PORTFOLIO OPTIMIZATION IN THE FRAMEWORK MEAN VARIANCE -VAR

Transcription

1 Lecturer Floret SERBAN, PhD Professor Vorca STEFANESCU, PhD Departmet of Mathematcs The Bucharest Academy of Ecoomc Studes Professor Massmlao FERRARA, PhD Departmet of Mathematcs Uversty of Reggo Calabra, Italy PORTFOLIO OPTIMIZATION IN THE FRAMEWORK MEAN VARIANCE -VAR Abstract. Ths paper proposes a model for portfolo optmzato. Frstly we compare the mea-varace method wth the mea -VaR method ad we search the lk betwee the mea-varace effcet set ad the mea-var effcet set. The we aalyze two portfolo optmzato approaches. The frst s a two-stage portfolo optmzato approach usg, order, both mea-varace ad mea-var approaches. The secod s a geeral mea-varace-var approach usg both varace ad VaR as a double-rsk measure smultaeously. Fally we cosder the case of a equty portfolo at the Itala Stock Market. We use data aalyze for portfolo selecto, the estmato of rsk for each stock ad after we solve the portfolo optmzato the framework mea var. Keywords: portfolo optmzato, rsk measures, mea-var aalyze, meavarace aalyze. JEL CLASSIFICATION : C, C6, G.. Itroducto Mea-rsk models are stll the most wdely used approach the practce of portfolo selecto. Wth mea-rsk models, retur dstrbutos are characterzed ad compared usg two statstcs: the epected value ad the value of a rsk measure. Thus, mea-rsk models have a ready terpretato of results ad most cases are coveet from a computatoal pot of vew. Sceptcs o the other had may questo these advatages sce the practce of descrbg a dstrbuto by ust two parameters volves great loss of formatos. It s evdet that the rsk measure used plays a mportat role makg the decsos. Varace was the frst rsk measure used mea-rsk models (Markowtz 95) ad, spte of crtcsm ad may proposals of ew rsk measures ( Koo ad Yamazak 99, Ogryczak ad Ruszczysk999,, Rockafellar ad Uryasev, ), varace s stll the most wdely used measure of rsk the practce of portfolo selecto.

2 Floret Serba,Vorca Stefaescu, Massmlao Ferrara For regulatory ad reportg purposes, rsk measures cocered wth the left tals of dstrbutos (etremely ufavourable outcomes) are used. The most wdely used rsk measure for such purposes s Value-at-Rsk (VaR). However, t s kow that VaR has udesrable theoretcal propertes (t s ot subaddtve) ad thus fals to reward dversfcato). I addto, optmzato of VaR leads to a o-cove NP-hard problem whch s computatoally tractable. I spte of a cosderable amout of research, optmzg VaR s stll a ope problem ( Larse et al. ). I ths paper, we propose a two-stage portfolo optmzato approach whch has all the stregths of the mea-var ad the mea-varace approaches, ad overcomes ther shortcomgs as the two stages complemet oe aother. Ths approach also uses more formato of the uderlyg dstrbuto of the portfolo retur. Here, varace ad VaR as rsk measures are used separately two stages accordg to a prorty order of the two rsk measures. I stage oe, we use a prmary rsk measure to collect all effcet portfolos. I stage two, we use a secodary rsk measure to re-evaluate (optmze) these effcet portfolos from stage oe. Ths approach provdes better results tha the mea-varace ad the mea-var approaches cosdered separately. The mea-varace-var effcet portfolo may ot be mea-varace effcet or mea-var effcet. We also show that the mea-varace ad the mea-var approaches are specal cases of the mea-varace-var approach. May papers have bee publshed that are related to ths work, the most related beg works by Aleader ad Baptsta [] ad Basak ad Shapro [999]. The frst study compares the mea-varace ad mea-var approaches for two specal cases: multvarate ormal dstrbuto ad multvarate t-dstrbuto. The secod study aalyzes optmal polces focusg o the VaR based rsk maagemet. Our work does ot oly compare the mea-varace ad mea-var approaches a geeral case, but also merges the two approaches to oe sgle approach. The rest of ths paper s orgazed to for sectos. I Secto, we revew the mea-rsk approach usg both varace as well as rsk measures VaR. I Secto 3, we propose a more geeral portfolo optmzato strategy: the mea-varace-var model. The usual mea-varace model ad the mea-var model are specal stuatos of ths model. Both varace ad VaR are used as the rsk measures durg the procedure of optmzato. I Secto, we cosder the case of a stocks portfolo lsted at the Itala Stock Market. We use data aalyze for portfolo selecto, the estmato of rsk wth VaR rsk measures for each stock ad fally we solve the portfolo optmzato the framework mea var.

3 Portfolo Optmzato the Framework Mea Varace -VaR. Mea-rsk models Mea-rsk models were developed the early 95s for the portfolo selecto problem. I hs semal work Portfolo selecto, Markowtz (95) proposed varace as a rsk measure. Sce the, may alteratve rsk measures have bee proposed. The questo of whch rsk measure s most approprate s stll the subect of much debate. I mea-rsk models, two scalars are attached to each radom varable: the epected value (mea) ad the value of rsk measure. Preferece s the defed usg a trade-off betwee the mea where a larger value s desrable ad rsk where a smaller value s desrable. I the mea-rsk approach wth the rsk measure deoted by, radom varable R domates (s preferred to) radom varable E R ER ad R y R y R y f ad oly f: wth least oe strct equalty. Alteratve, we ca say that portfolo domates portfolo y. I ths approach, the choce (or the radom varable R ) s effcet (o-domated) f ad oly f there s o other choce y such that y R has hgher epected value ad less rsk tha R. Ths meas that, for a gve level of mmum epected retur, R has the lowest possble rsk, ad, for a gve level of rsk, t has the hghest possble epected retur. Plottg the effcet portfolos a mea-rsk space gves the effcet froter. Thus, the effcet solutos a mea-rsk model are Pareto effcet solutos of a multple-obectve problem, whch the epected retur s mamzed ad the rsk s mmzed: ma ER R : A The problem of portfolo selecto wth oe vestmet perod s a eample of the geeral problem of decdg betwee radom varables whe larger outcomes are preferred. Decsos are requred o the amout (proporto) of captal to be vested each of a umber of avalable assets such that at the ed of the vestmet perod the retur s hgh as possble. Cosder a set of assets, wth asset,..., gvg a retur R at the ed of the vestmet perod. R s a radom varable, sce the future prce of the asset s ot kow. Let be the proporto of captal vested asset ( w / w where w s the captal vested asset ad w s the total amout of captal to be vested), ad let,..., represet the portfolo resultg from ths choce. Ths portfolo s retur s the radom varable: R R... R F r P R r that depeds o wth dstrbuto fucto,...,. the choce

4 Floret Serba,Vorca Stefaescu, Massmlao Ferrara To represet a portfolo, the weghts,..., must satsfy a set of costras that forms a feasble set M of decsos vectors. The smplest way to defe a feasble set s by the requremet that the weghts must sum to ad short sellg s ot allowed. For ths basc verso of the problem, the set of feasble decsos vectors s M,..., /,,,..., The et ssue s to cosder a practcal represetato for the radom varables that descrbe asset ad portfolo returs. We treat these radom varables as dscrete ad descrbe by realzatos uder T states of the world, geerated usg scearo,...,t occur wth geerato of fte samplg of hstorcal data. Let state T p probablty p,,,...,. Let r be the retur of asset uder scearo,...,t. Thus, the radom varable R represetg the retur of asset s ftely dstrbuted over r,..., r T wth the probabltes p,..., pt. The radom varable R represetg the retur of portfolo,..., s ftely dstrbuted overr,..., R T, where R r... r,,..., T. I the followg, we summarze: -Let there be assets S,,,..., ad let R be the radom varables represetg the rate of retur of S. Let be the proporto of the fud to be vested S. -The vector,,..., ) s called a portfolo, whch has to satsfy the followg codto: -Let (,,,,..., () R be the rate of retur of the portfolo: R R ad let r ad v be, respectvely the mea ad the rsk of R. The the mea-varace (MV) model s represeted as follows. m mze v MV subectto r (3) X where X R s a the set defed by (). Also, t may cota addtoal lear costrats. Ad s a costat to be specfed by a vestor. ()

5 Portfolo Optmzato the Framework Mea Varace -VaR Varyg ad repeatedly solvg the correspodg optmzato problem detfes the mmum rsk portfolo for each value of. Let, be a optmal soluto of the problem (3). The traectory of r, s called a effcet froter. By plottg the correspodg values of the obectves fucto ad of the epected retur respectvely a retur rsk space, we trace out the effcet froter. There are two alteratve represetatos of the mea varace model, amely ma mze r MV subectto v () X ma mze r MV (5) 3 subect to X All three represetatos are used terchageably sce they geerate the same effcet froter as we vary (MV ), (MV ) ad (MV 3 ). There are several measures of rsk uses for to assess the rsk such as : W E R E R ), lower sem-varace, partal absolute devato( momets, below-target rsk, value-at-rsk (VaR), codtoal value-at-rsk (CVaR). Most of these ecept VaR are cove fuctos of.. Mea -Varace Model The mea-varace approach s the earlest method to solve the portfolo selecto problem (Markowtz [95, 959]). The prcple of dversfcato s the foudato of ths method ad t stll has wde applcato rsk maagemet. However, there are some argumets agast t though ths approach has bee accepted ad apprecated by practtoers ad academcs for a umber of years (Kor [997]). The varace of the portfolo retur s the oly rsk measure of ths method. Cotrollg (mmzg) the varace does ot oly lead to low devato from the epected retur o the dow sde, but also o the up sde t may boud the possble gas too. I ths secto, we brefly revew the mea-varace approach. Suppose that there are securtes wth rates of retur X (,..., ). - The meas ad covaraces of these rates of retur are: E cov X, X,,,..., ad X - The portfolo vector s : ',..., R ad

6 Floret Serba,Vorca Stefaescu, Massmlao Ferrara -We defe that set W s a collecto of all possble portfolos: W R -The total retur of portfolo s R - Its mea ad varace are E R E X ad Var X There are two commo models that utlze the mea-varace prcple. The dea of the frst model s that for a gve upper boud for the varace of the portfolo retur ( ), select a portfolo, such that s mamum : ma s. t. X W (6) The secod model states that for a gve lower boud for the mea of the portfolo retur ( ), select a portfolo, such that s mmum : m W (7) s. t.. Mea-VaR Model I recet years, VaR has become a ew bechmark for maagg ad cotrol rsk (Dowd [998], Joo [997], RskMetrcs [995]). Ufortuately, VaR based rsk maagemet has two shortcomgs. Frst, VaR measures have dffcultes aggregatg dvdual rsks, ad sometmes dscourage dversfcato (Artzer et al [998]). Secod, the VaR based rsk maagemet s oly focusg o cotrollg the probablty of loss, rather tha ts magtude (Basak ad Shapro [999]). The epected losses, codtoal o the states where there are large losses, may be hgher sometmes. The mea-varace approach ecourages rsk dversfcato, but the mea-var approach dscourages rsk dversfcato sometmes. The mea-varace approach does ot oly cotrol the rsk of retur o the dow sde, but also bouds the possble

7 Portfolo Optmzato the Framework Mea Varace -VaR ga o the up sde whle the mea-var approach oly cotrols rsk of retur o the dow sde. Aother lmtato of both approaches s that the uderlyg dstrbuto of the rate of retur s ot well uderstood, ad there are o hgher degree formato s utlzed ecept meas, covaraces (varaces), or values of VaR I ths secto, we brefly revew the cocept of VaR ad the mea-var approach. The VaR measures the worst epected loss over a gve tme terval uder ormal market codtos at a gve cofdece level, ad provdes users a summary measure of market rsk. Precsely, the VaR at the % cofdece level of a portfolo for a specfc tme perod s the rate of retur q such that the probablty of that portfolo havg a rate of retur of P R q (8). q or less s : th Here q s also called the quatle of the dstrbuto of R. Smlar to the meavarace method, we defed two models for the mea-var prcple. The frst oe s that for a gve upper boud q for the VaR of the portfolo retur, select a portfolo, such that s the mamum wth q q : ma W s. t. q q The secod model states that for a gve lower boud for the mea of the retur, select a portfolo, such that ts VaR ( q ) s mmum wth : m q W s. t..3 Comparso of Mea -Varace ad Mea -VaR Models I ths secto, we compare the mea-var approach wth the mea-varace approach. The two approaches are usg completely dfferet rsk measures to optmze portfolos. The mea-varace approach oly uses of the mea ad varace of portfolo retur. The Mea-VaR approach oly uses the mea ad VaR of the portfolo retur. Both approaches have may advatages; however they do ot suffcetly use the formato from the dstrbuto of the portfolo retur. Eample shows that a mea-varace effcet portfolo s ot a mea-var effcet portfolo. Remark says that a mea-var effcet portfolo s ot a mea-varace effcet portfolo but uder (9) ()

8 Floret Serba,Vorca Stefaescu, Massmlao Ferrara the ormalty assumpto proposto ca show that a mea-var effcet portfolo s a mea-varace effcet portfolo. Eample : A mea-varace effcet portfolo s ot a mea-var effcet portfolo. We cosder a smple two-securty portfolo selecto problem. The rate of retur for the frst securty s X Z, where Z s the stadard ormal N, wth mea, varace ad VaR, X, X, ad q X z, where s the cofdece level (say, 5 ), ad z, s the z e the stadard ormal dstrbuto, such that z dz The rate of retur for the secod securty s X Z z varace, ad VaR, X z, X, ad X Var The correlato of X ad q ( th quatle of, wth mea, P X )., R X X s X s Corr X X CorrZ,Z z For ay portfolo, the varace of ts retur R Var X X -Ths varace reaches mmum value whe mea-varace effcet portfolo wth mea, varace, ad VaR, ad q z * * *. Therefore *, s a ** -But ths portfolo s ot mea-var effcet. Cosder portfolo, mea ad VaR of R ** X are better, ** X z * ad q q X q * ** Remark: A mea-var effcet portfolo s ot a mea-varace effcet portfolo.. Both Proposto: Uder the ormalty assumpto, a mea-var effcet portfolo s meavarace effcet. -Uder the ormalty assumpto, the portfolo retur N radom R s a varable wth VaR, q z () * -If s a mea-var effcet portfolo, the for ay portfolo, we have: q f * Usg result (), we have: q * z z q q * * *,

9 Portfolo Optmzato the Framework Mea Varace -VaR 3. Mea-Varace-VaR Model I ths secto, we propose a geeral mea-varace-var model for portfolo optmzato wth two varatos. We use both varace ad VaR as rsk cotrol measures. Our models cover both the mea-varace model ad the mea-var model. I other words, the two models are specal cases of our models. The frst model s that for a gve upper bouds ad q for the varace ad VaR of the portfolo retur respectvely, select a portfolo, such that s the mamum wth ad q q : ma W s. t. q q Comparg wth the mea-varace model or the mea-var model, we use double-rsk measures stead of oe sgle rsk measure. The mea-varace-var effcet portfolo may ot be mea-varace effcet or mea-var effcet. Moreover, the mea-varace (6) ad the mea-var models (9) are specal cases of our model (): - whe q, our model () becomes the mea-varace model (6); - whe, our model () becomes the mea-var model (9). The secod model s ot that for a gve lower boud for the mea of the portfolo retur, select a portfolo, such that the cove combato of varace ad VaR of the portfolo retur q s the mmum wth : m q W,,. (3) s. t. For the two etreme values of, we have - whe, our model (3) becomes the mea-varace model (7); - whe, our model (3) becomes the mea-var model (). Possble alteratves to the obectves fucto of model (3) are: q q.from the computatoal pot vew, ad q s better tha q sce square-root takes more computato tme tha square. We also ca substtute the obectves fucto of model f, q. (3) by a geeral utlty fucto ()

10 Floret Serba,Vorca Stefaescu, Massmlao Ferrara. Case study : Itala Stock Market.. Stage of selecto of assets I the cotet of owadays facal markets t s a huge amout of avalable facal data. It s therefore very dffcult to make use of such a amout of formato ad to fd basc patters, relatoshps or treds data. We apply data aalyss techques order to dscover formato relevat to facal data, whch wll be useful durg the selecto of assets ad decso makg. Cosder that we have collected formato o a umber S of assets, each wth P features, whch represet varous facal ratos, stll called varables. Deote by y the -th varable for stock. Multvarate data set wll be represeted by a matr, S, P Y ad ca be vewed as a set of S pots a P-dmesoal space. Prcpal compoets aalyss (PCA) s a useful techque for aalyzg data to fd patters of data a large-scale data space. PCA volves a mathematcal procedure that trasforms P varables, usually correlated a umber of p P ucorrelated varables called prcpal compoets. After applyg the PCA, each asset wll be characterzed by p varables, p represeted by a set of parameters y, y,..., y therefore, t s possble to form the p arrays Y y, y,..., y, S, p that we obtaed a data set Y y, y,..., y, S y, whch correspod to a set of S assets. Suppose ow,. We the use clusterg techques order to fd smlartes ad dffereces betwee the stocks uder cosderato. The dea of clusterg s a assgmet of the vectorsy, Y,..., YS T classes C, C,..., CT. Oce completed the selecto of actvtes, we costruct the tal portfolo by selectg low-rsk asset each class. We wll preset some of the most mportat facal dcators that we wll use study: - The P / E s calculated by dvdg the curret market prce to the value of et proft per share for the past four cosecutve quarters, et come per share s calculated by dvdg the total et proft eared by the compay durg the reportg perod (t s relevat to relate to the last moths) the umber of shares ssued ad outstadg. - The P / BV s calculated by dvdg the curret tradg prce to book value per share determed accordg to the latest facal reportg; accoutg value of a share s calculated by dvdg the total equty value of the compay to the total of t shares ssued ad outstadg; equty value s determed by deductg total labltes from total assets owed compay ad s "shareholder wealth", whch s what remas to be recovered f the assets ad labltes would be pad. - Dvy de measures the performace of dvded ad s calculated as the rato betwee the amout of the dvded ad book value or market value of the stock. ad assesses the effcecy of vestmet a asset.

11 Portfolo Optmzato the Framework Mea Varace -VaR - Volatlty s a measure for varato of prce of a facal strumet over tme. It s used to quatfy the rsk of the facal strumet over the specfed tme perod. - Evoluto of prce: to observe the prce level at a gve tme we take to acout the mamum prce ad mmum prce acheved the last 6 moths We used formato o a total of those shares represetg shares of FTSE MIB Ide traded o the Itala Stock Market. The am of our study s to fd smlartes ad dffereces betwee the curret aalyss ad buld a dversfed portfolo. We take to accout for each stock the 6 features descrbed above; We use data aalyss techques order to process ths vast amout of formato. Table lsts, for each of the stocks aalyzed, the values of the s features; we used the data avalable o the Itala Stock Market o 7 March. Table : The value of the 6 features No Compay P / E P / BV DIVY % Volatlty % P / Ma AA Asaldo Sts Atlata Autugrll Azmut Baca Popolare Bca Mps BcaPopEml Romaga BcaPop Mlao Buzz Ucem Campar DaSor Eel P / M

12 Floret Serba,Vorca Stefaescu, Massmlao Ferrara Eel Gree Power E Eor Fat Fat Idustral Fmeccaca Geeral Impreglo Itesa Sapaolo Lottomatca Luottca Medaset Medobaca Medolaum Parmalat Prell Prysma Sapem Salvatore Ferragamo Sam St

13 Portfolo Optmzato the Framework Mea Varace -VaR mcrolectroc s 35 Telecom Itala Tears Tera Tod S Ub Baca Ucredt Source: We apply data aalyss techques to dscover the smlartes ad dffereces betwee the stocks of the Bucharest Stock Echage, usg the package StatstXL.8. Fgure cotas the tree resulted from PCA (dedrogram). Dedrogram usually begs wth all assets as separate groups ad shows a combato of mergers to a sgle root. Stocks belogg to the same cluster are smlar terms of features take to accout. I order to buld a dversfed portfolo, we frst choose the umber of clusters, whch wll be take to accout. We wll the choose a stock from each group ad we get the tal portfolo. Fgure : Group of stocks Source: package programe STATISTI XL.8

14 Floret Serba,Vorca Stefaescu, Massmlao Ferrara.. Phase rsk estmato. We evaluate the performace of a asset usg epected future come, a dcator wdely used facal aalyss. Deote by S (t) the closg prce for a asset at tme t. Epected future come attached to the tme horzo [ t, t ] s gve by: R ( t) l S ( t ) l S ( t),, S. Smlary, we defe the loss radom varable, the varable L, for asset for [ t, t ] as L ( t) R ( t) l S ( t) l S ( t ),, S. Usg Rockafellar et al.,, defe the rsk measure VaR correspodg loss radom varable L L. Probablty of L ot to eceed a threshold z R sg ( z) PL z.. Value at rsk of loss radom varable level (, ) If L assocated wth the value of asset come ad correspodg probablty s: VaR L m z GL ( z) G s strctly creasg ad cotuous, L equato R or P ( X > VaR ) =. L (z) the L G L ( ). G VaR s the uque soluto of L VaR Oe of the most frequetly used methods for estmatg the rsk s the hstorcal smulato method. Ths rsk assessmet method s useful f emprcal evdece dcates that the radom varables questo may ot be well appromated by ormal dstrbuto or f we are ot able to make assumptos o the dstrbuto. Hstorcal smulato method calculates the value of a hypothetcal chages the curret portfolo, accordg to hstorcal chages rsk factors. The great advatage of ths method s that t makes o assumpto of probablty dstrbuto, usg the emprcal dstrbuto obtaed from aalyss of past data. Dsadvatage of ths method s that t predcts the future developmet based o hstorcal data, whch could lead to accurate estmates f the tred of the past o loger correspods. If L s the loss radom varable ad Ĝ s emprcal dstrbuto fucto of L ad (,) ˆ I. { L z} a fed level of probablty, theg z We ca prove that : VaRL m z R I. { L z} Oce completed the phase of groupg the assets T classes by the estg smlartes, we focus o the selecto of the assetsş of each class to have a mmal rsk. Cosder the loss radom varable correspodg to each asset each obtaed class C,, T : m VaR L( A ). We used the closg prce values daly for each share, correspodg k k to a tme horzo of 5 days to measure VaR for each stock. We used the data avalable

15 Portfolo Optmzato the Framework Mea Varace -VaR o the Itala Stock Market from 5 march - 3 aprl. The followg tables cota values of VaR for each stock ad three levels of probablty values.3 Optmzato portfolo phase the framework Mea - VaR We obta a tal portfolo comprsg a wde rage of stocks wth mmal rsk. We wll try to determe what percetage s the optmal composto of captal that eeds to be vested each of the assets uder cosderato, so that at the ed of the vestmet we have a mamum retur o vestmet. Thus, T s a set of stocks, wth stock that leads to epected come R,, T ;Epected come of portfolo s: R.The model to be solved s: T ma, wth, R T R VaR L where s the model parameter. As a cosequece of applyg the techque of selecto, we selected a portfolo of 5 stocks, each of them represetg the mmum rsk stock class correspodg VaR measured probablty level.99: Bca Mps,Telecom,Sapem,Ucredt, AA. We wll try to determe what percetage s the optmal composto of captal that eeds to be vested each of the stocks uder cosderato, so that at the ed of the vestmet we have a mamum retur o vestmet. I these codtos, the optmzato problem to be solved s:

16 Floret Serba,Vorca Stefaescu, Massmlao Ferrara ma f , There are several methods for solvg ths problem ( math programmg or software Scetfc Work Place). A soluto to ths problem remas a challege for future. 5.Cocluso : I ths paper we have dscussed ad compared the mea-varace approach wth the mea-var approach. The mea-varace-var approach uses varace ad VaR as a double-rsk measure smultaeously. The mea-varace ad the mea-var approaches are specal cases of ths approach. Fally we buld a portfolo wth stocks lsted at Itala Stock Market. Ackowledgmet : Ths work was supported from the Europea Socal Fud through Sectoral Operatoal Programme Huma Resources Developmet 7-3, proect umber POSDRU/89/.5/S/598 Performace ad ecellece postdoctoral research Romaa ecoomcs scece doma. 5 5 REFERENCES [] Aleader, G., Baptsta, A.(), Ecoomc Implcatos of Usg a MeaVaR Model for Portfolo Selecto: A Comparso wth Mea-Varace Aalyss. Workg Paper, Uversty of Mesota; []Basak, S., Shapro,A.(999), Value-at-Rsk Based Rsk Maagemet: Optmal Polces ad Asset Prces. Workg Paper, Uversty of Pesylvaa; [3] Dedu, S., Fulga, C. (), Value-at-rsk. Estmato Comparatve Approach wth Applcato to Optmzato Problem. Ecoomc Computato ad Ecoomc Cyberetcs Studes ad Research,5 (), 7-, ASE Publshg House, Bucharest; []Fulga, C., Dedu, S., Şerba, F. (9), Portfolo Optmzato wth Pror Stock Selecto. Ecoomc Computato ad Ecoomc Cyberetcs Studes ad Research, ASE Publshg House, Bucharest, 3 (), 57-7; [5] Fulga, C., Dedu, S. (), A New Approach Mult-Obectve Portfolo Optmzato Usg Value-at-Rsk Based Rsk Measure, Proceedgs of The Iteratoal Coferece o Iformato ad Facal Egeerg ICIFE, IEEE Computer Socety, Computer Socety Press, ;

17 Portfolo Optmzato the Framework Mea Varace -VaR [6] Markowtz, H.(95), Portfolo Selecto. Joural of Face, 7, 77-9; [7] Rockafellar, T., Uryasev, S.(), Optmzato of Codtoal Value-at-Rsk. Joural of Rsk, ; [8]Roma,D, Darby,K.,Mtra,G.(7), Mea-rsk Models Usg Two Rsk Measures : A Mult-obectve Approach. Quattatve Face, 7(),3-58; [9] Ştefăescu, V., Ferrara, M., Dedu, S. (8), Algorthms for Herarchcal Classfcato wth Applcatos Portfolo Maagemet, Ecoomc Computato ad Ecoomc Cyberetcs Studes ad Research, ASE Publshg,, 3-, 9-; []Ştefăescu, M.V, Şerba, F., Buşu, M., Ferrara, M. (), Portfolo Optmzato Usg Classfcato ad Fuctoal Data Aalyss Techques. Ecoomc Computato ad Ecoomc Cyberetcs Studes ad Research, (3), 93-8, ASE Publshg House, Bucharest; [] Wag, J.(7), Mea-Varace-VaR Based Portfolo Optmzato. Workg Paper, Valdosta Uversty.