Sectionwww.stockstar.com

www.stockstar.com  时间:2021-03-20  阅读:()
AFuzzyApproachtoPortfolioRebalancingwithTransactionCostsYongFang1,K.
K.
Lai2,andShou-YangWang31InstituteofSystemsScience,AcademyofMathematicsandSystemsSciences,ChineseAcademyofSciences,Beijing100080,Chinayfang@amss.
ac.
cn2DepartmentofManagementSciences,CityUniversityofHongKong,Kowloon,HongKongmskklai@cityu.
edu.
hk3InstituteofSystemsScience,AcademyofMathematicsandSystemsSciences,ChineseAcademyofSciences,Beijing100080,Chinaswang@mail.
iss.
ac.
cnAbstract.
Thefuzzysetisapowerfultoolusedtodescribeanuncertainnancialenvironmentinwhichnotonlythenancialmarketsbutalsothenancialmanagers'decisionsaresubjecttovagueness,ambiguityorsomeotherkindoffuzziness.
Basedonfuzzydecisiontheory,twoportfo-liorebalancingmodelswithtransactioncostsareproposed.
Anexampleisgiventoillustratethatthetwolinearprogrammingmodelsbasedonfuzzydecisionscanbeusedecientlytosolveportfoliorebalancingprob-lemsbyusingrealdatafromtheShanghaiStockExchange.
1IntroductionIn1952,Markowitz[8]publishedhispioneeringworkwhichlaidthefoundationofmodernportfolioanalysis.
Itcombinesprobabilitytheoryandoptimizationtheorytomodelthebehaviorofeconomicagentsunderuncertainty.
KonnoandYamazika[5]usedtheabsolutedeviationriskfunction,toreplacetheriskfunc-tioninMarkowitz'smodelthusformulatedameanabsolutedeviationportfoliooptimizationmodel.
Itturnsoutthatthemeanabsolutedeviationmodelmain-tainsthenicepropertiesofMarkowitz'smodelandremovesmostoftheprincipaldicultiesinsolvingMarkowitz'smodel.
Transactioncostisoneofthemainsourcesofconcerntoportfoliomanagers.
ArnottandWagner[2]foundthatignoringtransactioncostswouldresultinaninecientportfolio.
Yoshimoto'sempericalanalysis[12]alsodrewthesameconclusion.
Duetochangesofsituationinnancialmarketsandinvestors'pref-erencestowardsrisk,mostoftheapplicationsofportfoliooptimizationinvolvearevisionofanexistingportfolio,i.
e.
,portfoliorebalancing.
Usually,expectedreturnandriskaretwofundamentalfactorswhichinvestorsconsider.
Sometimes,investorsmayconsiderotherfactorsbesidestheexpectedSupportedbyNSFC,CAS,CityUniversityofHongKongandMADIS.
CorrespondingauthorP.
M.
A.
Slootetal.
(Eds.
):ICCS2003,LNCS2658,pp.
1019,2003.
Springer-VerlagBerlinHeidelberg2003returnandrisk,suchasliquidity.
Liquidityhasbeenmeasuredasthedegreeofprobabilityinvolvedintheconversionofaninvestmentintocashwithoutanysignicantlossinvalue.
Arenas,BilbaoandRodriguez[1]tookintoaccountthreecriteria:return,riskandliquidityandusedafuzzygoalprogrammingapproachtosolvetheportfolioselectionproblem.
In1970,BellmanandZadeh[3]proposedthefuzzydecisiontheory.
Ra-maswamy[10]presentedaportfolioselectionmethodusingthefuzzydecisiontheory.
AsimilarapproachforportfolioselectionusingthefuzzydecisiontheorywasproposedbyLeonetal.
[6].
Usingthefuzzydecisionprinciple,¨Ostermark[9]proposedadynamicportfoliomanagementmodelbyfuzzifyingtheobjectiveandtheconstraints.
Watada[11]presentedanothertypeofportfolioselectionmodelusingthefuzzydecisionprinciple.
Themodelisdirectlyrelatedtothemean-variancemodel,wherethegoalrate(orthesatisfactiondegree)foranex-pectedreturnandthecorrespondingriskaredescribedbylogisticmembershipfunctions.
Thispaperisorganizedasfollows.
InSection2,abi-objectivelinearpro-grammingmodelforportfoliorebalancingwithtransactioncostsisproposed.
InSection3,basedonthefuzzydecisiontheory,twolinearprogrammingmodelsforportfoliorebalancingwithtransactioncostsareproposed.
InSection4,anexampleisgiventoillustratethatthetwolinearprogrammingmodelsbasedonfuzzydecisionscanbeusedecientlytosolveportfoliorebalancingproblemsbyusingrealdatafromtheShanghaiStockExchange.
AfewconcludingremarksarenallygiveninSection5.
2LinearProgrammingModelforPortfolioRebalancingDuetochangesofsituationinnancialmarketsandinvestors'preferencesto-wardsrisk,mostoftheapplicationsofportfoliooptimizationinvolvearevisionofanexistingportfolio.
Thetransactioncostsassociatedwithpurchasinganewportfolioorrebalancinganexistingportfoliohaveasignicanteectonthein-vestmentstrategy.
Supposeaninvestorallocateshiswealthamongnsecuritiesoeringrandomratesofreturns.
Theinvestorstartswithanexistingportfolioanddecideshowtoreconstructanewportfolio.
Theexpectednetreturnontheportfolioafterpayingtransactioncostsisgivenbynj=1rj(x0j+x+jxj)nj=1p(x+j+xj)(1)whererjistheexpectedreturnofsecurityj,x0jistheproportionofthesecurityjownedbytheinvestorbeforeportfolioreblancing,x+jistheproportionofthesecurityjboughtbytheinvestor,xjistheproportionofthesecurityjsoldbytheinvestorduringtheportfoliorebalancingprocessandpistherateoftransactioncosts.
Denotexj=x0j+x+jxj,j=1,2,n.
Thesemi-absolutedeviationofreturnontheportfoliox=(x1,x2,xn)belowtheexpectedreturnoverthe11AFuzzyApproachtoPortfolioRebalancingwithTransactionCostspastperiodt,t=1,2,Tcanberepresentedaswt(x)=|min{0,nj=1(rjtrj)xj}|.
(2)whererjtcanbedeterminedbyhistoricalorforecastdata.
Theexpectedsemi-absolutedeviationofthereturnontheportfoliox=(x1,x2,xn)belowtheexpectedreturncanberepresentedasw(x)=1TTt=1wt(x)=1TTt=1|min{0,nj=1(rjtrj)xj}|.
(3)Usually,theanticipationofcertainlevelsofexpectedreturnandriskaretwofundamentalfactorswhichinvestorsconsider.
Sometimes,investorsmaywishtoconsiderotherfactorsbesidesexpectedreturnrateandrisk,suchasliquidity.
Liquidityhasbeenmeasuredasthedegreeofprobabilityofbeingabletoconvertaninvestmentintocashwithoutanysignicantlossinvalue.
Generally,investorsprefergreaterliquidity,especiallysinceinabullmarketforsecurities,returnsonsecuritieswithhighliquiditytendtoincreasewithtime.
Theturnoverrateofasecurityistheproportionofturnovervolumestotradablevolumesofthesecurity,andisafactorwhichmayreecttheliquidityofthesecurity.
Inthispaper,weassumethattheturnoverratesofsecuritiesaremodelledbypossibilitydistributionsratherthanprobabilitydistributions.
CarlssonandFuller[4]introducedthenotationofcrisppossibilisticmean(expected)valueandcrisppossibilisticvarianceofcontinuouspossibilitydistri-butions,whichareconsistentwiththeextensionprinciple.
Denotetheturnoverrateofthesecurityjbythetrapezoidalfuzzynumberlj=(laj,lbj,αj,βj).
Thentheturnoverrateoftheportfoliox=(x1,x2,xn)isnj=1lj.
Bythedenition,thecrisppossibilisticmean(expected)valueoftheturnoverrateoftheportfoliox=(x1,x2,xn)canberepresentedasE(l(x))=E(nj=1ljxj)=nj=1(laj+lbj2+βjαj6)xj.
(4)Assumethattheinvestordoesnotinvesttheadditionalcapitalduringtheport-foliorebalancingprocess.
Weusew(x)tomeasuretheriskoftheportfolioandusethecrisppossibilisticmean(expected)valueoftheturnoverratetomeasuretheliquidityoftheportfolio.
Assumetheinvestorwantstomaximizereturnonandminimizetherisktotheportfolioafterpayingtransactioncosts.
Atthesametime,herequiresthattheliquidityoftheportfolioisnotlessthanagivencon-stantthroughrebalancingtheexistingportfolio.
Basedontheabovediscussions,theportfoliorebalancingproblemisformulatedasfollows:12Y.
Fang,K.
K.
Lai,andS.
-Y.
Wang(P1)maxnj=1rj(x0j+x+jxj)nj=1p(x+j+xj)minTt=1|nj=1(rjtrj)xj|+nj=1(rjrjt)xj2Ts.
t.
nj=1(laj+lbj2+βjαj6)xj≥l,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n.
wherelisagivenconstantbytheinvestorandujrepresentsthemaximumproportionofthetotalamountofmoneydevotedtosecurityj,j∈S.
Eliminatingtheabsolutefunctionofthesecondobjectivefunction,theaboveproblemcanbetransformedintothefollowingproblem:(P2)maxnj=1rj(x0j+x+jxj)nj=1p(x+j+xj)min1TTt=1yts.
t.
nj=1(laj+lbj2+βjαj6)xj≥l,yt+nj=1(rjtrj)xj≥0,t=1,2,T,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n.
yt≥0,t=1,2,T.
wherelisagivenconstantbytheinvestor.
Theaboveproblemisabi-objectivelinearprogrammingproblem.
Onecanuseseveralalgorithmsofmultipleobjectivelinearprogrammingtosolveite-ciently.
3PortfolioRebalancingModelsBasedonFuzzyDecisionIntheportfoliorebalancingmodelproposedinabovesection,thereturn,theriskandtheliquidityoftheportfolioareconsidered.
However,investor'ssatisfactorydegreeisnotconsidered.
Innancialmanagement,theknowledgeandexperienceofanexpertareveryimportantindecision-making.
Throughcomparingthepresentproblemwiththeirpastexperienceandevaluatingthewholeportfoliointermsofriskandliquidityinthedecision-makingprocess,theexpertsmayestimatetheobjectivevaluesconcerningtheexpectedreturn,theriskandthe13AFuzzyApproachtoPortfolioRebalancingwithTransactionCostsliquidity.
Basedonexperts'knowledge,theinvestormaydecidehislevelsofaspirationfortheexpectedreturn,theriskandtheliquidityoftheportfolio.
3.
1PortfolioRebalancingModelwithLinearMembershipFunctionDuringtheportfoliorebalancingprocess,aninvestorconsidersthreefactors(theexpectedreturn,theriskandtheliquidityoftheportfolio).
Eachofthefactorsistransformedusingamembershipfunctionsoastocharacterizetheaspirationlevel.
Inthissection,thethreefactorsareconsideredasthefuzzynumberswithlinearmembershipfunction.
a)Membershipfunctionfortheexpectedreturnontheportfolior(x)=0ifE(r(x))r1wherer0representsthenecessityaspirationlevelfortheexpectedreturnontheportfolio,r1representsthesucientaspirationlevelfortheexpectedreturnoftheportfolio.
b)Membershipfunctionfortheriskoftheportfoliow(x)=1ifw(x)w1wherew0representsthenecessityaspirationlevelfortheriskoftheportfolio,w1representsthesucientaspirationlevelfortheriskoftheportfolio.
c)Membershipfunctionfortheliquidityoftheportfoliol(x)=0ifE(l(x))l1wherel0representsthenecessityaspirationlevelfortheliquidityoftheportfolio,l1representsthesucientaspirationlevelfortheliquidityoftheportfolio.
Thevaluesofr0,r1,w0,w1,l0andl1canbegivenbytheinvestorbasedontheexperts'knowledgeorpastexperience.
AccordingtoBellmanandZadeh'smaximizationprinciple,wecandeneλ=min{r(x),w(x),l(x)}.
Thefuzzyportfoliorebalancingproblemcanbeformulatedasfollows:(P3)maxλs.
t.
r(x)≥λ,w(x)≥λ,l(x)≥λ,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n,0≤λ≤1.
14Y.
Fang,K.
K.
Lai,andS.
-Y.
WangFurthermore,thefuzzyportfoliorebalancingproblemcanberewrittenasfollows:(P4)maxλs.
t.
nj=1rjxjnj=1p(x+j+xj)≥λ(r1r0)+r0,1TTt=1yt≤w1λ(w1w0),nj=1(laj+lbj2+βjαj6)xj≥λ(l1l0)+l0,yt+nj=1(rjtrj)xj≥0,t=1,2,T,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n,yt≥0,t=1,2,T,0≤λ≤1.
wherer0,r1,l0,l1,w0andw1areconstantsgivenbytheinvestorbasedontheexperts'knowledgeorpastexperience.
Theaboveproblemisastandardlinearprogrammingproblem.
Onecanuseseveralalgorithmsoflinearprogrammingtosolveiteciently,forexample,thesimplexmethod.
3.
2PortfolioRebalancingModelwithNon-linearMembershipFunctionWatada[11]employedalogisticfunctionforanon-linearmembershipfunctionf(x)=11+exp(α).
Wecanndthatatrapezoidalmembershipfunctionisanapproximationfromalogisticfunction.
Therefore,thelogisticfunctioniscon-sideredmuchmoreappropriatetodenoteavaguegoallevel,whichaninvestorconsiders.
Membershipfunctionsr(x),w(x)andl(x)fortheexpectedreturn,theriskandtheliquidityontheportfolioarerepresentedrespectivelyasfollows:r(x)=11+exp(αr(E(r(x))rM)),(5)w(x)=11+exp(αw(w(x)wM)),(6)l(x)=11+exp(αl(E(l(x))lM))(7)whereαr,αwandαlcanbegivenrespectivelybytheinvestorbasedonhisowndegreeofsatisfactionfortheexpectedreturn,thelevelofriskandtheliquidity.
rM,wMandlMrepresentthemiddleaspirationlevelsfortheexpectedreturn,15AFuzzyApproachtoPortfolioRebalancingwithTransactionCoststhelevelofriskandtheliquidityoftheportfoliorespectively.
ThevalueofrM,wMandlMcanbegottenapproximatelybythevaluesofr0,r1,w0,w1,l0andl1,i.
e.
rM=r0+r12,wM=w0+w12andlM=l0+l12.
Remark:αr,αwandαldeterminerespectivelytheshapesofmembershipfunc-tionsr(x),w(x)andl(x)respectively,whereαr>0,αw>0andαl>0.
Thelargerparametersαr,αwandαlget,thelesstheirvaguenessbecomes.
Thefuzzyportfoliorebalancingproblemcanbeformulatedasfollows:(P5)maxηs.
t.
r(x)≥η,w(x)≥η,l(x)≥η,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n,0≤η≤1.
Letθ=log11η,thenη=11+exp(θ).
Thelogisticfunctionismonotonouslyincreasing,somaximizingηmakesθmaximize.
Therefore,theaboveproblemmaybetransformedtoanequivalentproblemasfollows:(P6)maxθs.
t.
αr(nj=1rjxjnj=1p(x+j+xj))θ≥αrrM,θ+αwTTt=1yt≤αwwM,αlnj=1(laj+lbj2+βjαj6)xjθ≥αllM,yt+nj=1(rjtrj)xj≥0,t=1,2,T,nj=1xj=1,xj=x0j+x+jxj,j=1,2,n,0≤x+j≤uj,j=1,2,n,0≤xj≤x0j,j=1,2,n,yt≥0,t=1,2,T,θ≥0.
whereαr,αwandαlareparameterswhichcanbegivenbytheinvestorbasedonhisowndegreeofsatisfactionregardingthethreefactors.
Theaboveproblemisalsoastandardlinearprogrammingproblem.
Onecanuseseveralalgorithmsoflinearprogrammingtosolveiteciently,forexample,thesimplexmethod.
Remark:Thenon-linearmembershipfunctionsofthethreefactorsmaychangetheirshapeaccordingtotheparametersαr,αwandαl.
Throughselectingthevaluesoftheseparameters,theaspirationlevelsofthethreefactorsmaybede-scribedaccurately.
Ontheotherhand,deferentparametervaluesmayreect16Y.
Fang,K.
K.
Lai,andS.
-Y.
Wangdeferentinvestors'aspirationlevels.
Therefore,itisconvenientfordeferentin-vestorstoformulateinvestmentstrategiesusingtheaboveportfoliorebalancingmodelwithnon-linearmembershipfunctions.
4AnExampleInthissection,wegiveanexampletoillustratethemodelsforportfoliorebal-ancingbasedonfuzzydecisionasproposedinthispaper.
WesupposethataninvestorwantstochoosethirtydierenttypesofstocksfromtheShanghaiStockExchangeforhisinvestment.
Therateoftransactioncostsforstocksis0.
0055inthetwosecuritiesmarketsontheChinesemainland.
Assumethattheinvestorhasalreadyownedanex-istingportfolioandhewillnotinvesttheadditionalcapitalduringtheportfoliorebalancingprocess.
TheproportionsofthestocksarelistedinTable1.
Table1.
TheproportionsofstocksintheexistingportfolioStock1234567Proportions0.
050.
080.
050.
350.
100.
120.
25Suddenly,thenancialmarketsituationchanges,andtheinvestorneedstochangehisinvestmentstrategy.
Intheexample,weassumethattheupperboundoftheproportionsofStockjownedbytheinvestoris1.
Nowweusethefuzzyportfoliorebalancingmodelsinthispapertore-allocatehisassets.
Atrst,wecollecthistoricaldataofthethirtykindsofstocksfromJanuary,1999toJanuary,2002.
Thedataaredownloadedfromthewebsitewww.
stockstar.
com.
Thenweuseonemonthasaperiodtogetthehistoricalratesofreturnsofthirty-sixperiods.
Usinghistoricaldataoftheturnoverratesofthesecurities,wecanestimatetheturnoverratesofthesecuritiesasthetrapezoidalfuzzynumbers.
Inthefollowing,wewillgivetwokindscomputationalresultsaccordingtowhethertheinvestorhasaconservativeoranaggressiveapproach.
Atrst,weassumethattheinvestorhasaconservativeandpessimisticmind.
Thenthevaluesofr0,r1,l0,l1,w0,andw1whicharegivenbytheinvestormaybesmall.
Theyareasfollows:r0=0.
028,r1=0.
030,l0=0.
020,l1=0.
025,w0=0.
025andw1Table2.
Membershipgradeλ,obtainedrisk,obtainedreturnandobtainedliquiditywhenr0=0.
028,r1=0.
030,l0=0.
020,l1=0.
025,w0=0.
025andw1=0.
035.
λobtainedriskobtainedreturnobtainedliquidity0.
8350.
02660.
02970.
0301Consideringthethreefactors(thereturn,theriskandliquidity)asfuzzynumberswithnon-linearmembershipfunction,wegetaportfoliorebalancingstrategybysolving(P6).
17AFuzzyApproachtoPortfolioRebalancingwithTransactionCosts=0.
035.
Consideringthethreefactors(thereturn,theriskandliquidity)asfuzzynumberswithtrapezoidalmembershipfunction,wegetaportfoliorebalancingstrategybysolving(P4).
Themembershipgradeλ,theobtainedrisk,theob-tainedreturnandobtainedliquidityarelistedinTable2.
Intheexample,wegivethreedeferentvaluesofparametersαr,αwandαl.
Themembershipgradeη,theobtainedrisk,theobtainedreturnandobtainedliquidityarelistedinTable3.
Table3.
Membershipgradeη,obtainedrisk,obtainedreturnandobtainedliquiditywhenrM=0.
029,wM=0.
030andlM=0.
0225.
ηθαrαwαlobtainedriskobtainedreturnobtainedliquidity0.
8111.
4546008006000.
02820.
03140.
03040.
8061.
42550010005000.
02860.
03190.
03030.
7851.
29540012004000.
02890.
03220.
0302Secondly,weassumethattheinvestorhasanaggressiveandoptimisticmind.
Thenthevaluesofr0,r1,l0,l1,w0,andw1whicharegivenbytheinvestorarebig.
Theyareasfollows:r0=0.
028,r1=0.
036,l0=0.
021,l1=0.
031,w0=0.
032andw1=0.
036.
Consideringthethreefactors(thereturn,theriskandliquidity)asfuzzynumberswithtrapezoidalmembershipfunction,wegetaportfoliorebalancingstrategybysolving(P4).
Themembershipgradeλ,theobtainedrisk,theob-tainedreturnandobtainedliquidityarelistedinTable4.
Table4.
Membershipgradeλ,obtainedrisk,obtainedreturnandobtainedliquiditywhenr0=0.
028,r1=0.
036,l0=0.
021,l1=0.
031,w0=0.
032andw1=0.
036.
λobtainedriskobtainedreturnobtainedliquidity0.
8900.
03240.
03510.
0298Consideringthethreefactors(thereturn,theriskandliquidity)asfuzzynumberswithnon-linearmembershipfunction,wegetaportfoliorebalancingstrategybysolving(P6).
Intheexample,wegivethreedeferentvaluesofparametersαr,αwandαl.
Themembershipgradeη,theobtainedrisk,theobtainedreturnandobtainedliquidityarelistedinTable5.
Table5.
Membershipgradeη,obtainedrisk,obtainedreturnandobtainedliquiditywhenrM=0.
032,wM=0.
034andlM=0.
026.
ηθαrαwαlobtainedriskobtainedreturnobtainedliquidity0.
8491.
7266008006000.
03180.
03490.
02950.
8361.
63050010005000.
03240.
03530.
02930.
8021.
39640012004000.
03280.
03550.
0295Fromtheaboveresults,wecanndthatwegetthedierentportfoliorebal-ancingstrategiesbysolving(P6)inwhichthedierentvaluesoftheparameters(αr,αwandαl)aregiven.
Throughchoosingthevaluesoftheparametersαr,αwandαlaccordingtotheinvestor'sframeofmind,theinvestormaygetafavoriteportfoliorebalancingstrategy.
Theportfoliorebanlancingmodelwiththenon-linearmembershipfunctionismuchmoreconvenientthantheonewiththelinearmembershipfunction.
18Y.
Fang,K.
K.
Lai,andS.
-Y.
Wang5ConclusionConsideringtheexpectedreturn,theriskandliquidity,alinearprogrammingmodelforportfoliorebalancingwithtransactioncostsisproposed.
Basedonfuzzydecisiontheory,twofuzzyportfoliorebalancingmodelswithtransactioncostsareproposed.
Anexampleisgiventoillustratethatthetwolinearpro-grammingmodelsbasedonfuzzydecision-makingcanbeusedecientlytosolveportfoliorebalancingproblemsbyusingrealdatafromtheShanghaiStockExchange.
Thecomputationresultsshowthattheportfoliorebanlancingmodelwiththenon-linearmembershipfunctionismuchmoreconvenientthantheonewiththelinearmembershipfunction.
Theportfoliorebalaningmodelwithnon-linearmembershipfunctioncangenerateafavoriteportfoliorebalancingstrategyaccordingtotheinvestor'ssatisfactorydegree.
References1.
Arenas,M.
,Bilbao,A.
,Rodriguez,M.
V.
:AFuzzyGoalProgrammingApproachtoPortfolioSelection.
EuropeanJournalofOperationalResearch133(2001)287–297.
2.
Arnott,R.
D.
,Wanger,W.
H.
:TheMeasurementandControlofTradingCosts.
FinancialAnalystsJournal46(6)(1990)73–80.
3.
Bellman,R.
,Zadeh,L.
A.
:DecisionMakinginaFuzzyEnvironment.
ManagementScience17(1970)141–164.
4.
Carlsson,C.
,Fuller,R.
:OnPossibilisticMeanValueandVarianceofFuzzyNum-bers.
FuzzySetsandSystems122(2001)315–326.
5.
Konno,H.
,Yamazaki,H.
:MeanAbsolutePortfolioOptimizationModelandItsApplicationtoTokyoStockMarket.
ManagementScience37(5)(1991)519–531.
6.
Leon,T.
,Liern,V.
,Vercher,E.
:ViabilityofInfeasiblePortfolioSelectionProblems:aFuzzyApproach.
EuropeanJournalofOperationalResearch139(2002)178–189.
7.
Mansini,R.
,Speranza,M.
G.
:HeuristicAlgorithmsforthePortfolioSelectionProb-lemwithMinimumTransactionLots.
EuropeanJournalofOperationalResearch114(1999)219–233.
8.
Markowitz,H.
M.
:PortfolioSelection.
JournalofFinance7(1952)77–91.
9.
¨Ostermark,R.
:AFuzzyControlModel(FCM)forDynamicPortfolioManagement.
FuzzySetsandSystems78(1996)243–254.
10.
Ramaswamy,S.
:PortfolioSelectionUsingFuzzyDecisionTheory,WorkingPaperofBankforInternationalSettlements,No.
59,1998.
11.
Watada,J.
:FuzzyPortfolioModelforDecisionMakinginInvestment.
In:Yoshida,Y.
(eds.
):DynamicalAsspectsinFuzzyDecisionMaking.
Physica-Verlag,Heidel-berg(2001)141–162.
12.
Yoshimoto,A.
:TheMean-VarianceApproachtoPortfolioOptimizationSubjecttoTransactionCosts.
JournaloftheOperationalResearchSocietyofJapan39(1996)99–117.
19AFuzzyApproachtoPortfolioRebalancingwithTransactionCosts

昔日数据月付12元起,湖北十堰机房10M带宽月付19元起

昔日数据怎么样?昔日数据是一个来自国内服务器销售商,成立于2020年底,主要销售国内海外云服务器,目前有国内湖北十堰云服务器和香港hkbn云服务器 采用KVM虚拟化技术构架,湖北十堰机房10M带宽月付19元起;香港HKBN,月付12元起; 此次夏日活动全部首月5折促销,有需要的可以关注一下。点击进入:昔日数据官方网站地址昔日数据优惠码:优惠码: XR2021 全场通用(活动持续半个月 2021/7...

RAKsmart秒杀服务器$30/月,洛杉矶/圣何塞/香港/日本站群特价

RAKsmart发布了9月份优惠促销活动,从9月1日~9月30日期间,爆款美国服务器每日限量抢购最低$30.62-$46/月起,洛杉矶/圣何塞/香港/日本站群大量补货特价销售,美国1-10Gbps大带宽不限流量服务器低价热卖等。RAKsmart是一家华人运营的国外主机商,提供的产品包括独立服务器租用和VPS等,可选数据中心包括美国加州圣何塞、洛杉矶、中国香港、韩国、日本、荷兰等国家和地区数据中心(...

爱用云互联租用服务器租美国、日本、美国、日本、购买2天内不满意可以退换,IP可免费更换!

爱用云互联怎么样?爱用云是一家成立于2018年的老牌商家旗下的服务器销售品牌,是正规持证IDC/ISP/IRCS商家,主要销售国内、中国香港、国外服务器产品,线路有腾讯云国外线路、自营香港CN2线路等,都是中国大陆直连线路,非常适合免备案建站业务需求和各种负载较高的项目,同时国内服务器也有多个BGP以及高防节点。专注为个人开发者用户,中小型,大型企业用户提供一站式核心网络云端服务部署,促使用户云端...

www.stockstar.com为你推荐
蓝瘦香菇被抢注蓝瘦香菇这梗是怎么火起来的?怎么觉得火得莫名其妙?站酷zcool有那位知道从哪个网站能下到广告素材蓝色骨头手机宠物的一个蓝色骨头代表多少级,灰色又代表多少级,另外假如有骨头又代表多少级rawtools闪迪32Gsd卡,无法格式化,显示只有30M,并且是raw格式。如何恢复?8090lu.com《8090》节目有不有高清的在线观看网站啊?www.vtigu.com如图所示的RT三角形ABC中,角B=90°(初三二次根式)30 如图所示的RT三角形ABC中,角B=90°,点p从点B开始沿BA边以1厘米每秒的速度向A移动;同时,点Q也从点B开始沿BC边以2厘米每秒的速度向点C移动。问:几秒后三角形PBQ的面积为35平方厘米?PQ的距离是多少百度指数词百度指数我创建的新词se95se.com现在400se就是进不去呢?进WWW怎么400se总cOM打开一半,?求解www.5any.comwww.qbo5.com 这个网站要安装播放器555sss.com拜求:http://www.jjj555.com/这个网站是用的什么程序
花生壳免费域名 万网免费域名 淘宝抢红包攻略 新加坡主机 紫田 seovip linux空间 qq对话框 鲁诺 备案空间 成都主机托管 万网注册 后门 privatetracker 挂马检测工具 电脑主机打不开 主机托管 如何架设服务器 八度空间论坛 2000元电脑主机配置 更多