Robust-based Random Fuzzy Mean-Variance Model Using a Fuzzy Reasoning Method

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1 Robubaed Rado Fuzzy MeaVaace Model Ug a Fuzzy Reaog Mehod Takah Hauke, Hdek Kaag, Mebe, IAEN, ad Hoh Tuda Abac Th pape code a obubaed ado fuzzy eavaace pofolo eleco poble ug a fuzzy eaog ehod, paculaly a gle pu ype fuzzy eaog ehod. Capal Ae Pcg Model oduced a a fuue eu of each ecuy, ad he ake pofolo aued o be a ado fuzzy vaable whoe ea deved fo a fuzzy eaog ehod. Fuheoe, ude eval pu of fuzzy eaog ehod, a obu pogag appoach oduced ode o ze he wo cae of he oal vaace. The popoed odel equvalely afoed o he deec olea pogag poble, ad o he oluo ep o oba he eac opal pofolo ae developed. Ide Te Pofolo eleco poble, Rado fuzzy pogag, Fuzzy eaog ehod, Robu pogag. I. INTRODUCTION The deco of opal ae allocao aog vaou ecue called pofolo eleco poble, ad oe of he o poa eeach hee vee ad facal eeach feld ce he eavaace odel wa popoed by Makowz [9]. The, afe h ouadg eeach, ueou eeache have cobued o he develope of ode pofolo heoy (cf. Elo ad ube [], Luebege [8]), ad ay eeache have popoed eveal ype of pofolo odel eedg Makowz odel; eaabolue devao odel (Koo [], Koo, e al. [3]), afeyf odel [], Value a Rk ad codoal Value a Rk odel (Rockafella ad Uyaev []), ec.. A a eul, owaday coo pacce o eed hee clacal ecooc odel of facal vee o vaou ype of pofolo odel becaue veo coepod o pee cople ake. I pacce, ay eeache have bee yg dffee aheacal appoache o develop he heoy of pofolo odel. Paculaly, Capal Ae Pcg Model (CAPM), whch a gle faco odel popoed by Shape [3], Le [5], ad Mo [], ha bee oe of he o ueful ool he vee feld ad alo ued he Takah Hauke wh aduae School of Ifoao Scece ad Techology, Oaka Uvey, Japa (coepodg auho o povde phoe: ; fa: ; eal: hauke@.oakau.ac.p). Hdek Kaag wh aduae School of Egeeg, Hoha Uvey, Japa. (eal: kaagh@hohau.ac.p). Hoh Tuda wh Depae of Maheacal Scece, Faculy of Scece ad Egeeg, Dohha Uvey, Japa (eal: huda@al.dohha.ac.p). Th wok wa uppoed by The My of Educao, Culue, Spo, Scece ad Techology (MEXT), aad fo Youg Sce (B) (733). pefoace eaue of fuue eu fo pofolo ad he ae pcg heoy. I uch pevou eeache, epeced fuue eu ad vaace of each ae ae aued o be kow. The, pevou ay ude he ee of aheacal pogag fo he vee, fuue eu ae aued o be couou ado vaable accodg o oal dbuo. Howeve, veo eceve effecve o effecve foao fo he eal ake ad ecooc aaly, ad abguou faco uually e. Fuheoe, veo ofe have he ubecve pedco fo fuue ake whch ae o deved fo he acal aaly of hocal daa, bu he loge epeece of vee. The, eve f veo hold a lo of foao fo he eal ake, dffcul ha he pee o fuue ado dbuo of each ae cly e. Coequely, we eed o code o oly ado codo bu alo abguou ad ubecve codo fo pofolo eleco poble. A ece ude aheacal pogag, oe eeache have popoed vaou ype of pofolo odel ude adoe ad fuzze. Thee pofolo odel wh pobable ad poble ae cluded ochac pogag poble ad fuzzy pogag poble, epecvely, ad hee ae oe bac ude ug ochac pogag appoache, goal pogag appoache, ad fuzzy pogag appoache o deal wh abguou faco a fuzzy e (Iuguch ad Rak [8], Leo, e al. [4], Taaka ad uo [5], Taaka e al. [6], Veche e al. [7], Waada [8]). Fuheoe, oe eeache have popoed aheacal pogag poble wh boh adoe ad fuzze a fuzzy ado vaable (fo ace, Kaag e al. [, ]). I he ude [, ], fuzzy ado vaable wee elaed wh he abguy of he ealzao of a ado vaable ad deal wh a fuzzy ube ha he cee value occu accodg o a ado vaable. O he ohe had, fuue eu ay be deal wh ado vaable deved fo he acal aaly, whoe paaee ae aued o be fuzzy ube due o he deco ake ubecvy,.e., ado fuzzy vaable whch Lu [6] defed. Thee ae a few ude of ado fuzzy pogag poble (Hauke e al. [3, 4], Huag [7], Kaag e al. [9]). Mo ecely, Hauke e al. [4] popoed eveal pofolo eleco odel cludg ado fuzzy vaable ad developed he aalycal oluo ehod. Howeve, [4], each ebehp fuco of fuzzy ea value of fuue eu wa e by he veo, ad he aheacal deal of eg he ebehp fuco. Of coue, alo poa o deee he fuzzy ea value of fuue eu wh he veo loge epeece ad ecoocal aaly effecve foao.

2 Theefoe, ode o volve he eceay foao o ea value of fuue eu aheacally, we oduce a fuzzy feece o eaog ehod baed o fuzzy fhe ule. The fuzzy eaog ehod he o poa appoach o eac ad decde effecve ule ude fuzze aheacally. Sce ouadg ude of Mada [] ad Takag ad Sugeo [4], ay eeache have eeded hee pevou appoache, ad popoed ew fuzzy eaog ehod. Paculaly, we focu o a gle pu ype fuzzy eaog ehod popoed by Hayah e al. [5, 6]. Th ehod e up ule odule o each pu e, ad he fal feece eul obaed by he weghed aveage of he degee of he aecede pa ad coeque pa of each ule odule. Nevehele h appoach oe of he ple aheacal appoache fuzzy eaog ehod, he fal feece eul la o he ohe adad appoache. Theefoe, h pape, we popoed a ado fuzzy eavaace odel oducg CAPMbaed fuue eu ad Hayah gle pu ype fuzzy eaog ehod fo he ea value of ake pofolo of CAPM. The popoed ado fuzzy eavaace odel o foulaed a a welldefed poble due o fuzze, we eed o e oe cea opzao ceo o a o afo o welldefed poble. I h pape, aug he eval value a a pecal cae of fuzzy ube ad oducg he cocep of obu pogag, we afo he a poble o a obu pogag poble. Recely, he obu opzao poble becoe a oe acve aea of eeach, ad hee ae oe ude of obu pofolo eleco poble deeg opal vee aegy ug he obu appoach (Fo eaple, oldfab ad Iyega [], Lobo [7]). I obu pogag, we oba he eac opal pofolo. Th pape ogazed he followg way. I Seco, we oduce aheacal cocep of ado fuzzy vaable, Capal Ae Pcg Model, ad a gle pu ype fuzzy eaog ehod. I Seco 3, we popoe a ado fuzzy pofolo eleco poble wh ea value deved fo he fuzzy eaog ehod. Pefog he deec equvale afoao, we oba a facoal pogag poble wh oe vaable. Fally, Seco 4, we coclude h pape. II. MATHEMATICAL DEFINITION AND NOTATION I ay eg ude of pofolo eleco poble, fuue eu ae aued o be ado vaable o fuzzy ube. Howeve, ce hee ae few ude of he eaed a he CAPM wh ado fuzzy vaable ad fuzzy eaog ehod, ulaeouly. Theefoe, h eco, we epla defo ad aheacal foulao of ado fuzzy vaable, CAPM, ad gle pu ype fuzzy eaog ehod popoed by Hayah e al. [5, 6]. A. Rado Fuzzy Vaable F of all, we oduce a ado fuzzy vaable defed by Lu [6] a follow. Defo (Lu [6]) A ado fuzzy vaable a fuco fo a colleco of ado vaable R o [, ]. A deoal ado fuzzy veco ξ = ( ) a uple of ado,,..., fuzzy vaable,,...,. Tha, a ado fuzzy vaable a fuzzy e defed o a uveal e of ado vaable. Fuheoe, he followg ado fuzzy ahec defo oduced. Defo (Lu [6]) Le,,..., be ado fuzzy vaable, ad f : R R be a couou fuco. The, = f (,,..., ) a ado fuzzy vaable o he poduc pobly pace (, P,Po) qq (,,..., q) = f ( ( q), ( q),..., ( q) ) fo all ( q q q ) ÎQ.,,..., Q Q, defed a Fo hee defo, he followg heoe deved. Theoe (Lu[6]) Le be ado fuzzy vaable wh ebehp fuco, =,,...,, epecvely, ad f : R R be a couou fuco. The, = f,,..., a ado fuzzy vaable whoe ebehp fuco h = up ( h) h= f ( h, h,..., h) hî R, { } fo all h Î R, whee R = f h, h,..., h h Î R, =,,...,. { } B. Capal Ae Pcg Model I pofolo odel, Capal Ae Pcg Model (CAPM) popoed by Shape [3], Le [5], ad Mo [] ha bee ued ay paccal vee cae by o oly eeache bu alo paccal veo. The a advaage of CAPM o deal wh he elao bewee eu of each ae ad ake pofolo uch a NASDAQ ad TOPIX a he he followg ple lea foulao; = d + d whee he eu of ake pofolo. The, d ad d ae hee value deved fo hocal daa vee feld. Howeve, ake pofolo o eely equal o NASDAQ ad TOPIX, ad o alo poble o obeve eacly he vee feld. Fuheoe, he cae ha he deco ake pedc he fuue eu ug CAPM, obvou ha ake pofolo alo occu accodg o a ado dbuo wh he veo ubecvy. Theefoe, hee uao, we popoe a

3 ado fuzzy CAPM odel. I h odel we aue ha a ado fuzzy vaable, ad he "dah above" ad "wave above",.e., ad ~, deoe adoe ad fuzze of he coeffce, epecvely. I h pape, occu accodg o a ado dbuo wh fuzzy ea value ad coa vaace. To plfy, we aue ha each fuzzy epeced eu a eval value = é, ù ê ë ú û deved fo a fuzzy eaog ehod he e ubeco. C. Sgle Ipu Type Fuzzy Reaog Mehod May eeache have popoed vaou fuzzy feece ad eaog ehod baed o o eedg Mada [] o Takag ad Sugeo [4] ouadg ude. I h pape, a a aheacally ple appoach, we oduce a gle pu ype fuzzy eaog ehod popoed by Hayah e al. [5, 6]. I h ehod, we code he followg ule odule: Rule: { z = A = }, ( =,,..., ) S = whee z ad ae he h pu ad coeque daa, epecvely. The, he eal value of oupu fo he coeque pa. A he fuzzy e of he h ule of he Rule, ad S he oal ube of ebehp fuco of A. The degee of he aecede pa he h ule of Rule obaed a h A( z ) =. I Hayah gle pu ype fuzzy eaog ehod, he feece eul calculaed a follow: S S S h + + h h = = = = = = S S S h + + h h = = = = Paculaly, f ebehp fuco of all fuzzy e A ae agle fuzzy ube, foula () a lea facoal fuco o pu colu veco z. I h pape, ug h fuzzy eaog ehod, we oba he ea value of ake pofolo. We aue ha pu colu veco z ea poa facal ad ocal faco o decde he ea value of ake pofolo. Howeve, dffcul o e pu colu veco z a coa value éz, z ù êë ú. û Theefoe, we e each pu z a a eval value, ad ode o oba he au ad u value of ude eval value of éz, z ù êë ú, we oduce he û followg aheacal pogag: () Maze( Mze ) S = = S = = ubec o z z z, ( =,,..., ) h Each poble a facoal lea pogag poble ude agle fuzzy ube A, ad o we oba he opal oluo. Le U ad L be he opal oluo azg ad zg he obec, epecvely. III. FORMULATION OF PORTFOLIO SELECTION PROBLEM WITH RANDOM FUZZY RETURNS The pevou ude o ado ad fuzzy pofolo eleco poble ofe have codeed adad eavaace odel o afey f odel oducg pobably o fuzzy chace coa baed o ode pofolo heoe (e.g. Hauke e al. [4]). Howeve, hee o udy o he ado fuzzy ea vaace odel ug he fuzzy eaog ehod o oba he eval ea value of ake pofolo. Theefoe, h pape, we eed he pevou ado fuzzy eavaace odel o a obu pogagbaed odel ug he fuzzy eaog ehod. F, we deal wh he followg o ple pofolo eleco poble volvg he ado fuzzy vaable baed o he adad ae allocao poble o aze oal fuue eu. Maze = ubec o =, ³, =,,, = whee he oao of paaee ued h pape a follow: : Fuue eu of he h facal ae aued o be a ado fuzzy vaable, whoe fuzzy epeced value h () (3) ad vaacecovaace a V, epecvely. The, we deoe adoe ad fuzze of he coeffce by he "dah above" ad "wave above",.e., ad ~, epecvely. : Tage oal eu : Toal ube of ecue : Budgeg allocao o he h ecuy I [4], we code eveal odel ad oluo appoache baed o adad afeyf odel of pofolo eleco poble. Howeve, ode o olve he pevou odel aalycally, we u aue ha each eu occu accodg o he oal dbuo he ee of adoe. Th aupo a lle eced. Theefoe, h pape, we do o aue cea ado dbuo fo fuue eu. Aleavely, we oduce he followg pofolo odel zg he wo oal vaace,.e., azg he oal vaace, a a obu pofolo odel:

4 Mze Îé, ù êë ú û î = = = a æ ö ubec o E ³, è = ø = ì í ü ý þ By aug h obu pogag poble, he veo ay be able o avod he lae k cludg he wo cae of fuue eu. I ode o olve h poble, we f he value of eval ea value of ado fuzzy ake pofolo a ( w ),.e., each fuue eu alo oly a ado vaable a w = d + d ( w). Theefoe, poble (4) afoed o he followg adad eavaace pofolo odel: Mze = = æ ö ubec o E ( w) ³, è = ø whee = = ( w) = + ( w) d d Th poble a cove quadac pogag poble due o pove defe a, ad o we oba he eac opal pofolo by ug he followg ep olea pogag. F we oduce he Lagage fuco fo poble (5) a follow: L= æ ö æ ö + l ( w) + = = = è ø è = ø (6) whee l ad ae Lagage ulple. The, by ug KauhKuhTucke (KKT) codo, we oba he followg equao o each vaable : ì L = l w =, =,,..., = í ( w) = = = î = I ode o olve equao deved fo KKT codo, we e he veco oao, ad oba he oluo of a follow: (4) (5) (7) Vl w I = ælö V A =, è æ ( w) ( w) ö æ ö = A è è ælö V A è = By ubug h oluo o ecod ad hd equao KKT codo (7), we oba he followg opal value of Lagage ulple: æ lö æ æ ö ö AV A, = = è è è ø (9) ælö = è AV A Coequely, we oba he opal pofolo * ad he * obecve value * V a follow: * = V A ( AV A ) = = ( AV A ) AV A ( AV A ) = AV A ( ) * * V AV A AV V V A AV A (8) () * Fo ad he opal obecve value, we ecodly code a obu pogagbaed pofolo odel,.e. he wo cae of he oal vaace: Maze AV A Îé, ù ê ë ú û () ubec o é Î, ù ê ë ú û I obecve fuco AV A, vee a AV A calculaed a he followg fo: AV æ ö = = = = A = è = = = = ì æ ö = = = = ( AV A ) = D í è = = = = ø æ öæ ö æ ö D = = = = = î è øè ø è = = ø ()

5 Theefoe, obecve fuco AV A alo calculaed a follow: AV A æ ö (3) = + D è = = = = = = ø Coequely, poble () equvalely afoed o he followg poble: æ Maze D è = = ö + = = = = ø (4) ubec o = d + d, Îé, ù ê ë ú û æ æ ö ö whee æ ö æ ö D = è = = ø = = è è ø è = = ø I h poble, we ubue ( w) = d + d ( w) deved fo CAPM, ad he ueao ad deoao of obecve fuco ae calculaed a follow: + = = = = = = = dd = = + ( dd + d d d ) = = + ( coa value) = p+ p + p ææ ö ö D = æ ö dd è = = ø è è = = ø æ öæ ö + dd dd + = = è øè = = ø æ ö k kdd è k= = = ø æ ö k dd è k= = = ø + ( coa value) = q + q + q (5) Theefoe, he a poble epeeed a he followg aheacal pogag poble wh vaable : p+ p + p Maze q+ q + q (6) ubec o Î é, ù ê ë ú û Th poble a olea facoal pogag poble, ad o geeally dffcul o oba he opal oluo. Howeve, h poble ha oly oe vaable, ad o we ca oba he opal oluo by ug adad olea pogag appoache o lluag he obecve fuco decly. IV. CONCLUSION I h pape, we have popoed a obubaed eavaace pofolo eleco poble wh ado fuzzy CAPM ug a gle pu ype fuzzy eaog ehod. I ode o deal wh he ake pofolo of CAPM a a ado eval vaable, ad o pefo he deec equvale afoao, he popoed odel ha bee olea pogag poble wh oly oe vaable. Theefoe, we have obaed he eac opal pofolo ug adad olea pogag appoache. A fuue ude, we eed o develop he oluo algoh cae of geeal fuzzy ube cludg eval value. Fuheoe, we alo eed o code ado fuzzy pofolo odel deved fo o oly a gle pu ype fuzzy eaog ehod bu alo oe geeal fuzzy eaog ehod. REFERENCES [] E.J. Elo ad M.J. ube, Mode Pofolo Theoy ad Ivee Aaly, Wley, New Yok, 995. [] D. oldfab ad. Iyega, Robu pofolo eleco poble, Maheac of Opeao Reeach 8, pp. 38, 3. [3] T. Hauke, H. Kaag, ad H. Ih, Mulobecve ado fuzzy lea pogag poble baed o he pobly azao odel, Joual of Advaced Copuaoal Iellgece ad Iellge Ifoac 3(4), pp , 9. [4] T. Hauke, H. Kaag, ad H. Ih, Pofolo eleco poble wh ado fuzzy vaable eu, Fuzzy Se ad Sye6, pp , 9. [5] K. Hayah, A. Oubo, ad K. Shaa, Realzao of olea ad lea PID cool ug plfed dec feece ehod, IEICE Ta. Fudaeal (Japaee Edo), J8A(7), pp.884, 999. [6] K Hayah, A. Oubo, ad K. Shahaa, Ipovee of coveoal ehod of PI fuzzy cool, IEICE Ta. Fudaeal, E84A(6), pp ,. [7] X. Huag, Two ew odel fo pofolo eleco wh ochac eu akg fuzzy foao, Euopea Joual of Opeaoal Reeach 8, pp , 7. [8] M. Iuguch ad J. Rak, Poblc lea pogag: A bef evew of fuzzy aheacal pogag ad a copao wh ochac pogag pofolo eleco poble, Fuzzy Se ad Sye, pp. 38,. [9] H. Kaag, T. Hauke, H.Ih, ad I. Nhzak, Rado Fuzzy Pogag Model baed o Poblc Pogag, Poceedg of he 8 IEEE Ieaoal Cofeece o Sye, Ma ad Cybeec (o appea). [] H. Kaag, H. Ih, ad M. Sakawa, O fuzzy ado lea kapack poble, Ceal Euopea Joual of Opeao Reeach, pp. 597, 4. [] H. Kaag, M. Sakawa, ad H. Ih, A udy o fuzzy ado pofolo eleco poble ug pobly ad ecey eaue, Sceae Maheacae Japoocae 65, pp , 5 [] H. Koo ad H. Yaazak, MeaAbolue Devao Pofolo Opzao Model ad I Applcao o Tokyo Sock Make, Maagee Scece 37, pp. 5953, 99. [3] H. Koo, H. Shakawa, ad H. Yaazak, A eaabolue devaokewe pofolo opzao odel, Aal of Opeao Reeach 45, pp. 5, 993. [4] R T. Leo, V. Le, ad E. Veche, Valdy of feable pofolo eleco poble: fuzzy appoach, Euopea Joual of Opeaoal Reeache 39, pp. 7889,.

6 [5] B J. Le, Valuao of ky ae ad he eleco of ky vee ock pofolo ad capal budge, Rev. Ecoo. Sa. 47, pp. 337, 965. [6] B. Lu, Theoy ad Pacce of Ucea Pogag, Phyca Velag,. [7] M.S. Lobo, Robu ad cove opzao wh applcao face, Doco he of he Depae of Eleccal Egeeg ad he Coee o aduae Sude, Safod Uvey,. [8] D.. Luebege, Ivee Scece, Ofod Uv. Pe, 997. [9] H.M. Makowz, Pofolo eleco, The Joual of Face 7(), pp. 779, 95. [] E.H. Mada, Applcao of fuzzy algoh fo cool of ple dyac pla, Poceedg of IEE, (), pp , 974. [] J. Mo, Equlbu capal ae ake, Ecooeca 34(4), pp , 966. [] R.T. Rockafella ad S. Uyaev, Opzao of codoal valueak, Joual of Rk (3), pp.,. [3] W.F. Shape, Capal ae pce: A heoy of ake equvale ude codo of k, Joual of Face 9(3), pp. 4544, 964 [4] T. Takag ad M. Sugeo, Fuzzy defcao of ye ad applcao o odelg ad cool, IEEE Ta. Sy. Ma. & Cybe. SMC5(), pp. 63, 985. [5] H. Taaka ad P. uo, Pofolo eleco baed o uppe ad lowe epoeal pobly dbuo, Euopea Joual of Opeaoal Reeache 4, pp. 56, 999. [6] H. Taaka, P. uo ad I.B. Tuke, Pofolo eleco baed o fuzzy pobable ad pobly dbuo, Fuzzy Se ad Sye, pp ,. [7] E. Veche, J.D. Beúdez, ad J.V. Segua, Fuzzy pofolo opzao ude dowde k eaue, Fuzzy Se ad Sye 58, pp , 7. [8] J. Waada, Fuzzy pofolo eleco ad applcao o deco akg, Taa Moua Mah. Pub. 3, pp. 948, 997.