"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA STOCK INVESTMENT MANAGEMENT NDER NCERTAINTY Madaa Ecatera ANDREICA 1 Mar ANDREICA ABSTRACT Ths paper presets a stock vestet aageet probe uder ucertaty soved by appyg a portfoo seecto agorth for terva attrbutes. The curret top 10 ost traded stocks o the Bucharest Stock Exchage Market were take to cosderato ths study. The resuts dcated that accordg to the decso aker rsk atttude there are severa dfferet fa portfoo structures. For stace, for the case of a rsk eutra decso aker, the structure was sar to the case of a rsk adverse decso aker, whe the case of a rsk over decso aker, however, the fa stock portfoo had a ore dfferet structure. KEYWORDS: terva data, stock portfoo seecto, ucertaty JE CASSIFICATION: G11, D81 1. INTRODCTION Stock arkets are turg to very ustabe vestet feds uder ecooc recesso ad faca stabty. I ths cotext stock vestet aageet shoud tur to a true busess chaege for ay vestor. That s why ew ad effcet agorths are expected order to sove copex decso probes. The terature revew s qute vast ad offers varous effcet decso akg techques uder copete forato (see Resteau et a., 007, uder rsk (see Adreca et a., 008, or uder ucertaty (Ar et a., 008; Adreca et a., 010; Che et a., 009; Ye ad, 009. Oe of the best kow decso akg optzato ethods are ut-attrbute ad utobectve decso akg, fuzzy decso rues (Stoca et a., 008 ad dyac prograg. Accordg to Adreca et a. (010 ut-attrbute decso akg (MADM refers to akg preferece decsos over the avaabe ateratves that are characterzed by utpe, geeray cofctg attrbutes. I cassc MADM probes, ost of the put varabes are assued to be crsp data. However, sce ost cases t s qute dffcut to precsey detere the exact vaue of the attrbutes uder copete forato ad ucertaty, ther vaues are better descrbed usg terva data. I ths paper we sove a stock portfoo seecto probe uder ucertaty by appyg a portfoo seecto agorth for terva data attrbutes. The paper s structured as foows. I secto we suarze the portfoo seecto ethod for terva data proposed by Adreca et a., 010, whe a stock portfoo decso probe w be soved Secto 3. Secto 4 cocudes. 1 Acadey of Ecooc Studes ad The Natoa Scetfc Research Isttute for abour ad Soca rotecto, Roaa, adaa.adreca@ga.co Acadey of Ecooc Studes ad The Coerca Acadey of Satu Mare, Roaa, aradreca@ga.co 655
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA. THE ORTFOIO SEECTION AGORITHM FOR INTERVA DATA I ths secto we suarze the exteso of a portfoo seecto agorth prevousy proposed by Adreca et a. 010 for the case of terva data. The portfoo seecto probe cossts how to aocate a aout of oey to a uber of goods or stocks order to brg a ost proftabe retur for vestors. Therefore, the agorth pes usg both a TOSIS ad a EECTRE III ethod for terva data order to seect the best portfoo structure. The reaso for choosg these partcuar ethods cossts the fact that the TOSIS ethod s a effectve ethod to detere the rakg of decso ateratves, but caot, however, dstgush the dfferece degree betwee two decso ateratves easy. O the other had, athough the EECTRE III ethod ca easy copare the degree of dfferece aog a ateratves, t caot aways provde tota orderg. That s why, whe cobg the ethods, a fa portfoo wth proved characterstcs s obtaed. The geera MADM probe s preseted the for of a atrx, whch there are rows, represetg dfferet ateratves ad cous, represetg the crtera specfyg the propertes of the ateratves. I the terva data approach the assesset of ateratve A wth respect to [x, x ] crtero C s represeted by the tervas ad the vector of weghts s repaced wth a vector whose copoets are tervas w [ w1, w1 ],[ w, w ],...,[ w, w ]. The decso atrx s as foows: C C... C 1 A [ x, x ] [ x, x ]... [ x, x ] A [ x, x ] [ x, x ] [ x, x ] A [ x, x ] [ x, x ] [ x, x ] 1 11 11 1 1 1 1... 1 1... 1 1.1. The TOSIS ethod for terva data The basc cocept of TOSIS ethod s that the chose ateratve shoud have the shortest dstace fro the postve dea souto ad the farthest dstace fro the egatve dea souto. As preseted Jahashahoo et a. (006 the exteded TOSIS ethod for terva data has the foowg steps: Step 1. Frst cacuate the orazed decso atrx, usg the foruas for each terva: x x, 1,.. ; 1,.., 1,.. ; 1,.. [( x ( x ] 1 [( x ( x ] 1 Step. Appyg a Iprecse Shao s Etropy ethod (see otf et a., 010 order to w detere the average obectve crsp weghts for each crtero as: w w, 1,.. w ( w w /, 1,.. w 1. The weghts are the oraze as: so that 1 w 1. After that, the weghted orazed terva decso atrx s costructed as: v w, 1,.. ; 1,.. v w, 1,.. ; 1,.. 656
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA Step 3. The postve ad the egatve dea soutos are detered as: A {(ax v B,( v C} A {( v B,(ax v C} where B s assocated wth the beeft crtera ad C wth the cost crtera. Step 4. The separato of each ateratve fro the postve ad the egatve dea soutos are cacuated as: 1 B C d { ( v v ( v v }, 1,.. 1 B C d { ( v v ( v v }, 1,.. Step 5. A coseess coeffcet s defed to detere the rakg order of a ateratves a d CC, 1,.. descedg order: d d.. EECTRE III ethod for terva data The EECTRE III ethod deas wth pseudo-crtera stead of true crtera, aowg the foowg types of prefereces betwee ateratves: strog preferece, weak preferece ad dfferece. I order to do that, t uses a preferece threshod p, a dfferece threshod q ad a veto threshod v. The EECTRE III ethod exteso for terva data preseted by Adreca et a. (010 pes the foowg steps: Step 1. A rsk atttude factor for the decso aker s frst troduced, sary to (Ye ad, 009, order to trasfor a terva vaue to a exact vaue. I case of a beeft crtero, the x x xˆ x xˆ exact vaue x s obtaed as:, where s the dde vaue of the terva ad s xˆ x x the wdth of the terva, easured as:, whe case of a cost crtero, the exact x x xˆ vaue x s obtaed as:. The rsk factor represets the rsk atttude of the decso aker ad takes vaues betwee -0.5 ad 0.5. If the decso aker s rsk adverse, the the rage of the rsk factor s 0.5 0, whe f the decso aker s rsk over, the rsk factor s 0 0.5. The case whch the decso aker s rsk eutra pes that 0. C (, Step. The cocordace dex s cacuated for each ad wth respect to each crtero as: 1, q p C (,, q p p q 0, p D (, Step 3. The dscordace dex s the cacuated for each par of ateratves wth respect to each crtero: 1, v p D (,, p v v p 0, p Step 4. The overa cocordace dex for each ad s detered as: 657
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA C(, w C (, 1 Step 5. The credbty atrx S(, of each par of ateratves s cacuated as: C(,, D (, C(,, S(, 1 (, D C(,, othewse SC(, 1 (, C where SC(, D (, C(, s the set of crtera for whch: Step 6. The the cocordace credbty ad dscordace credbty degrees are defed as: ( S(,,, 1,.., ( S(,,, 1,.. where the cocordace credbty degree represets that the degree of the ateratve s at east as good as a the other ateratves, whe the dscordace credbty degree represets that the degree of a the other ateratves s at east as good as the ateratve. Based o these two dcators, a et credbty degree for each ateratve s defed as: ( ( ( whch has hgher vaues whe the ateratve s cosdered ore attractve coparso to the other ateratves. Step 7. Fay, a outrakg dex OTI, s defed for each ateratve the foowg aer: ( 1 OTI( 1 Based o the outrakg dex the fa orderg of the ateratves s obtaed..3 ortfoo seecto agorth The portfoo seecto ethod uder ucertaty proposed by Adreca et a. (010 requres a cobato of the resuts obtaed fro the two exteded versos of TOSIS ad EECTRE III ethods for terva data. It ca actuay be assued that TOSIS ad EECTRE III ethods represet two decso akers of the portfoo seecto. That s why the best decso w be ade whe takg to cosderato both experts opos regardg the set of ateratves that ay ead to best portfoo. The portfoo seecto agorth for terva data has the foowg steps: Step 1. Appy the exteded TOSIS ethod for terva data ad detfy the coseess coeffcet CC for each ateratve. CC 1 TOSIS Step. Detere the threshod ad the detfy the vestet portfoo set of T { CC TOSIS } the TOSIS ethod as:. Step 3. Appy the exteded EECTRE III ethod for terva data ad detfy the outrakg dex OTI for each ateratve. OTI ( 1 EECTRE Step 4. Detere the threshod ad the detfy the vestet portfoo set E { OTI ( EECTRE } of the EECTRE III ethod as: Step 5. The decso upo the fa vestet portfoo set pes the tersecto of the two portfoo sets that resuted based o the exteded TOSIS ad EECTRE III ethods for terva 658
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA data, T E. Accordg to the coseess coeffcet, the vestet portfoo ratos for the TOSIS probe are cacuated as: CC(, CC( T _ ( 0, otherwse Whe accordg to the outrakg dex, the vestet portfoo ratos for the EECTRE III probe are cacuated as: OTI(, OTI( E _ ( 0, otherwse Step 6. Fay, the rsk atttude of the decso aker s take to cosderato by assug that the decso aker ca ether be rsk adverse, rsk eutra or rsk over. Accordg to the rsk atttude of the decso aker, the fa portfoo ratos of the strct vestet portfoo set are detered based o the foruas: ( (, ( T _ E _ RA ( T _ (, E _ (, RN R ax( T _ (, E _ ( ax( T _ (, E _ ( ( ( ( T _ E _ ( ( ( T _ E _ where RA represets the fa portfoo ratos case the decso aker s rsk adverse, RN for the rsk eutra case, whe R represets the fa portfoo ratos case the decso aker s rsk over. 3. THE STOCK INVESTMENT MANAGEMENT ROBEM A decso aker wats to vest a su of oey to the Bucharest Stock Exchage Market ad takes to cosderato the top 10 ost traded stocks o the Roaa capta arket: TV, F, SN, TGN, TE, BIO, BRK, BRD, BVB ad EMA, for whch othy rsk ad retur for the perod Jauary 01 Septeber 01 are kow. I order to cacuate each stock s rsk ad retur for these oths of the year 01, we frst coected data regardg each stock prce evouto fro the Karket webste. We the detfed the u ad the axu vaue for each stock rsk ad retur ad suarzed the resuts Tabe 1. We cosdered that both decso crtera cocerg rsk ad retur are of equa portace the portfoo decso probe. The terva decso atrx for the stock portfoo seecto probe s preseted Tabe 1. The portfoo agorth for terva data was the apped. Frst, the TOSIS ethod for terva data was used order to detere the portfoo set of stocks that are the cosest to the dea postve souto ad the farthest to the dea egatve souto. It resuted the foowg copete order of stocks: F (0.601, SN (0.600, BIO (0.581, BRK (0.559, TE (0.518, BRD (0.518, BVB (0.499, TV (0.494, TGN (0.484 ad EMA (0.476 out of whch oy the frst 4 stocks were seected to be the best by havg hgher coseess coeffcet tha the average vaue TOSIS of 0,533. The TOSIS stock portfoo structure s the foowg: F (5.66%, SN (5.65%, BIO (4.8% ad BRK (3.87%. 659
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA Tabe 1. The terva decso atrx RETRN (ax RISK ( TV [-1.16%, 0.73%] [1.40%,.81%] F [-0.55%, 0.88%] [0.88%,.04%] SN [-0.6%,1.00%] [0.70%,.4%] TGN [-1.03%, 0.40%] [0.74%,.91% ] TE [-0.8%, 0.34%] [0.71%,.01%] BIO [-0.38%, 0.55%] [0.53%,,77%] BRK [-0.46%, 0.65%] [1.09%, 3.13%] BRD [-0.7%, 0.3%] [0.7%,.6%] BVB [-1.4%, 0.9%] [0.9%,.5%] EMA [-1.1%, 0.6%] [1.3%, 3.5%] Source: ade by authors usg data fro www.karket.ro ad www.bvb.ro The, the exteded verso of EECTRE III ethod for terva data was apped, order to detfy the secod stock portfoo set, based o par coparsos of each cobato of stocks. For that, we frst had to decde upo the eve of the paraeter. We used -0.5 case the decso aker s rsk adverse, 0 for the rsk eutra case ad 0.5 case the decso aker s rsk over. Secody, the threshod eves of the paraeters p, q ad v were predetered. Sar to Che ad Hug s approach (009 whch q= 1/6; p=/6 ad v=3/6, we used the foowg foruas for coputg the threshods. et MD be the axu dfferece betwee two ateratves for crtero. We set the dfferece threshod q to 1/6 * MD, the preferece threshod p to be /6 * MD ad the veto threshod v to 3/6 * MD. After that we coputed the cocordace ad dscordace dex order to detere the credbty atrx ad the cocordace ad dscordace credbty degrees. Based o that, we were abe to cacuate the OTI vaues for each ateratve ad to estabsh the fa rakg of the ateratves for each possbe decso aker s rsk atttude, as preseted Tabe, where betwee brackets are the OTI vaues hgher tha EECTRE =0.5. Tabe. The resuts of EECTRE III ethod RANK RISK RISK RISK ADVERSE NETRA OVER 1 F (0.87 SN (0.90 SN (0.90 SN (0.81 F (0.89 F (0.8 3 TE (0.78 BIO (0.84 BVB (0.81 4 BIO (0.67 BRK (0.55 BIO (0.64 5 BRD (0.61 TE (0.55 6 BRK (0.5 beta=0,5 beta=0,5 beta=0,5 Source: ade by authors usg data fro www.karket.ro The fa portfoo structures whe cosderg the three types of decso aker s rsk atttude are descrbed Fgure 1. Oe ca otce that there are severa dffereces the fa rakg of the 660
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA ateratves based o the decso aker rsk atttude, whe tersectg the portfoo sets of the two MADM ethods. For stace, the case of rsk adverse decso aker, the best cobato of stock vestet cossts 7.7% SN stocks, 7.7% F stocks, 5.1% BIO stocks ad 19.5% BRK stocks. I case of rsk eutra decso aker, the structure s sar to the case of rsk adverse decso aker, but oe shoud vest 0.8% ess SN stocks as we as 0.8% ess F stocks, 0.5% ore BIO stocks ad 1.1% ore BRK stocks. Fgure1. The fa stock portfoo structure Source: ade by the authors I the case of a rsk over decso aker, however, the fa stock portfoo s ade up of oy three stocks as coparatve to the prevous resuts ad has the foowg structure: SN a proporto of 36.3%, F a proporto of 3.8% ad BIO a proporto of 30.9%. 4. CONCSIONS I ths paper we sove a stock portfoo seecto probe uder ucertaty by appyg a portfoo seecto agorth for terva data attrbutes, such as: rsk ad retur. The partcuar portfoo seecto agorth that was apped ths paper has the advatage of aowg descrbg ucertaty based o terva data attrbutes. The resuts dcated dfferet fa portfoo structures accordg to the decso aker rsk atttude. For stace, for the case of a rsk eutra decso aker, the structure was sar to the case of a rsk adverse decso aker, whe the case of a rsk over decso aker, however, the fa stock portfoo had a ore dfferet structure tha the prevous resuts, dcatg that the BRK stocks are ess favorabe the fa stock portfoo of a rsk over decso aker, sce t brgs ower returs. 661
"AROACHES IN ORGANISATIONA MANAGEMENT" 15-16 Noveber 01, BCHAREST, ROMANIA REFERENCES Ar, M., Nosrata, N.E., Jashd A. & Kaze A. (008 Deveopg a ew EECTRE ethod wth Iterva Data Mutpe Attrbute Decso Makg robes. Jouras of Apped Sceces 8(, 4017-408. Adreca, M.E., Dobre,I., Adreca, M.I. & Resteau, C. (010 A New ortfoo Seecto Method based o Iterva Data, Studes Iforatcs ad Cotro Joura 19, Issue 3, 53-6 Adreca M.E., Dobre, I., Adreca, M., Nţu, B. & Adreca, R., (008 A New Approach of the Rsk roect fro Maagera erspectve. Ecooc Coputato ad Ecooc Cyberetcs Studes ad Research Joura, vo. 4, o.1-, 11-130. Che, C.T. & Z.Hug,W. (009 A New Decso-Makg Method for Stock ortfoo Seecto Based o Coputg wth gustc Assesset. Joura of Apped Matheatcs ad Decso Sceces Voue. Jahashahoo, G.R., otf, F.H. & Izadkhah, M. (006 A agorthc ethod to exted TOSIS for decso-akg probes wth terva data. Apped Matheatcs ad Coputato 175, 1375 1384. otf, F.H. & Faahead, R. (010 Iprecse Shao s Etropy ad Mut Attrbute Decso Makg. Etropy 1, 53-6. Resteau C., Sood, M., Adreca, M. & Mta, E. (007 Dstrbuted ad arae Coputg MADM Doa usg the OTCHOICE Software. roceedgs of the 7 th WSEAS Iteratoa Coferece o Apped Coputer Scece, 376-384. Stoca, M., Ncoae, D., gureau, M. A., Adreca, A. & Adreca, M.E. (008 Fuzzy Sets ad Ther Appcatos. roceedgs of the WSEAS Iteratoa Coferece o Matheatca ad Coputg Busess ad Ecoocs, 197-0. Ye, F. &, Y.N. (009 Group ut-attrbute decso ode to parter seecto the forato of vrtua eterprse uder copete forato. Expert Systes wth Appcatos 36, 9350 9357. www.karket.ro www.bvb.ro 66