Portfolio Performance Evaluation in a Modified Mean-Variance-Skewness Framework with Negative Data

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1 Available lie at It. J. Data Evelpmet Aalysis (ISSN X) Vl., N.3, Year 04 Article ID IJDEA-003,3 pages Research Article Iteratial Jural f Data Evelpmet Aalysis Sciece ad Research Brach (IAU) Prtfli Perfrmace Evaluati i a Mdified Mea-Variace-Skewess Framewrk with Negative Data Sh.Baihashemi a*,m.saei b, M.Azizi a (a) Departmet f Mathematics ad Cmputer Sciece, Faculty f Ecimics Allameh Tabataba i Uiversity. (b) Departmet f Applied Mathematics, Islamic Azad Uiversity, Cetral Tehra Brach, Tehra, Ira. Abstract Received 8 May 04, Revised 5 July 04, Accepted 6 September 04 The preset study is a attempt tward evaluatig the perfrmace f prtflis usig meavariace-skewess mdel with egative data. Mea-variace -liear framewrk ad meavariace-skewess - liear framewrk had bee prpsed based Data Evelpmet Aalysis, which the variace f the assets had bee used as a iput t the DEA ad expected retur ad skewess were the utput. Cvetial DEA mdels assume -egative values fr iputs ad utputs. Hwever, we kw that ulike retur ad skewess, variace is the ly variable i the mdel that takes -egative values. This paper fcuses the evaluati prcess f the prtflis i a mea-variace-skewess mdel with egative data. The prblem csists f chsig a ptimal set f assets i rder t miimize the risk ad maximize retur ad psitive skewess. This methd is illustrated by applicati i Iraia stck cmpaies ad extremely efficiecies are btaied via mea-variace-skewess -liear framewrk with egative data fr makig the best prtfli. The fidig culd be used fr cstructig the best prtfli i stck cmpaies, i varius fiace rgaizati ad public ad private sectr cmpaies. Keywrds: Prtfli, Data Evelpmet Aalysis (DEA), Skewess, Efficiecy, Negative data.. Itrducti I fiacial literature, a prtfli is a apprpriate mix ivestmets held by a istituti r private idividuals. Evaluati f prtfli perfrmace has created a large iterest amg emplyees als academic researchers because f huge amut f mey are beig ivested i fiacial markets. The * Crrespdig authr: shbaihashemi@atu.ac.ir. Tel: Fax:

2 04 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) thery f mea variace, Markwitz [] is csidered the basis f may curret mdels ad this thery is widely used t select prtflis. This mdel is due t the ature f the variace i quadratic frm.. Due t quadratic frm, a study by Arditti [], Kae [9] ad H ad Cheug [7] shw that ivestrs prefer skewess which meas that utility fuctis f ivestrs are t quadratic. Other prblem i Markwitz mdel is that icreasig the umber f assets will be develped the cvariace matrix f asset returs ad will be added t the ctet calculati. Due t these prblems sharp efactr mdel is prpsed by Sharp [6]. This methd reduces the umber f calculatis required ifrmati fr the decisi. Data evelpmet aalysis (DEA) has prved the efficiecy fr assessig the relative efficiecy f Decisi Makig Uits (DMUs) that emplyig multiple iputs t prduce multiple utputs (Chares et al [4]). Mrey ad Mrey [3] prpsed mea variace framewrk based Data Evelpmet Aalysis, which the variace f the prtflis is used as a iput t the DEA ad expected retur is the utput. Jr& Na [8] itrduced mea - variace skewess framewrk ad skewess f returs are als csidered as a utput. Briec et al. [3] itrduced shrtage fucti. This shrtage fucti btais a efficiecy measure that lks fr imprvemets i bth mea ad skewess ad decreases i variace. Kerstece et al. [0] itrduced a gemetric represetati f the MVS frtier related t a ew tl itrduced i the literature by Briec. Mhiri ad priget [] aalyze the prtfli ptimizati prblem by itrducig the higher mmets f the mai fiacial idex returs. I ew mdels istead f estimatig the whle efficiet frtier, ly the precti pits f the assets are calculated. I these mdels are used a -liear DEA-like framewrk where the crrelati structure amg the uits is take it accut. Cvetial DEA mdels assume -egative values fr iputs ad utputs. These mdels cat be used fr the case i which DMUs iclude bth egative ad psitive iputs ad/r utputs. Pltera et al. [4] csider a DEA mdel which ca be applied i the cases where iput/ utput data take psitive ad egative values.the ther mdels slve egative data such as Mdified slacks-based measure mdel (MSBM) [006], semi-rieted radial measure (SORM) [00] ad etc. The prtfli ptimizati prblem is a well-kw prblem i fiacial real wrld. The ivestr s bective is t get the maximum pssible retur a ivestmet with the miimum pssible risk. Als the ivestrs prefer t maximize psitive skewess. Sice there are a large umber f assets t ivest i, this bective leads t select the best assets via mea-variace-skewess -liear mdel with egative data. The rest f the paper is rgaized as fllws: Secti briefly reviews the prtfli perfrmace literature. Secti 3 explaismea-variace RDM ad mea-variace-skewess RDM -liear mdels. Secti 4 presets cmputatial results usig Iraia stck cmpaies data ad fially cclusis are give i secti 5.

3 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) Backgrud Prtfli thery t ivestig is published by Markwitz []. This apprach starts by assumig that a ivestr has a give sum f mey t ivest at the preset time. This mey will be ivested fr a time as the ivestr s hldig perid. The ed f the hldig perid, the ivestr will sell all f the assets that were bught at the begiig f the perid ad the either csume r reivest. Sice prtfli is a cllecti f assets, it is better that t select a ptimal prtfli frm a set f pssible prtflis. Hece the ivestr shuld recgize the returs (ad prtfli returs), expected (mea) retur ad stadard deviati f retur. This meas that the ivestr wats t bth maximize expected retur ad miimize ucertaity (risk). Rate f retur (r simply the retur) f the ivestr s wealth frm the begiig t the ed f the perid is calculated as fllws: (ed f perid wealth) (begiig f perid wealth) Retur = begiig f perid wealth () Sice Prtfli is a cllecti f assets, its retur r p ca be calculated i a similar maer. Thus accrdig t Markwitz, the ivestr shuld view the rate f retur assciated t ay e f these prtflis as what is called i statistics a radm variable. These variables ca be described expected the retur (mi r r p retur are calculated as fllws: r p i i p i i i i ) ad stadard deviati f retur. Expected retur ad deviati stadard f r, () Where: =the umber f assets i the prtfli / r p =The expected retur f the prtfli i r i =The prprti f the prtfli s iitial value ivested i asset i =The expected retur f asset i = The deviati stadard f the prtfli p i = The cvariace f the returs betwee asset i ad asset I the abve, ptimal prtfli frm the set f prtflis will be chse that maximum expected retur fr varyig levels f risk ad miimum risk fr varyig levels f expected retur(sharp [7] ). Data Evelpmet Aalysis is a parametric methd fr evaluatig the efficiecy f systems with multiple iputs ad multiple utputs. I this secti we preset sme basic defiitis, mdels ad

4 04 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) ccepts that will be used i ther sectis i DEA. They will t be discussed i details. Csider DMU, (,..., ) where each DMU csumes m iputs t prduce s utputs. Suppse that the bserved iput ad utput vectrs f DMU are X ( x,..., x m ) ad Y ( y,..., ) y s respectively, ad let X 0 ad X 0, Y 0 ad Y 0. A basic DEA frmulati i iput rietati is as fllws: s m mi ( s s ) r i r i s. t x s x i i i i,..., m, (3) y s y r r r r,..., s,, s, s 0, 0 Where is a -vectr f variables, s as-vectr f utput slacks, s a m-vectr f iput slacks ad set is defied as fllws: { R { R, } { R, } with cstat returs t scale, with -icreasig returs t scale, with variable returs t scale (4) Nte that subscript refers t the uit uder the evaluati. A DMU is efficiet iff ad all slack variables s, s equal zer; therwise it is iefficiet (Chares et al. [4]). I the DEA frmulati abve, the left had sides i the cstraits defie a efficiet prtfli. θ is a multiplier defies the distace frm the efficiet frtier. The slack variables are used t esure that the efficiet pit is fully efficiet. This mdel is used fr asset selecti. The prtfli perfrmace evaluati literature is vast. I recet years these mdels have bee used t evaluate the prtfli efficiecy. Als i the Markwitz thery, it is required t characterize the whle efficiet frtier but the prpsed mdels by Jr & Na d t eed t characterize the whle efficiet frtier but ly the precti pits.

5 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) The distace betwee the asset ad its precti which meas the rati betwee the variace f the precti pit ad the variace f the asset is csidered as a efficiecy measure ( ). I this framewrk, there is assets, is the weight f asset i the precti pit, r is the expected retur f asset, ad are the expected retur ad variace f the asset uder evaluati respectively. Efficiecy measure ca be slved via fllwig mdel: mi ( s s ) s. t. E, r s E ( ( )) r s 0 5 Mdel (5) is revealed by the -parametric efficiecy aalysis Data Evelpmet Aalysis (DEA). Jr ad Na [8] exteded the described apprach i (5) it mea-variace-skewess framewrk where is the skewess f the asset uder evaluati. The efficiecy measure ca be slved thrugh usig the fllwig mdel: mi ( s s s 3 ) s. t. E r s, E ( ( r )) s 3 E ( ( r )) s Mdel (6) prects the asset with the efficiet frtier by fixig the expected retur ad skewess levels ad miimizig the variace.

6 00 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) B D 0 5 C A B Assets Precti Pits Fig. Differet prectis (iput rieted, utput rieted, cmbiati rieted) Fig illustrates differet precti that csist f iput rieted, utput rieted ad cmbiati rieted i mdels f data evelpmet aalysis. C is the precti pit btaied via fixig expected retur ad miimizig variace, B via maximizig retur ad miimizig variace simultaeusly, ad D via fixig variace ad maximizig retur. I the cvetial DEA mdels, each DMU (,..., ) is specified by a pair f -egative m s iput ad utput vectrs ( x, y ) R, i which iputs x ( i,..., m) are utilized t prduce utputs, y ( r,..., s). These mdels cat be used fr the case i which DMUs iclude bth r egative ad psitive iputs ad/r utputs. Pltera et al.[4] csider a DEA mdel which ca be applied i the cases where iput/ utput data take psitive ad egative values. Rag Directial Measure (RDM) mdel prpsed by Plera et al. ges as fllws: i max st x x R i,..., m, i i i y y R r,..., s, (7) r r r, 0,...,. Ideal pit ( I ) withi the presece f egative data, is : I (max { y : r,..., s},mi { x : i,..., m} where r i

7 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) R x mi { x :,..., }, i,..., m, i i i R max { y :,..., } y r,..., s. (8) r r r, The ther mdels slve egative data such as Mdified slacks-based measure mdel (MSBM), Emruzead [6], semi-rieted radial measure (SORM), Sharp et al. [5] ad etc. 3. Mdified mdels i the presece f egative data I mdel (6) if retur ad skewess is csidered psitive the the results are crrect, hece the prblem ca happe ly if retur ad skewess ca take bth psitive ad egative values. As we kw that ulike retur ad skewess, variace is ly a -egative umber. Als, Bhattacharyya et al. [] predicted that fr assets with egative expected returs, expected retur will be a decliig ad cvex fucti f skewess. Assume the basic prblem is t select a prtfli frm fiacial assets. A prtfli (,..., ) is a vectr f prprtis i each f these fiacial assets with i. Excludig shrt sales, e must impse the cditi i the set f admissible prtfli is writte as fllws: { R : i, i 0} i. Assets are characterized by a expected retur E[ i cv(r i,r ) E[(ri E[r i])(r E[r ])] (9) Ad by a c-skewess tesr f rak three with: r i i 0 ]. By a cvariace matrix with i E[(ri E[r i])(r E[r ])(rk E[r k ])] (0). The, we have: E[r( )] E[r ] i i i V ar[r( )] E[(r( ) E[r( )]) ] i i () i, 3 Sk[r( )] E[(r( ) E[r( )]) ] T cdese tati, the fucti i,,k ( ) (E[r( )], V ar[r( )],Sk[r( )]) i k i 3 : R defied by: fr all i {,..., }. I geeral,

8 04 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) is itrduced t represet the expected retur, variace ad skewess f a give prtfli.i the remider, a elemet f 3 R is called a MVS pit. Thus, a MVS pit ca be the image by f a prtfli, r ay arbitrary pit i this three-dimesial space. It is useful t defie the MVS image f as the image ( ) ( ) { ( ); }., with This set ca be exteded by defiig a MVS dispsal represetati set via: DR ( ) (R R R ). Fr the purpse f gaugig prtfli efficiecy, a subset f this represetati set the weakly ad strgly efficiet frtier must be defied as: Defiiti : I the MVS space, the weakly efficiet frtier is defied as: w ' ' ' ' ' ' ( ) {(M, V,S) DR;( M, V, S ) ( M, V, S) (M, V,S ) DR}. Defiiti : I the MVS space, the strgly efficiet frtier, is defied as: s ' ' ' ' ' ' ' ' ' ( ) {(M, V,S) DR;( M, V, S ) ( M, V, S)ad ( M, V, S ) ( M, V, S) (M, V,S ) DR} Extremely, we preset fllwig -liear mea-variace RDM mdel the basis f egative data: max s.t. E r R E ( (r )) R () 0 Ideal pit ( I ) withi the presece f egative data, is I (mi { }, max { }) where, R max { :,..., } (3) R mi { :,..., }. The abve mdel ca be expressed as fllwig: max s.t. E r( ) R Var r( ) R (4) 0

9 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) Als, we preset fllwig -liear mea-variace-skewess RDM mdel the basis f egative data: max s.. t E r R E r R ( (5) 3 E ( ) r R 0 (5) Ideal pit ( I ) withi the presece f egative data, is I (mi { }, max {, }) Where R R mi (6) R max max The abve mdel ca be expressed as fllwig: max s. t. E r( ) R Var r ( ) R (7) Sk r( ) R 0 The methdlgy i this paper starts with asset selecti via perfrmace evaluati i presece f egative data. The data used fr this methdlgy is frm 0 Iraia stck cmpaies. I may cases similar t this example there are a lt f assets. It is better that starts with asset selecti via perfrmace evaluati. The chice f the asset ca be radm r discrete. The radm chice f assets is usually biased ad d t prmise a ptimum prtfli; hece it is mre ratial t have a bective chice while selectig the assets t be icluded i the prtfli. Perfrmace evaluati is calculated by usig mdels 4 ad 7.

10 04 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) Applicati i Iraia Stck Cmpaies We illustrate ur apprach i -liear mea-variace-skewess mdel fr a data set 0 Iraia stck cmpaies. A list f stcks used is prvided i Table. I this reprt, there is expected retur, variace, skewess f stcks which expected retur ad skewess are csidered as utput ad variace is as iput. The example is received frm Iraia stck cmpaies ad is abut prtfli perfrmace evaluati i a mea-variace-skewess RDM framewrk. Thus, we kw that ulike retur ad skewess, variace is the ly variable i the mdel that takes -egative values. I the aalysis, the variace f the stcks is used as a iput t the DEA ad expected retur ad skewess are used as utput. Table. Descriptive statistics f the Iraia stck cmpaies Iraia stck cmpaies Expected retur Variace Skewess VNVIN VPARS VBHMN VPASAR DGABR STRAN FBAHNR FMLI FVLAD KCHINI VTVSA VLSAPA VNFT VTGART VKHARZM VSAKHT KHSAPA VSINA RTKG VBMLT Liear MV ad liear MVS with RDM mdel are calculated at table. Als, liear MV efficiecy measure ad -liear MVS efficiecy measure usig mdels 4 ad 7 are calculated at table.

11 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) Table. Efficiecy measure f the Iraia stck cmpaies Iraia stck cmpaies N-liear meavariace MV RDM N-liear mea-variace- MVS RDM mdel liear skewess RDM mdel RDM (mdel 4) (mdel 7) liear VNVIN VPARS VBHMN VPASAR DGABR STRAN FBAHNR FMLI FVLAD KCHINI VTVSA VLSAPA VNFT VTGART VKHARZM VSAKHT KHSAPA VSINA RTKG VBMLT Table represets the calculated ad cmpared the results f efficiecy f mdel 4 ad mdel 7 t liear MV RDM mdel ad liear MVS RDM mdel. As see i Table, mdel 4 ad mdel 7 scres are as a cservative estimate f the liear MV RDM ad liear MVS RDM scres. I this example all the liear DEA scres are greater tha the -liear mdel.als, we cmpare the results liear MV RDM ad liear MVS RDM. Because f, skewess equati is icreased the results are t better. But, we kw that the RDM mdel gives iefficiecy scre. Therefre, the efficiecy scre liear MVS RDM mdel is greater tha liear MV RDM. But, thse havig psitive skewess, have mre efficiecy icreasig. Fr example, STRAN havig the mst psitive skewess, Firstly liear MV RDM efficiecy scre has 0.4 ad liear MVS RDM efficiecy scre,secdly becmes. But thse havig egative skewess, d t chage i efficiecy scre. N-liear MVS RDM efficiecy

12 044 Sh.Baihashemi,et al /IJDEA Vl., N.3, (04) scres are becme early better tha -liear MV RDM efficiecy scres. But thse havig psitive skewess, have mre efficiecy icreasig. Fr example, STRAN ad KCHINE. The results are btaied by Geeral Algebraic Mdelig System (GAMS) sftware. 5. Cclusi This paper itrduced a measure fr prtfli perfrmace usig -liear mea-variaceskewess RDM mdel. Jr ad Na had prpsed mdels fr evaluatig prtfli efficiecy i which Data Evelpmet Aalysis mdel was emplyed. I these mdels was used a -liear DEA-like framewrk where the crrelati structure amg the uits was take it accut. We have applied mdel 4, ad mdel 7 with retur ad skewess as utput ad the variace as the iput t 0 stcks. The detailed results are preseted i Table. I the umerical example is als bserved that cmpared with liear, these mdels are highly exact i all the uits, that is, all the liear DEA scres are greater tha the -liear mdels. This meas that the DEA frtier is always dmiated via the -liear mdified mea-variace frtier. But, thse havig psitive skewess, have mre efficiecy icreasig. Fr example, STRAN havig the mst psitive skewes. Firstly, liear MV RDM efficiecy scre has 0.4 ad liear MVS RDM efficiecy scre, Secdly becmes. But thse havig egative skewess, d t chage i efficiecy scre. N-liear MVS RDM efficiecy scres are early better tha -liear MV RDM efficiecy scres. But, thse havig psitive skewess, have mre efficiecy icreasig. Fr example, STRAN ad KCHINE. Ackwledgemet The authrs sicere thaks g t the kw ad ukw frieds ad reviewers wh meticulusly cvered the article ad prvided us with valuable isights. Referece [] Arditti, Skewess ad ivestrs_ decisis: A reply, Jural f Fiacial ad Quatitative Aalysis, vl. 0, pp , 975. [] Na. Bhattacharyya, Th. garrett, St. Luis, Why peple chse egative expected retur assets- a emprical examiati f a utility Theretic explaati,006. [3] W. Briec, K. Kerstes ad O. Jkug, Mea-Variace-Skewess Prtfli Perfrmace Gaugig: A Geeral Shrtage Fucti ad Dual Apprach, Maagemet Sciece, vl. 53,pp 35 49, 007. [4] A. Chares, W.W. Cper ad E. Rhdes, Measurig Efficiecy f Decisi Makig Uits, Eurpea Jural f Operatial Research, vl., pp , 978.

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