderng of all decson alternatves. Therefore, the ELECTRE and TOPSIS methods are combned to determne the fnal nvestment rato. In the proposed model, the subjectve opnons of experts can be expressed by dfferent 2-tuple lngustc varables n accordance wth ther habtual knowledge and experence. After aggregatng opnons of all experts, the lngustc TOPSIS and lngustc ELECTRE methods are appled to obtan the nvestment portfolo sets Ω t and Ω e, respectvely. The strct stock portfolo set Ω p s determned by ntersecton Ω t wth Ω e. In general, the rsk preference of nvestor can be dvded nto three types such as rsk-averter, rsk-neutral, and rsk-lovng. Consderng the rsk preference of nvestor, we can calculate the nvestment rato of each

6 6 Journal of Appled Mathematcs and Decson Scences Experts choose dfferent type of lngustc varables to express ther opnons. Transfer experts opnons to the same type of lngustc valuable. Aggregate experts opnons. Usng lngustc TOPSIS to obtan the nvestment portfolo set Ω t Usng lngustc ELECTRE to obtan the nvestment portfolo set Ω e The strct nvestment portfolo set Ω p s determned n accordance wth the ntersecton Ω t wth Ω e. Preference Rsk-averter Rsk-neutral The nvestment rato of each stock n Ω p s calculated based on rsk preference of fnal decson-maker. Rsk-lovng Fgure 2: The decson-makng process of the proposed method. stock n strct stock portfolo set Ω p. The decson process of the proposed method s shown as n Fgure 2. In general, a stock portfolo decson may be descrbed by means of the followng sets: a set of experts or decson-makers called E {E 1,E 2,...,E K }; a set of stocks called S {S 1,S 2,...,S m }; a set of crtera C {C 1,C 2,...,C n } wth whch stock performances are measured; v a weght vector of each crteron W W 1,W 2,...,W n ; v a set of performance ratngs of each stock wth respect to each crteron called S j, 1, 2,...,m, j 1, 2,...,n. Accordng to the aforementoned descrpton, there are K experts, m stocks and n crtera n the decson process of stock portfolo. Experts can express ther opnons by dfferent 2-tuple lngustc varables. The kth expert s opnon about the performance ratng of th stock wth respect to jth crteron can be represented as S k j Sk j,αk j.thekth expert s opnon about the mportance of jth crteron can be represented as W jk S w jk,αw jk. The aggregated lngustc ratng S j of each stock wth respect to each crteron can be calculated as ( 1 K S j Δ Δ 1( S k j K j) ),αk ( ) S j,α j. 3.1 k 1

7 Journal of Appled Mathematcs and Decson Scences 7 The aggregated lngustc weght w j of each crteron can be calculated as ( 1 K W j Δ Δ 1( S w jk K jk) ),αw k 1 ( ) S w j,αw j Lngustc TOPSIS Method Consderng the dfferent mportance of each crteron, the weghted lngustc decson matrx s constructed as Ṽ ṽ j m n, 1, 2,...,m, j 1, 2,...,n, 3.3 where ṽ j x j w j Δ Δ 1 S j,α j Δ 1 S w j,αw j Sv j,αv j. Accordng to the weghted lngustc decson matrx, the lngustc postve-deal soluton LPIS, S and lngustc negatve-deal soluton LNIS, S can be defned as S ( ṽ 1, ṽ 2 n),...,ṽ, S ( ṽ 1, ) 3.4 ṽ 2,...,ṽ n, where ṽ j max { S v j,αv j } and ṽ j mn { S v j,αv j }, 1, 2,...,m, j 1, 2,...,n. The dstance of each stock S 1, 2,...,m from S and S can be currently calculated as d d S,S n d ( ) n ( {( )} ṽ j, ṽ j (Δ ) ( ) ) 2 1 max S v j,αv j Δ 1 S v j,αv j, d j 1 j 1 j 1 d ( S,S ) n d ( ) n ( ) ( )} ṽ j, ṽ j (Δ 1 S v j,αv j Δ 1 mn {(S )) 2 v j,αv j. j A closeness coeffcent s defned to determne the rankng order of all stocks once d and d of each stock S 1, 2,...,m have been calculated. The closeness coeffcent represents the dstances to the lngustc postve-deal soluton S and the lngustc negatve-deal soluton S smultaneously by takng the relatve closeness to the lngustc postve-deal soluton. The closeness coeffcent CC of each stock s calculated as d CC d, 1, 2,...,m. 3.6 d The hgher CC means that stock S relatvely close to postve deal soluton, the stock S has more ablty to compete wth each others. If the closeness coeffcent of stock S s greater than the predetermned threshold value β t, we consder stock S s good enough to choose n the nvestment portfolo set. Accordng to closeness coeffcent of each stock, the

8 8 Journal of Appled Mathematcs and Decson Scences nvestment portfolo set Ω t can be determned based on nvestment threshold value β t as Ω t {S CC β t }. Fnally, the nvestment rato of each stock n Ω t can be calculated as CC S P t S S Ω t CC S, S Ω t, 0, S / Ω t, 3.7 where P t S s the nvestment rato of each stock by lngustc TOPSIS method Lngustc ELECTRE Method Accordng to the ELECTRE method, the concordance ndex C j S,S l s calculated for S and S l / l,, l 1, 2,...,m wth respect to each crteron as 1, Δ 1( s ) ( s ) j Δ 1 lj qj, C j S,S l Δ 1( s ) ( s ) j Δ 1 lj pj, Δ 1( s ) lj qj Δ 1( s ) ( s ) j Δ 1 lj pj, p j q j 0, Δ 1( s ) ( s ) j Δ 1 lj pj, 3.8 where q j and p j are ndfference and preference threshold values for crteron C j,p j >q j. The dscordance ndex D j S,S l s calculated for each par of stocks wth respect to each crteron as 1, Δ 1( s ) ( s ) j Δ 1 lj vj, D j S,S l Δ 1( s ) lj pj Δ 1( s ) j, Δ 1( s ) lj pj Δ 1( s ) ( s ) j Δ 1 lj vj, v j p j 0, Δ 1( s ) ( s ) j Δ 1 lj pj, 3.9 where v j s the veto threshold for crteron C j,v j >p j. Calculate the overall concordance ndex C S,S l as n C S,S l Δ 1( ) w j Cj S,S l. j The credblty matrx S S,S l of each par of the stocks s calculated as C S,S l, S S,S l C S,S l j J S,S l f D j S,S l C S,S l j, 1 D j S,S l C S,S l, otherwse, where J S,S l s the set of crtera for whch D j S,S l >C S,S l,/ l,, l 1, 2,...,m.

9 Journal of Appled Mathematcs and Decson Scences 9 The concordance credblty and dscordance credblty degrees are defned as 32 φ S / l S S,S l, φ S / l S S l,s The concordance credblty degree represents that the degree of stock S s at least as good as all the other stocks. The dscordance credblty degree represents that the degree of all the other stocks s at least as good as stock S. Then, the net credblty degree s defned as φ S φ S φ S. If the net credblty degree of stock S s hgher, then t represents a hgher attractveness of stock S.In order to determne the nvestment rato, the outrankng ndex of stock S can be defned as OTI S φ S / m Property 3.1. Accordng to the defnton of OTI S, we can fnd 0 OTI S 1. Proof. Because φ S φ S φ S / l S S,S l / l S l S,/ l,, l 1, 2,...,m. If the stock S s better than S l wth respect to each crteron, the best case s S S,S l S l,s m / l / l If the stock S s worse than S l wth respect to each crteron, the worst case s S S,S l S l,s m 1. / l / l 3.15 Therefore, m 1 φ S m 1. Then, 1 φ S / m 1 1. Fnally, we can prove 0 φ S / m 1 1 /2 OTI S 1. The OTI S denotes the standardzaton result of the net credblty degree. Accordng to the defnton, t s easy to understand and transform the net credblty degree nto nterval 0, 1. If the outrankng ndex of stock S s greater than the predetermned threshold value β e, we consder stock S s good enough to choose n the nvestment portfolo set. Accordng to the outrankng ndex of each stock, the nvestment portfolo set Ω e can be determned based on nvestment threshold value β e as Ω e {S OTI S β e }. Fnally, the nvestment rato of each stock n Ω e can be calculated as OTI S P e S S Ω e OTI S, S Ω e, 0, S / Ω e, 3.16 where P e S s the nvestment rato of each stock by usng lngustc ELECTRE method.

10 10 Journal of Appled Mathematcs and Decson Scences 3.3. Stock Portfolo Decson We can consder Lngustc TOPSIS and Lngustc ELECTRE methods as two fnancal experts to provde nvestment rato of each stock, respectvely. Smart nvestor wll make a stock portfolo decson by consderng the suggestons of nvestment rato of each stock smultaneously. Therefore, the portfolo set Ω p s defned as strct stock portfolo set Ω p Ω t Ω e. Accordng to the closeness coeffcent, the nvestment rato of each stock n strct stock portfolo set Ω p can be calculated as CC S P t p S S Ω p CC S, S Ω p, 0, S / Ω p Accordng to the outrankng ndex, the nvestment rato of each stock n strct stock portfolo set Ω p can be calculated as OTI S P e p S S Ω p OTI S, S Ω p, 0, S / Ω p In general, the nvestment preference of nvestors can be dvded nto three types such as rsk-averter RA, rsk-neutral RN, and rsk-lovng RL. If a person s rsk-averter, he/she wll consder the smaller nvestment rates between P t p S and P e p S. Therefore, the fnal rato of each stock n strct portfolo set can be calculated as P RA S mn ( P t p S,P e p S ) S Ω p mn ( ) P t p S,P e p S If a person s rsk-neutral, he/she wll consder the average nvestment rates between P t p S and P e p S. Therefore, the fnal rato of each stock n strct portfolo set can be calculated as ( Pt p S P e p S ) /2 P RN S (( S Ω Pt p p S P e p S ) ) /2 If a person s rsk-lovng, he/she wll consder the bgger nvestment rates between P t p S and P e p S. Therefore, the fnal rato of each stock n portfolo set can be calculated as P RL S max ( P t p S,P e p S ) S Ω p max ( ) P t p S,P e p S

11 Journal of Appled Mathematcs and Decson Scences 11 Table 1: Ten stocks of semconduct ndustry n Tawan. S Tawan Semconductor Manufacturng Co. 1 Ltd. S 2 Unted Mcroelectroncs Corp. S 3 Advanced Semconductor Engneerng, Inc. S 4 Va Technologes, Inc. S 5 MedaTek Inc. S 6 Kng Yuan Electroncs Co. Ltd. S 7 Tawan Mask Corp. S 8 Wnbond Electroncs Corp. S 9 SunPlus Technology Co. Ltd. S 10 Nanya Technology Corporaton 4. Numercal Example An example wth ten stocks of semconduct ndustry n placecountry-regon, Tawan, wll be consdered to determne the nvestment rato of each stock n ths paper. Ten stocks are shown as Table 1. A commttee of three fnancal experts E {E 1,E 2,E 3 } has been formed to evaluate the performance of each stock. They are famous professors of a department of fnance at well-known unversty n country-regonplace, Tawan. Ther knowledge and experences are enough to evaluate the stock performance of each company for ths example. In the process of crtera selecton, they consdered the quanttatve and qualtatve factors to deal wth the portfolo selecton. After the serous dscusson and selecton by three fnancal experts, sx crtera are consdered to determned the nvestment rato of each stock such as proftablty C 1, asset utlzaton C 2, lqudty C 3, leverage C 4, valuaton C 5,growth C 6. Proftablty (C 1 ) The goal of enterprse s tomakeaproft. There are some ndexes to evaluate the proftablty of a company such as earnngs per share EPS, net proft margn, return on assets ROA, and return on equty ROE. The proftablty of a company wll nfluence the performance of each stock. Asset Utlzaton (C 2 ) Asset utlzaton means the effcency of usng company s resource n a perod. A good company wll promote the resource usng effcency as more as possble. Experts evaluate the asset utlzaton of the company based on recevables turnover, nventory turnover, and asset turnover. Lqudty (C 3 ) Lqudty wll focus on cash flow generaton and a company s ablty to meet ts fnancal oblgatons. When company s transfer assets 1 and, factory buldngs, equpment, patent, goodwll to currency n a short perod, there wll have some loss because the company s manager do not have enough tme to fnd out the buyer who provde the hghest prce. An approprate lqudty rato debt to equty rato, current rato, quck rato wll both prevent lqudty rsk and mnmze the workng captal.

12 12 Journal of Appled Mathematcs and Decson Scences s 5 0 s5 1 s5 2 s5 3 s5 4 Fgure 3: Membershp functons of lngustc varables at type 1 t 1. Table 2: Dfferent types of lngustc varables. Type Lngustc varable Fgure 1 Performance Extremely Poor s 5 0, Poor s5 1,Far s5 2, Good s5 3, Extremely Good s 5 4 Fgure 3 Weght Extremely Low s 5 0,Low s5 1,Far s5 2, Hgh s5 3, Extremely Hgh s Performance Extremely Poor s 7 0, Poor s7 1, Medum Poor s7 2,Far s7 3, Medum Good s 7 4, Good s7 5, Extremely Good s7 6 Fgure 4 Weght Extremely Low s 7 0,Low s7 1, Medum Low s7 2,Far s7 3, Medum Hgh s 7 4, Hgh s7 5, Extremely Hgh s7 6 3 Performance Extremely Poor s 9 0,Very Poor s9 1, Poor s9 2, Medum Poor s 9 3,Far s9 4, Medum Good s9 5, Good s9 6, Very Good s9 7, Extremely Good s 9 8 Fgure 5 Weght Extremely Low s 9 0,Very Low s9 1,Low s9 2, Medum Low s9 3, Far s 9 4, Medum Hgh s9 5, Hgh s9 6, Very Hgh s9 7, Extremely Hgh s 9 8 Leverage (C 4 ) When the return on assets s greater than lendng rate, t s tme for a company to lend money to operate. But ncreasng the company s debt wll ncrease rsk f the company does not earn enough money to pay the debt n the future. A sutable leverage rato s one of the crtera to evaluate the performance of each stock. Valuaton (C 5 ) Book value means the currency whch all of the company s assets transfer to, stock value means the prce f you want to buy now, earnngs before amortzaton, nterest and taxes rato EBAIT means the company earns n ths year, expert must consder the best tme pont to buy the stock by Techncal Analyss TA and Tme Seres Analyss TSA. So, valuaton s also one of the crtera to evaluate the performance of each stock. Growth (C 6 ) If the scale of a company was expanded year by year, EBAIT wll ncrease whch s lke compound nterest. Because of economes of scale, the growth of the company wll promote asset utlzaton and then rase the EBAIT and EPS. Accordng to the proposed method, the computatonal procedures of the problem are summarzed as follows.

13 Journal of Appled Mathematcs and Decson Scences 13 s 7 0 s7 1 s7 2 s7 3 s7 4 s7 5 s7 6 Fgure 4: Membershp functons of lngustc varables at type 2 t 2. s 9 0 s9 1 s9 2 s9 3 s9 4 s9 5 s9 6 s9 7 s Fgure 5: Membershp functons of lngustc varables at type 3 t 3. Table 3: Evaluaton decsons the ratngs of the all stocks under all crtera by three experts. C 1 C 2 C 3 C 4 C 5 C 6 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 E 1 E 2 E 3 S 1 F F G EG MG VG G MG VG P F EG F G VG P MG VG S 2 P F MG F MG G F F G EP F MG P MG VG P MG G S 3 F F G G F MG F MG G F F MG P MG G F MG MG S 4 F G MG G G G F MG MG F G G F MG MG P EG MG S 5 F MG EG G MG VG F G G G MG VG G MG VG P G G S 6 P F G G F VG F F VG P MG VG F F G F F G S 7 G F G P MG VG F F G F F VG P MG VG F MG VG S 8 EP MG G F F VG EP F VG EP MG EG EP MG VG P MG VG S 9 G MG VG F MG G F F VG F MG VG F MG VG F G G S 10 EP G G F G G F MG MG EP MG G EP F MG EP MG MG Step 1. Each expert selects the sutable 2-tuple lngustc varables to express ther opnons. Expert 1 uses lngustc varables wth 5 scale of lngustc term set to express hs opnon, expert 2 uses lngustc varables wth 7 scale of lngustc term set and expert 3 uses lngustc varables wth 9 scale of lngustc term set, respectvely see Table 2. Step 2. Each expert expresses hs opnon about the performance of each stock wth respect to each crteron as shown n Table 3. Step 3. Each expert expresses hs opnon about the mportance of each crteron as shown n Table 4. Step 4. Transform the lngustc ratngs nto the lngustc varables of type 2 and aggregate the lngustc ratngs of each stock wth respect to crtera as Table 5.

14 14 Journal of Appled Mathematcs and Decson Scences Table 4: Evaluaton decsons the weghtngs of all crtera by three experts. C 1 C 2 C 3 C 4 C 5 C 6 E 1 EH H H H EH F E 2 EH H H MH H H E 3 EH EH VH EH VH H Table 5: Transfer to the lngustc varable of type 2. Stock Crteron E 1 E 1 E 1 Average S 1 S 7 3, S7 3, S7 5, S7 4, S 2 S 7 2, S7 3, S7 4, S7 3, S 3 S 7 3, S7 3, S7 5, S7 4, S 4 S 7 3, S7 5, S7 4, S7 4, C 1 C 2 C 3 S 5 S 7 3, S7 4, S7 6, S7 4, S 6 S 7 2, S7 3, S7 5, S7 3, S 7 S 7 5, S7 3, S7 5, S7 4, S 8 S 7 0, S7 4, S7 5, S7 3, S 9 S 7 5, S7 4, S7 5, S7 5, S 10 S 7 0, S7 5, S7 5, S7 3, S 1 S 7 6, S7 4, S7 5, S7 5, S 2 S 7 3, S7 4, S7 5, S7 4, S 3 S 7 5, S7 3, S7 4, S7 4, S 4 S 7 5, S7 5, S7 5, S7 5, S 5 S 7 5, S 4, S 7 5, S7 5, S 6 S 7 5, S7 3, S7 5, S7 4, S 7 S 7 2, S7 4, S7 5, S7 4, S 8 S 7 3, S7 3, S7 5, S7 4, S 9 S 7 3, S 4, S 7 5, S7 4, S 10 S 7 3, S7 5, S7 5, S7 4, S 1 S 7 5, S7 4, S7 5, S7 5, S 2 S 7 3, S7 3, S7 5, S7 4, S 3 S 7 3, S7 4, S7 5, S7 4, S 4 S 7 3, S7 4, S7 4, S7 4, S 5 S 7 3, S7 5, S7 5, S7 4, S 6 S 7 3, S7 3, S7 5, S7 4, S 7 S 7 3, S7 3, S7 5, S7 4, S 8 S 7 0, S7 3, S7 5, S7 3, S 9 S 7 3, S7 3, S7 5, S7 4, S 10 S 7 3, S7 4, S7 4, S7 4,

15 Journal of Appled Mathematcs and Decson Scences 15 Table 5: Contnued. Stock Crteron E 1 E 1 E 1 Average S 1 S 7 2, S7 3, S7 6, S7 4, S 2 S 7 0, S7 3, S7 4, S7 2, S 3 S 7 3, S7 3, S7 4, S7 3, S 4 S 7 3, S7 5, S7 5, S7 4, C 4 C 5 S 5 S 7 5, S7 4, S7 5, S7 5, S 6 S 7 2, S7 4, S7 5, S7 4, S 7 S 7 3, S7 3, S7 5, S7 4, S 8 S 7 0, S7 4, S7 6, S7 3, S 9 S 7 3, S7 4, S7 5, S7 4, S 10 S 7 0, S7 4, S7 5, S7 3, S 1 S 7 3, S7 5, S7 5, S7 4, S 2 S 7 2, S7 4, S7 5, S7 4, S 3 S 7 2, S7 4, S7 5, S7 3, S 4 S 7 3, S7 4, S7 4, S7 4, S 5 S 7 5, S7 4, S7 5, S7 5, S 6 S 7 3, S7 3, S7 5, S7 4, S 7 S 7 2, S7 4, S7 5, S7 4, S 8 S 7 0, S7 4, S7 5, S7 3, S 9 S 7 3, S7 4, S7 5, S7 4, S 10 S 7 0, S7 3, S7 4, S7 2, S 1 S 7 2, S7 4, S7 5, S7 4, S 2 S 7 2, S7 4, S7 5, S7 3, S 3 S 7 3, S7 4, S7 4, S7 4, S 4 S 7 2, S7 6, S7 4, S7 4, C 6 S 5 S 7 2, S7 5, S7 5, S7 4, S 6 S 7 3, S7 3, S7 5, S7 4, S 7 S 7 3, S7 4, S7 5, S7 4, S 8 S 7 2, S7 4, S7 5, S7 4, S 9 S 7 3, S7 5, S7 5, S7 4, S 10 S 7 0, S7 4, S7 4, S7 3, Table 6: Transfer to the lngustc varable of type 2. Crteron E 1 E 2 E 3 Average C 1 S 7 6, S7 6, S7 6, S7 6, C 2 S 7 5, S7 5, S7 6, S7 5, C 3 S 7 5, S7 5, S7 5, S7 5, C 4 S 7 5, S7 4, S7 6, S7 5, C 5 S 7 6, S7 5, S7 5, S7 5, C 6 S 7 3, S7 5, S7 5, S7 4,

16 16 Journal of Appled Mathematcs and Decson Scences Table 7: The weghted lngustc decson matrx. C 1 C 2 C 3 C 4 C 5 C 6 S S S S S S S S S S Table 8: Lngustc postve-deal soluton LPIS, S and lngustc negatve-deal soluton LNIS, S. C 1 C 2 C 3 C 4 C 5 C 6 S S Table 9: Calculate the dstance from S and the dstance from S, the closeness coeffcentofeachstock. S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 d d CC Table 10: The overall concordance matrx. S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S S S S S S S S S S Step 5. Transform the lngustc evaluatons of weght of each crteron nto the lngustc varables of type 2 and aggregate the lngustc weght of each crteron as Table 6. Step 6. Calculate the weghted lngustc decson matrx V v j m n as Table 7. Step 7. Calculate the lngustc postve-deal soluton LPIS, S and lngustc negatve-deal soluton LNIS, S as Table 8.

17 Journal of Appled Mathematcs and Decson Scences 17 Table 11: The credblty matrx. S 1 S 2 S 3 S 4 S 5 S 6 S 7 S 8 S 9 S 10 S S S S S S S S S S Table 12: The concordance credblty degree, the dscordance credblty degree, the net credblty degree, and the outrankng ndex. Stock φ S φ S φ S OTI S S S S S S S S S S Table 13: Compute the rato of nvestment n accordance wth the rsk preference. Rank P t S P e S P RA S P RN S P RL s 1 S 5, S 5, S 5, S 5, S 5, S 9, S 9, S 9, S 9, S 9, S 1, S 1, S 1, S 1, S 1, S 4, S 4, S 4, S 4, S 4, S 7, S 7, S 7, S 7, S 7, S 6, S 3, Step 8. Calculate the dstance of each stock from S and the dstance from S,andthe closeness coeffcent of each stock as Table 9. Step 9. Defne nvestment threshold value as the average of the closeness coeffcent β t n 1 CC S /n, so the nvestment portfolo set s Ω t {S 1,S 4,S 5,S 7,S 9 } n accordance wth TOPSIS. The rato of nvestment based on TOPSISmethod s shown as Table 13.

18 18 Journal of Appled Mathematcs and Decson Scences Step 10. The ndfference threshold, preference threshold, and veto threshold values of each crteron can be determned n accordance wth the lngustc varables of type 2 as q j Δ 1( ) S 7 1 Δ 1( ) S , p j Δ 1( ) S 7 2 Δ 1( ) S , v j Δ 1( ) S 7 3 Δ 1( ) S , j 1,...,6. Step 11. Calculate the concordance matrx and the dscordance matrx of each par stock wth respect to each crteron.then, calculate the overall concordance matrx as Table 10 and the credblty matrx as Table 11. Step 12. Calculate the concordance credblty degree, the dscordance credblty degree, the net credblty degree, and the outrankng ndex as Table 12. Step 13. Defne nvestment threshold value as the average of the outrankng ndex β e n 1 OTI S /n, so the nvestment portfolo set s Ω e {S 1,S 3,S 4,S 5,S 6,S 7,S 9 } n accordance wth ELECTRE method. The rato of nvestment based on ELECTRE method s shown as Table 13. Step 14. Compute strct stock portfolo set as Ω p Ω t Ω e {S 1,S 4,S 5,S 7,S 9 }. Step 15. Accordng to the nvestment preference of nvestor, the result of the rato of nvestment based on combnng lngustc ELECTRE wth TOPSIS can be calculated as Table 13. Accordng to the result of numercal example, experts consdered that the proposed method s useful to help nvestor determne the stock portfolo. 5. Concluson In general, the stock portfolo decson problem adheres to uncertan and mprecse data, and fuzzy set theory s adequate to deal wth t. In ths proposed model, dfferent types of 2-tuple lngustc varables are appled to express the subjectve judgment of each expert. Expert can easly express hs opnon by dfferent types of 2-tuple lngustc varables. The generalzed translaton method of dfferent types of 2-tuple lngustc varables s appled to aggregate the subjectve judgment of each expert. It s a flexble way to aggregate the opnons of all experts. Then, a new decson-makng method has been presented n ths paper by combnng the advantages of ELECTRE wth TOPSIS methods. Accordng to the experts opnons, the lngustc ELECTRE method and lngustc TOPSIS method are used to derve the closeness coeffcent and the outrankng ndex of each stock, respectvely. Based on the closeness coeffcent, the outrankng ndex, and selecton threshold, we can easly obtan three type of the nvestment rato n accordance wth dfferent nvestment preference of fnal decson-maker. It s a reasonable way n real decson envronment. In other words, the proposed method provdes a flexble way to determne the stock portfolo under the uncertan envronment. In the future, the concept of combng dfferent decson methods for decdng stock portfolo wll be appled to dfferent felds such as R&D projects nvestment, bonus dstrbuton n a company. A decson support system wll be developed based on the proposed method for dealng wth the stock selecton problems n the future.

19 Journal of Appled Mathematcs and Decson Scences 19 References 1 X. Huang, Portfolo selecton wth a new defnton of rsk, European Journal of Operatonal Research, vol. 186, no. 1, pp , H. Markowtz, Portfolo selecton, Journal of Fnance, pp , C. A. Magn, Correct or ncorrect applcaton of CAPM? Correct or ncorrect decsons wth CAPM? European Journal of Operatonal Research, vol. 192, no. 2, pp , F. Tryak and M. Ahlatcoglu, Fuzzy stock selecton usng a new fuzzy rankng and weghtng algorthm, Appled Mathematcs and Computaton, vol. 170, no. 1, pp , E. F. Fama and K. R. French, The cross-secton of expected stock returns, Journal of Fnance, vol. 47, pp , C. D. Fensten and M. N. Thapa, Notes: a reformulaton of a mean-absolute devaton portfolo optmzaton model, Management Scences, vol. 39, no. 12, pp , W.F.Sharpe,G.J.Alexander,andJ.F.Baley,Investments, Prentce-Hall, Englewood Clffs, NJ, USA, 6th edton, T. L. Saaty, P. C. Rogers, and R. Bell, Portfolo selecton through herarches, Journal of Portfolo Manage, pp , N. C. P. Edrsnghe and X. Zhang, Generalzed DEA model of fundamental analyss and ts applcaton to portfolo optmzaton, Journal of Bankng & Fnance, vol. 31, no. 11, pp , F. Ben Abdelazz, B. Aoun, and R. El Fayedh, Mult-objectve stochastc programmng for portfolo selecton, European Journal of Operatonal Research, vol. 177, no. 3, pp , E. Ballestero, Stochastc goal programmng: a mean-varance approach, European Journal of Operatonal Research, vol. 131, no. 3, pp , B. Aoun, F. Ben Abdelazz, and J.-M. Martel, Decson-maker s preferences modelng n the stochastc goal programmng, European Journal of Operatonal Research, vol. 162, no. 3, pp , C. L. Hwang and K. Yoon, Multple Attrbute Decson Makng Methods and Applcatons, vol. 186 of Lecture Notes n Economcs and Mathematcal Systems, Sprnger, Berln, Germany, A. Papadopoulos and A. Karagannds, Applcaton of the mult-crtera analyss method Electre III for the optmsaton of decentralsed energy systems, Omega, vol. 36, no. 5, pp , C.-T. Chen, Extensons of the TOPSIS for group decson-makng under fuzzy envronment, Fuzzy Sets and Systems, vol. 114, no. 1, pp. 1 9, F. Herrera and E. Herrera-Vedma, Lngustc decson analyss: steps for solvng decson problems under lngustc nformaton, Fuzzy Sets and Systems, vol. 115, no. 1, pp , F. Herrera and L. Martnez, A 2-tuple fuzzy lngustc representaton model for computng wth words, IEEE Transactons on Fuzzy Systems, vol. 8, no. 6, pp , Z. Xu, Devaton measures of lngustc preference relatons n group decson makng, Omega, vol. 33, no. 3, pp , E. Herrera-Vedma, O. Cordón,M.Luque,A.G.Lopez,andA.M.Muñoz, A model of fuzzy lngustc IRS based on mult-granular lngustc nformaton, Internatonal Journal of Approxmate Reasonng, vol. 34, no. 2-3, pp , F. Herrera, L. Martnez, and P. J. Sanchez, Managng non-homogeneous nformaton n group decson makng, European Journal of Operatonal Research, vol. 166, no. 1, pp , L. A. Zadeh, Fuzzy sets, Informaton and Control, vol. 8, no. 3, pp , R. R. Yager, An approach to ordnal decson makng, Internatonal Journal of Approxmate Reasonng, vol. 12, no. 3-4, pp , C.-T. Ln and Y.-T. Chen, Bd/no-bd decson-makng a fuzzy lngustc approach, Internatonal Journal of Project Management, vol. 22, no. 7, pp , R.-C. Wang and S.-J. Chuu, Group decson-makng usng a fuzzy lngustc approach for evaluatng the flexblty n a manufacturng system, European Journal of Operatonal Research, vol. 154, no. 3, pp , E. Herrera-Vedma and E. Pes, Evaluatng the nformatve qualty of documents n SGML format from judgements by means of fuzzy lngustc technques based on computng wth words, Informaton Processng and Management, vol. 39, no. 2, pp , 2003.

20 20 Journal of Appled Mathematcs and Decson Scences 26 L. A. Zadeh, The concept of a lngustc varable and ts applcaton to approxmate reasonng I, Informaton Scences, vol. 8, pp , L. A. Zadeh, The concept of a lngustc varable and ts applcaton to approxmate reasonng II, Informaton Scences, vol. 8, pp , A. Kaufmann and M. M. Gupta, Introducton to Fuzzy Arthmetc: Theory and Applcatons, Van Nostrand Renhold, New York, NY, USA, F. Herrera and L. Martnez, A model based on lngustc 2-tuples for dealng wth multgranular herarchcal lngustc contexts n mult-expert decson-makng, IEEE Transactons on Systems, Man, and Cybernetcs, Part B, vol. 31, no. 2, pp , W.-S. Ta and C.-T. Chen, A new evaluaton model for ntellectual captal based on computng wth lngustc varable, Expert Systems wth Applcatons, vol. 36, no. 2, pp , E. Herrera-Vedma, F. Herrera, L. Martínez, J. C. Herrera, and A. G. López, Incorporatng flterng technques n a fuzzy lngustc mult-agent model for nformaton gatherng on the web, Fuzzy Sets and Systems, vol. 148, no. 1, pp , H.-F. L and J.-J. Wang, An mproved rankng method for ELECTRE III, n Internatonal Conference on Wreless Communcatons, Networkng and Moble Computng (WCOM 07), pp , 2007.

21 Advances n Operatons Research Advances n Decson Scences Journal of Appled Mathematcs Algebra Journal of Probablty and Statstcs The Scentfc World Journal Internatonal Journal of Dfferental Equatons Submt your manuscrpts at Internatonal Journal of Advances n Combnatorcs Mathematcal Physcs Journal of Complex Analyss Internatonal Journal of Mathematcs and Mathematcal Scences Mathematcal Problems n Engneerng Journal of Mathematcs Dscrete Mathematcs Journal of Dscrete Dynamcs n Nature and Socety Journal of Functon Spaces Abstract and Appled Analyss Internatonal Journal of Journal of Stochastc Analyss Optmzaton