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AFuzzyIndexTrackingPortfolioSelectionModelYongFangandShou-YangWangInstituteofSystemsScience,AcademyofMathematicsandSystemsScience,ChineseAcademyofSciences,Beijing100080,Chinayfang@amss.
ac.
cnswang@iss.
ac.
cnAbstract.
Theinvestmentstrategiescanbedividedintotwoclasses:passiveinvestmentstrategiesandactiveinvestmentstrategies.
Anindextrackinginvestmentstrategybelongstotheclassofpassiveinvestmentstrategies.
Theindextrackingerrorandtheexcessreturnareconsideredastwoobjectivefunctions,abi-objectiveprogrammingmodelisproposedfortheindextrackingportfolioselectionproblem.
Furthermore,basedonfuzzydecisiontheory,afuzzyindextrackingportfolioselectionmodelisalsoproposed.
Anumericalexampleisgiventoillustratethebehavioroftheproposedfuzzyindextrackingportfolioselectionmodel.
1IntroductionInnancialmarkets,theinvestmentstrategiescanbedividedintotwoclasses:passiveinvestmentstrategiesandactiveinvestmentstrategies.
Investorswhoareadoptingactiveinvestmentstrategiescarryoutsecuritiesexchangeactivelysothattheycanndprotopportunityconstantly.
Activeinvestorstakeitforgrantedthattheycanbeatmarketscontinuously.
Investorswhoareadopt-ingpassiveinvestmentstrategiesconsiderthatthesecuritiesmarketisecient.
Thereforetheycannotgobeyondtheaveragelevelofmarketcontinuously.
Indextrackinginvestmentisakindofpassiveinvestmentstrategy,i.
e.
,investorspur-chaseallorsomesecuritieswhicharecontainedinasecuritiesmarketindexandconstructanindextrackingportfolio.
Thesecuritiesmarketindexisconsideredasabenchmark.
Theinvestorswanttoobtainasimilarreturnasthatofthebenchmarkthroughtheindextrackinginvestment.
In1952,Markowitz[6,7]proposedthemeanvariancemethodologyforport-folioselection.
Ithasservedasabasisforthedevelopmentofmodernnancialtheoryoverthepastvedecades.
KonnoandYamazaki[5]usedtheabsolutedeviationriskfunctiontoreplacetheriskfunctioninMarkowitz'smodeltofor-mulateameanabsolutedeviationportfoliooptimizationmodel.
Roll[8]usedSupportedbytheNationalNaturalScienceFoundationofChinaunderGrantNo.
70221001.
Correspondingauthor.
V.
S.
Sunderametal.
(Eds.
):ICCS2005,LNCS3516,pp.
554–561,2005.
cSpringer-VerlagBerlinHeidelberg2005AFuzzyIndexTrackingPortfolioSelectionModel555thesumofthesquareddeviationsofreturnsonareplicatingportfoliofrombenchmarkasthetrackingerrorandproposedameanvarianceindextrackingportfolioselectionmodel.
Clarke,KraseandStatman[2]denedalineartrack-ingerrorwhichistheabsolutedeviationbetweenthemanagedportfolioreturnandthebenchmarkportfolioreturn.
Basedonthelinearobjectivefunctioninwhichabsolutedeviationsbetweenportfolioandbenchmarkreturnsareused,Rudolf,WolterandZimmermann[9]proposedfouralternativedenitionsofatrackingerror.
Furthermore,theygavefourlinearoptimizationmodelsforin-dextrackingportfolioselectionproblem.
ConsiglioandZenios[3]andWorzel,Vassiadou-ZeniouandZenios[11]studiedthetrackingindexesofxed-incomesecuritiesproblem.
Inthispaper,wewillusetheexcessreturnandthelineartrackingerrorasobjectivefunctionsandproposeabi-objectiveprogrammingmodelfortheindextrackingportfolioselectionproblem.
Furthermore,weusefuzzynumberstodescribeinvestors'vagueaspirationlevelsfortheexcessreturnandthetrackingerrorandproposeafuzzyindextrackingportfolioselectionmodel.
Thepaperisorganizedasfollows.
InSection2,wepresentabi-objectiveprogrammingmodelfortheindextrackingportfolioselectionproblem.
InSection3,regardinginvestors'vagueaspirationlevelsfortheexcessreturnandlineartrackingerrorasfuzzynumbers,weproposeafuzzyindextrackingportfolioselectionmodel.
InSection4,anumericalexampleisgiventoillustratethebehavioroftheproposedfuzzyindextrackingportfolioselectionmodel.
SomeconcludingremarksaregiveninSection5.
2Bi-objectiveProgrammingModelforIndexTrackingPortfolioSelectionInthispaper,weassumethataninvestorwantstoconstructaportfoliowhichisrequiredtotrackasecuritiesmarketindex.
Theinvestorallocateshis/herwealthamongnriskysecuritieswhicharecomponentstockscontainedinthesecuritiesmarketindex.
Weintroducesomenotationsasfollows.
rit:theobservedreturnofsecurityi(i=1,2,n)attimet(t=1,2,T);xi:theproportionofthetotalamountofmoneydevotedtosecurityi(i=1,2,n);It:theobservedsecuritiesmarketindexreturnattimet(t=1,2,T).
Letx=(x1,x2,xn).
Thenthereturnofportfolioxattimet(t=1,2,T)isgivenbyRt(x)=ni=1ritxi.
Anexcessreturnisthereturnofindextrackingportfolioxabovethereturnontheindex.
Theexcessreturnofportfolioxattimet(t=1,2,T)isgivenbyEt(x)=Rt(x)It.
556Y.
FangandS.
-Y.
WangTheexpectedexcessreturnofindextrackingportfolioxisgivenbyE(x)=Tt=11T(Rt(x)It).
Roll[8]usedthesumofsquareddeviationsbetweentheportfolioandbench-markreturnstomeasurethetrackingerrorofindextrackingproblem.
Rudolf,WolterandZimmermann[9]usedlineardeviationsinsteadofsquareddeviationstogivefourdenitionsofthelineartrackingerrors.
Weadoptthetrackingerrorbasedonthemeanabsolutedownsidedeviationstoformulatetheindextrackingportfolioselectionmodelinthispaper.
ThetrackingerrorbasedonthemeanabsolutedownsidedeviationscanbeexpressedasTDMAD(x)=Tt=11T|min{0,Rt(x)It}|.
Generally,intheindextrackingportfolioselectionproblem,thetrackerrorandtheexcessreturnaretwoimportantfactorswhichareconsideredbyin-vestors.
Aninvestortriestomaximizetheexpectedexcessreturn.
Atthesametime,theinvestorhopesthatthereturnofportfolioequalsthereturnoftheindexapproximativelytosomeextentintheinvestmenthorizon.
Hence,theex-pectedexcessreturnandthetrackingerrorcanbeconsideredastwoobjectivefunctionsoftheindextrackingportfolioselectionproblem.
Inmanynancialmarkets,thesecuritiesarenoshortselling.
Soweaddthefollowingconstraints:x1,x2,xn≥0,i=1,2,n.
Weassumethattheinvestorpursuestomaximizetheexcessreturnofport-folioandtominimizethetrackingerrorunderthenoshortsellingconstraint.
Theindextrackingportfolioselectionproblemcanbeformallystatedasthefollowingbi-objectiveprogrammingproblem:(BP)maxE(x)minTDMAD(x)s.
t.
ni=1xi=1,x1,x2,xn≥0,i=1,2,n.
Theproblem(BP)canbereformulatedasabi-objectivelinearprogrammingproblembyusingthefollowingtechnique.
Notethatmin{0,a}=12a12aforanyrealnumbera.
Thus,byintroducingauxiliaryvariablesb+t,bt,t=1,2,Tsuchthatb+t+bt=Rt(x)It2,AFuzzyIndexTrackingPortfolioSelectionModel557b+tbt=Rt(x)It2,(1)b+t≥0,bt≥0,t=1,2,T,(2)wemaywriteTDMAD(x)=Tt=12btT.
Hence,wemayrewriteproblem(BP)asthefollowingbi-objectivelinearpro-grammingproblem:(BLP)maxE(x)minTt=12btTs.
t.
(1),(2)andallconstraintsof(BP).
Thustheinvestormaygettheindextrackinginvestmentstrategiesbycomputingecientsolutionsof(BLP).
Onecanuseoneoftheexistingalgorithmsofmultipleobjectivelinearprogrammingtosolveiteciently.
3FuzzyIndexTrackingPortfolioSelectionModelInaninvestment,theknowledgeandexperienceofexpertsareveryimportantinaninvestor'sdecision-making.
Basedonexperts'knowledge,theinvestormaydecidehis/herlevelsofaspirationfortheexpectedexcessreturnandthetrackingerrorofindextrackingportfolio.
In[10],Watadaemployedanon-linearSshapemembershipfunction,toexpressaspirationlevelsofexpectedreturnandofriskwhichtheinvestorwouldexpectandproposedafuzzyactiveportfolioselectionmodel.
TheSshapemembershipfunctionisgivenby:f(x)=11+exp(αx).
Inthebi-objectiveprogrammingmodelofindextrackingportfolioselectionproposedinSection2,thetwoobjectives,theexpectedexcessreturnandthetrackingerror,areconsidered.
Sincetheexpectedexcessreturnandthetrack-ingerrorarevagueanduncertain,weusethenon-linearSshapemembershipfunctionsproposedbyWatadatoexpresstheaspirationlevelsoftheexpectedexcessreturnandthetrackingerror.
ThemembershipfunctionoftheexpectedexcessreturnisgivenbyE(x)=11+exp(αE(E(x)EM)),whereEMisthemid-pointwherethemembershipfunctionvalueis0.
5andαEcanbegivenbytheinvestorbasedonhis/herowndegreeofsatisfactionforthe558Y.
FangandS.
-Y.
WangFig.
1.
Membershipfunctionofthegoalforexpectedexcessreturnexpectedexcessreturn.
Figure1showsthemembershipfunctionofthegoalfortheexpectedexcessreturn.
ThemembershipfunctionofthetrackingerrorisgivenbyT(x)=11+exp(αT(TDMAD(x)TM)),whereTMisthemid-pointwherethemembershipfunctionvalueis0.
5andαTcanbegivenbytheinvestorbasedonhis/herowndegreeofsatisfactionregardingtheleveloftrackingerror.
Figure2showsthemembershipfunctionofthegoalforthetracingerror.
Fig.
2.
MembershipfunctionofthegoalfortrackingerrorRemark1:αEandαTdeterminetheshapesofmembershipfunctionsE(x)andT(x)respectively,whereαE>0andαT>0.
ThelargerparametersαEandαTget,thelesstheirvaguenessbecomes.
AccordingtoBellmanandZadeh'smaximizationprinciple[1],wecandeneλ=min{E(x),T(x)}.
Thefuzzyindextrackingportfolioselectionproblemcanbeformulatedasfol-lows:AFuzzyIndexTrackingPortfolioSelectionModel559(FP)maxλs.
t.
E(x)≥λ,T(x)≥λ,andallconstraintsof(BLP).
Letη=log11λ,thenλ=11+exp(η).
Thelogisticfunctionismonotonouslyincreasing,somaximizingλmakesηmaximize.
Therefore,theaboveproblemcanbetransformedtoanequivalentproblemasfollows:(FLP)maxηs.
t.
αE(E(x)EM)η≥0,αT(TDMAD(x)TM)+η≤0,andallconstraintsof(BLP),whereαEandαTareparameterswhichcanbegivenbytheinvestorbasedonhis/herowndegreeofsatisfactionregardingtheexpectedexcessreturnandthetrackingerror.
(FLP)isastandardlinearprogrammingproblem.
Onecanuseoneofseveralalgorithmsoflinearprogrammingtosolveiteciently,forexample,thesimplexmethod.
Remark2:Thenon-linearSshapemembershipfunctionsofthetwofactorsmaychangetheirshapeaccordingtotheparametersαEandαT.
Throughselectingthevaluesoftheseparameters,theaspirationlevelsofthetwofactorsmaybedescribedaccurately.
Ontheotherhand,dierentparametervaluesmayreectdierentinvestors'aspirationlevels.
Therefore,itisconvenientfordierentin-vestorstoformulateinvestmentstrategiesbyusingtheproposedfuzzyindextrackingportfolioselectionmodel.
4NumericalExampleInthissection,wewillgiveanumericalexampletoillustratetheproposedfuzzyindextrackingportfolioselectionmodel.
WesupposethattheinvestorconsidersShanghai180indexasthetrackinggoal.
WechoosethirtycomponentstocksformShanghai180indexastheriskysecurities.
WecollecthistoricaldataofthethirtystocksandShanghai180indexfromJanuary,1999toDecember,2002.
Thedataaredownloadedfromtheweb-sitewww.
stockstar.
com.
Weuseonemonthasaperiodtogetthehistoricalratesofreturnsoffortyeightperiods.
ThevaluesoftheparametersαE,αT,EMandTMcanbegivenbytheinvestoraccordinghis/heraspirationlevelsfortheexpectedexcessreturnandthetrackingerror.
Intheexample,weassumethatαE=500,αT=1000,EM=0.
010andTM=0.
009.
Usingthehistoricaldata,wegetanindextrackingportfolioselectionstrategybysolving(FLP).
AllcomputationswerecarriedoutonaWINDOWSPCusingtheLINDOsolver.
Table1showstheobtainedexpectedexcessreturnandtrackingerrorofportfoliobysolving(FLP).
Table2showstheinvestmentratiooftheobtainedfuzzyindextrackingportfolio.
560Y.
FangandS.
-Y.
WangTable1.
Membershipgradeλ,obtainedexpectedexcessreturnandobtainedtrackingerrorληexcessreturntrackingerror0.
94312.
80950.
01520.
0062Table2.
InvestmentratiooftheobtainedfuzzyindextrackingportfolioStock12345678910Ratio0.
00000.
00000.
06200.
02540.
00000.
04080.
01800.
13890.
03240.
0082Stock11121314151617181920Ratio0.
14400.
14880.
01300.
00000.
00000.
00000.
18890.
00000.
00000.
0000Stock21222324252627282930Ratio0.
02760.
00000.
00000.
01240.
10010.
00000.
03950.
00000.
00000.
000001020304050600.
20.
100.
10.
20.
30.
40.
5January,1999March,2003ReturnIndextrackingportfolioShanghai180indexFig.
3.
ThedeviationsbetweenthereturnsoftheobtainedindextrackingportfolioandthereturnsonthebenchmarkShanghai180indexFigure3showsthedeviationsbetweenthereturnsoftheobtainedindextrackingportfolioandthereturnsonthebenchmarkShanghai180indexforeachmonthfromJanuary,1999toMarch,2003.
FromFigure3,wecanndthattheobtainedfuzzyindexportfoliobysolving(FLP)tracksShanghai180indexeciently.
AFuzzyIndexTrackingPortfolioSelectionModel5615ConclusionRegardingtheexpectedexcessreturnandthetrackingerrorastwoobjectivefunctions,wehaveproposedabi-objectiveprogrammingmodelfortheindextrackingportfolioselectionproblem.
Furthermore,investors'vagueaspirationlevelsfortheexcessreturnandthetrackingerrorareconsideredasfuzzynum-bers.
Basedonfuzzydecisiontheory,wehaveproposedafuzzyindextrackingportfolioselectionmodel.
Anexampleisgiventoillustratethattheproposedfuzzyindextrackingportfolioselectionmodel.
Thecomputationresultsshowthattheproposedmodelcangenerateafavoriteindextrackingportfoliostrat-egyaccordingtotheinvestor'ssatisfactorydegree.
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