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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.
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