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TRY3'3' 3' Letter3'T    (XtXX~Xr @0(5X(XtCOMPARISONOFCOMPOSITEPERFORMANCEMEASURES: $  @ 9ANEMPIRICALINVESTIGATION  8  #(XtX(5#@``E BY  $   CherylJ.Frohlich    AssociateProfessorofFinance    UniversityofNorthFlorida     ForCorrespondence:CherylJ.FrohlichDepartmentofAccountingandFinanceUniversityofNorthFlorida4567St.JohnsBluffRoadSouthJacksonville,Florida322249046202630email:cfrohlic@unf.edu +'* Л(5X(Xt@0COMPARISONOFCOMPOSITEPERFORMANCEMEASURES:  @ 9ANEMPIRICALINVESTIGATION   #(XtX(5#r@rrB(5X(Xt ABSTRACT#(XtX(5#   ̜  X+XX(XtThisstudyexaminessevencompositeperformancemodelstoexploretheimpactthemodel   hasupontheperformancerankingsofmutualfundswhenfundtype(objectiveofafund)isϜconsidered.ThestudycomparesrankingsofmutualfundsbythestandardCAPMJensenmodel,theConnorKorajczykAPTJensenlikeperformancemodel,theTreynormodel,theSharpemodel,theAngChuamodel,PrakashBearmodel,andtheStephenProffittmodel.   Theempiricalevidenceshowsthatthemodelsdifferintheirperformancemeasurementofamutualfund.Inaddition,theFriedmantestwasappliedtotherankingorderofthefundsbythemodelsandthenullhypothesiswasrejected.Resultsshowadifferenceinrankingorderinrankingsbetweenthemodels.Theseresultemphasizestheimportanceofknowingtheappropriatemodelforriskandreturn.Thestudy'sresultsshowstatisticallysignificantmeasuredabnormalperformanceinallmodels.Unlikepreviousstudies,thisstudy'sresultsshowthatontheaveragethefundsperform  \ aswellasthemarket.Thatis,alackofoverwhelmingstatisticallysignificantnegativeinterceptsis 4 presentintheresults.Moreover,whenthirdandfourthmomentmodels(AngChua,PrakashBear,andStephenProffittmodels)areusedmostfundshadstatisticallysignificantpositiveintercepts.  #! Л  7?TJOa KO oOa  XX+Ԝ7+XXdXXd7  COMPARISONOFCOMPOSITEPERFORMANCEMEASURES:  @@t t 9ANEMPIRICALINVESTIGATION#X+X #       ӜXX+INTRODUCTION #X+X #  Л      Financiersmustoptforoneoftheavailableperformancemodelstoevaluatetheperformanceofaportfolio.SomefavorthemoretraditionalmeanvarianceperformancemodelssuchastheCapitalAssetPricingModelwhileothersfindthemorerecentmodelsthatincludetheskewnessandkurtosisofthedata(thirdandfourthmomentmodelsrespectively)tobemorereliable.Theselectionofaperformancemodelhasseriousimplicationstoafundmanagerifcompensationistiedtoeither(1)thegenerationofexcessreturnsonthefundor(2)thefund'srankingamongotherfunds.Thepertinentquestionbecomes:Aretheperformanceresultsofpreviousmutualfundstudiesduetothepoorperformanceofthefunds,thuspoorfundmanagementonthesideofthefundmanager,ortheuseofaninadequateperformancemodel?Toanswerthisquestion,theperformanceresultsofrandomlychosenmutualfundsareexaminedandcomparedusingamodifiedArbitragePricingTheoryModel,therewardtovolatilityindexbyTreynor,therewardtovariabilitybySharpe,theAngChuaModel,thePrakashBearmodel,andtheStephenProffittmodelasanalternativetotheCapitalAssetPricingModel(CAPM).  AnearlybyproductofthemeanvarianceportfolioframeworkdevelopedandrefinedbyϜMarkowitz(1952),Tobin(1958)andSharpeLitner(1966,1965),wasthecompositemeanvarianceperformancemeasureknownastheCapitalAssetPricingModel(CAPM).TheoriginalCAPMϜperformancemeasuresdevelopedbySharpe(1966),Treynor(1965)andJensen(1968)werearguedtobeinadequateowingtotheassumptionofasymmetryintheratesofreturnprobabilitydistribution (Arditti,1971;BowerandWippern,1969;Jean,1971;Leland,1999). *D&)w   Roll(1978)attemptedtoaddresstheshortcomingsoftheoriginalmeanvariancemodelsby developingtheArbitragePricingTheory(APT)Model.Othermodels,knownasThirdMomentModels,weredevelopedtoaddresstheasymmetricissueofthedataprobabilitydistributionbyincludingthedatasskewness.TheyweredevelopedandexploredbysuchresearchersasJean(1971),Rubinstein(1973),KrausandLitzenberger(1986),AngChua(1979),PrakashandBear(1986),Diacogiannis(1994),Lee,MoyandLee(1996),andChunhachinda,Dandapani,Hamid,Prakash(1997).ThedevelopmentofFourthMomentModels(StephenandProffitt(1991))furtherexploredthedatasϜasymmetricalnaturebyincludingthekurtosisofthedatasprobabilitydistribution.  Inthefirstsectionofthisstudy,performancemeasurementmodelsarediscussed.Inthefollowingsection,thedataispresented.Themethodologyisinthethirdsection.Thefourthsectioncontainstheempiricalresults.Thelastsectionassertstheconclusionsandpossibleavenuesforfurtherresearch.̛Q XX+  PERFORMANCEMEASURES#X+X#    М/ XX+ CapitalAssetPricingModel#X+X#  H   Traditionally,theCapitalAssetPricingModel(CAPM)oramodifiedCAPMhasbeenusedinperformancemodels.TheCAPMemploysaoneparameterrisk/returnbenchmarksuchastheonedevelopedbyJensen(1968and1969)andlaterrefinedbyBlack,JensenandScholes(1972),andBlumeandFriend(1973).Assumingtherisklevelsoftheexaminedportfoliostoremainstationarythroughtime,performancestudiesusingtheCAPMhaveeffectivelyfocusedonlyonafundmanager'ssecurityselectionskills.Fama(1972)wasthefirsttopointouttheempiricalmeasurementproblemsthatinvolve properlyevaluatingthecomponentsoftimingandselectivity. +%(   AlthoughFama(1972)andJensen(1972)werethefirsttoidentifyempiricalmeasurement problemswithCAPM,otherresearchersusingtheCAPMhaveacknowledgedproblemsassociatedwiththemodel.AseriousproblemusingtheCAPMasaperformancemodelwasidentifiedbyDybvigandRoss'(1985aand1985b)andRoll's(1978)empiricalresultsthatsuggestedtheSMLasabenchmarkforperformancemeasurementcannotbereliedupontogiveaccuraterankings(DybvigandRoss,1985aand1985b;Roll,1978).  AdditionalproblemswiththeCAPMincludeunrealisticassumptions(Diacogiannis,1994),incorrectspecificationoftheindexportfolio(GrinblattandTitman,1994),therelianceoftheCAPMtheoryuponmeanvarianceanalysis,thenonconstantbeta,theerrorsinvariableproblemcausedbyusinganestimatedtimeseriesofbetacoefficients,mismeasurementofriskbybeta(Leland,1999),andtheissueofmarkettiming.  However,perhaps,themostcompellingreasonfornotusingtheCAPMasthenormalbenchmarkcomesfromargumentsbyRoll(1977and1978).HecontendsthatthisapplicationoftheCAPMislogicallyinconsistentundertheassumptionsofthemodel.Roll'sreasoningisthatanymeasuredabnormalperformancecanoccuronlywhenthemarketproxyisinefficient.Price/earningsratios(Basu,1977;Reinganum1981),firmsize(Banz,1981;Chan,1997),Januaryeffect(Keim,1983),anddividendyield(LitzenbergerandRamaswamy,1979andKeim,1985)arefactorsindeterminingabnormalperformance.Thesestudiessuggestthatthemeanvarianceportfolioanalysisapproach,withtheusualmarketindices,resultininefficientestimators.TheseresultsleadonetoquestionseriouslytheuseofthetraditionalCAPMasanappropriateperformancemeasurementmodel.Thedesignofthe CAPMframeworkseemstolackthefeaturestoaddressalltheseconcernsadequately. D+$(   Sincemanyfinancialanalystsandportfoliomanagersperformancesareevaluatedusinga CAPMoramodifiedCAPMperformancemeasurementmodel,theproblemsinvolvedintheCAPMmayhaveaseriousfinancialimpactuponthoseanalystsandmangers.Therefore,financialanalystsandportfoliomanagersmaywishtobeevaluatedbyanotherperformancemodelthatavoidsmanyoftheproblemsingrainedintheCAPMandresultsinamoreaccuratefundperformancemeasurement.TheArbitragePricingTheory(APT)ModelwasthefirstalternativeperformancemodeltotheCAPMtobeconsidered. XX+ArbitragePricingTheory#X+X (#  `    TheArbitragePricingTheory(APT)wasdevelopedbyRoss(1977and1976).FactoranalysisistheusualprocedureappliedinRoss'APTmodel.ThemajorcriticismsoftheAPTmodelinvolvethisprocedure.Sincestandardfactoranalysisincurssuchprohibitivecomputationalcosts,manystudiesfinditnecessarytodecreasethenumberofcrosssectionalobservations.Shanken(1982)andDhrymes,FriendandGultekin(1984)criticizedtheArbitragePricingTheory.DybvigandRoss(1985c)andRollandRoss(1980)addressedmanyoftheAPTissuesraisedbyShanken(1982)andDhrymes,FriendandGultekin(1984).However,theissuesraisedinvolvingthefactoranalysisprocedurewereunresolveduntilConnorandKorajczyk(1986).  AnalternativetofactoranalysisistheConnorKorajczyk(CK)asymptoticprincipalcomponentprocedurethatisamoreappropriateapplicationwhenthenumberofsecurities,n,ismuchlargerthanthenumberoftimeperiods,T.TheCK'sproceduretreatsallthecrosssectionaldatasimultaneously thusavoidingthenecessityofreducingthenumberofobservations.Theresultingstudyisnotonly *D$' morecosteffective,butalsomoremarketinclusive.Thatis,themarketindexincludesbothstocksandbonds.   However,neitherthetraditionalCAPMnortheAPTmodelsconsideredriskadjustedreturns.Therefore,modelsthatassessedtheperformanceofindividualstocksorportfoliosbymeasuringtheirriskadjustedreturnsweredeveloped.Onemeasureofriskisthebeta,whichmeasuresthesensitivityofthestocks(portfolios)returnstomarketreturns.Analternativeriskmeasureisthestandarddeviationofthestocks(portfolios)historicalreturns.Twocommonlyusedtechniquestomeasureriskadjustedreturnsare:TreynorIndex(rewardtovolatilityratio)andtheSharpeIndex(rewardtovariability).XX+}5TreynorIndex/SharpeIndex#}5\0##X+X=0# 8 Ӏ  Treynor(1965)developedariskadjustedmeasureofperformancecalledtheTreynorIndexor 8 therewardtovolatilityratio(RVOL).TheRVOLdistinguishedbetweentotalriskandsystematicriskimplicitlyassumingthatportfoliosarewelldiversified;thus,ignoringanydiversifiablerisk.Therefore,ifthebetaistheappropriatetypeofrisk,astocks(portfolios)riskadjustedreturnscanbedeterminedbytheTreynorIndex.  Sharpe(1966)introducedariskadjustedmeasureofportfolioperformancecalledtheSharpeIndexortherewardtovariabilityratio.Iftotalvariabilityisthoughttobetheappropriatemeasureofrisk,astocks(portfolios)riskadjustedreturnscanbecomputedusingtheSharpeIndex.  TheSharpeandTreynorIndexeliminatedtheproblemofonlyconsideringreturnasameasureofperformance.However,neitherratioisindependentofthetimeperiodoverwhichitismeasured.Thismeansthattheratiocanchangefromoneperiodtoanotherwithdifferentresults.Moreover,bothratiosalsoignorethecorrelationofafundwithotherassets,liabilities,orpreviousrealizationsofits ,&) ownreturn(Hodges,Taylor,andYoder,1997).Furthermore,ananalysisofthesemeasures(FriendandBlume,1970;Gaumnitz,1970)revealasystematicbiasedrelationshipwithriskmeasures.FriendandBlume(1970)concludedthatthisbiaswasduetotheinvalidityofanassumptionusedinthedevelopmentoftheperformancemeasure;theassumptionbeingthatoftheexistenceofequallendingandborrowingopportunitiesforallinvestors.Inaddition,neitherratioalwaystakesintoconsiderationthetransactioncostsnorexpensesassociatedwiththepurchaseandsaleofassets(Murthi,Choi,andDesai,1997).Sincetransactioncostsareimportantinmeasuringafundsperformance,theratiocalculatedwithouttransactioncostsmaybebiased(Pettengill,Sundaram,Mathur,1995).XX+y5ThirdMomentModels#y5+9##X+X 9# "   Eversincethemeanvariancebasedperformancemeasureswereproposed,adebateinvolvingthefailureofthemeanvariancemodelstoconsidertheskewness(thirdmoment)orkurtosis(fourthmoment)ofthereturnsandtheirdistributionshasbeenofconcern.Researchershavearguedthathighermomentscouldnotbeneglectedinperformancemodelsunlesstherewasareasontobelievethattheassetreturnswerenormallydistributedandthattheutilityfunctionwasquadratic,orthathighermomentsareirrelevanttotheinvestorsdecision.However,asearlyasthe1970's,itwasknownthatinvestorspreferred(positive)skewnessandhadanaversiontorisk,asmeasuredbystandarddeviation(Arditti,1971).TheAngChuamodel(1979)buildingonKrausandLitsenbergs(1976)workusedthelinearrelationshipbetweentheexpectedreturnonasecurityorportfoliowithitssystematicriskandϜitssystematicskewnesstodevelopanexcessreturnindextomeasureperformance.Afewyearslater PrakashandBear(1986)builtathirdmomentmodelbaseduponKrausandLitsenbergsworkthat *0$( incorporatedskewnessintothemodelwithoutbringingazerosystematicriskbutanonzerosystematicskewnessportfoliointotheanalysis.XX+ <   StephensProffittModel #X+X>#     StephensandProffitt(1991)extendedtheworkofPrakashandBear(1986)developingageneralizedperformanceevaluationmodelallowingformultiplemomentsofutility.Theirstudyoninternationalfundsdidnotfindkurtosis(fourthmoment)tobeasignificantvariableinexplainingthefundperformance,butskewness(thirdmoment)wassignificant.Fundrankingsobtainedfromthismeasurearedifferentfromthoseobtainedfromthemoretraditionaltwomomentmodels.Chunhachinda,Dandapani,Hamid,andPrakash(1994)studycomparedthetraditionalSharpe,andTreynormeasurestothePrakashBearandStephensProffittmeasures.Theyfoundvastlydifferentperformancerankingsamongthemeasures.Theirstudyalsoaddedsupportforthesignificanceofskewnessandkurtosisvariablesinaperformancemodel.̛  @@D XX+DATA #X+XC# J B  U.S.governmentsecuritiesaccountfor35%,corporatebondsfor15%,andequitiesfortheremaining50%ofthe1984yearendoutstandingbondandequitymarketvalue. * )  1      ׀Twohundredfifty J  fourrandomlychosensecuritieswerechosentorepresentthisdatamixfortheprincipalcomponentanalysis.Stocks(127)represent50%,corporatebonds(40)represents15%,andgovernmentbondsandnotes(87)represent35%ofthedatamix.Thedataforthestocks,governmentnotesandbonds,ϜandcorporatebondsusedtoderivetheprincipalcomponentfactorsforthisstudycovertheperiodfromJanuary1977,throughMarch1984.ThisdataisextractedfromCRSPstocktapes,CRSPgovernment bondtapes,Moody'sBondGuide,andStandardandPoor'sBondGuide.Thisstudyextendstheusual Z+$( marketproxyusedinperformancemeasurementmodelstoincludeboththestockandbondportionofthe"true"market.   Theannualizedmonthlyreturnsfortheindividualstocks,governmentbondsandnotes,andcorporatebondsareusedinthecomponentanalysistechnique.Theadjustedreturnisusedforthereturnsonthegovernmentbonds.Theadjustedreturnisthepriceequivalentofthetotalreturnonacommonstock,wherethevariabilityinreturnsduetothevariabilityinthetimebetweenquotationdateshasbeenminimized.Thegovernmentnoteandbondreturnsareconstructedsoacontinuousflowofreturnsoccurredduringthisperiodfor87datapoints.Ifduringthisperiodagovernmentnoteorbondwithlessthan87periodsuntilmaturityleavesthedatasample,thatobservationismatchedwithasimilarissue(notetonote,bondtobond)toensureacontinuousflowofreturns.Thatis,anotematuringinJanuary1979,ismatchedwithanotestartinginFebruary1979,andcontinuingthroughMarch1984.Thistechniquegeneratestheprincipalcomponentsthatserveasaproxyforthevaryingmarketfactorsthatinfluencethemarketreturns.  Theperformanceresultsof91randomlychosenmutualfundsareexaminedandrankedusingamodifiedArbitragePricingTheoryModel,therewardtovolatilityindexbyTreynor,therewardtovariabilitybySharpe,theAngChuaModel,thePrakashBearmodel,andtheStephenProffittmodelasanalternativetotheCAPM.TherankorderofthefundsbythemodelsaretestedfordifferencesinlocationbythenonparametricFriedmantest.TheannualizedmonthlymutualfundreturnsareextractedfromWeisenberger'sInvestmentCompanies - )  2      ׀database. '0!$  -&* 7?T H    @@((A XX+METHODOLOGY#X+XN#  d N XX+ConnorKorajczykAsymptoticPrincipalComponentProcedure  d #X+XSO#  TheConnorKorajczyk(CK)asymptoticprincipalcomponentstechnique . )  3      ׀toderivetheAPT d  factorsisadirectapplicationofthecompetitiveequilibriumversionofRoss'APT.Afterthefifthprincipalcomponent,thecurvebecomesalmostastraightline.Theadditionalfactorsoverthefirstfivefactorsdonotcontributeanyfurtherinsight.Therefore,thisstudy'sAPTempiricalresultsinvolveafivefactormodel.AnAPTJensenlikeperformancemodel,developedusingatimeseriesanalysis,follows:   %%mWh9)%`|0ya  `E  zK-m   `  ̜ %%  where   ` Rit= Theithmutualfund'sexcessreturn; 8 Л   ` 0t= Intercepttermallowingforreturndifferentfromthenormalrisk $    `  premium.ThisinterceptistheAPTanaloguetoJensen'smeasure   `  ofabnormalperformance;   ` Gjt= Eigenvectorsasproxiesforthefactorsthatinfluencesthereturns;     ` ij= SensitivityoffundI'sreturnstomovementinGjt;     ` it= Afinitevarianceresidualerrorterm.Thecovariancebetweenit      `  andGjtisassumedtobezero./ќ < Л̜ XX+TraditionalJensenPerformanceMeasure#X+X#W#  ! М  xX+XXX+ThemarketproxyusedintheCAPMisthevalueweightedNewYorkStockIndexforbalanced "N andstockfunds.Forthebondfunds,themarketproxyusedintheCAPMistheDowJonesBondAverages. 0 )  4      לThetraditionalCAPMJensenperformancemodel,developedusingthesametimeseries b&" Мanalysis,follows: %%%%mW9)%`|0E%w `OE+ww+ m %%%% .J(+  JOa / H  where   ` Rit= Theithmutualfund'sexcessreturn; P    ` 0t= Intercepttermallowingforreturndifferentfromthenormalrisk <    `  premium.ThisinterceptisthetraditionalJensen'smeasureof   `  abnormalperformance;X   ` Rmt=ReturnonvalueweightedNewYorkStockIndexintimeperiodt; 0  X   `  ReturnonDowJonesBondAverageintimeperiodt;      ` Rft= OnemonthTreasurybillrate;      ` it= SensitivityoffundIsreturnstomovementinthemarket;      ` it= Afinitevarianceresidualerrorterm.Thecovariancebetweeni󛀀      `  andthemarketisassumedtobezero.  Iftheportfoliomanagerhassuperior(inferior)ability,theintercept0inequation(2)will   bepositive(negative).Theinterceptrepresentstheaverageincrementalrateofreturnontheportfolioperunitoftime.Thisincrementalrateofreturnisduesolelytothemanager'sabilitytoforecastfuturesecurityprices.Anaiverandomselectionbuyandholdpolicyyieldsazerointercept.  Tomakeinferencesregardingthefundmanager'sability,knowingthestandarderroroftheestimateoftheperformancemeasureisnecessary.Leastsquaresregressiontheoryprovidesanestimateofthedispersionofthesamplingdistributionoftheintercept0.Inaddition,the  samplingdistributionoftheestimate0isastudenttdistributionwiththeappropriatedegreesof  ` freedom.H  Ifthefund'srisklevelisconstant,thealphaoftheJensenlikeperformancemodelcanbeareasonableindicationofitsstockselectionability(discountingbenchmarkerror).However,markettimingattemptsstillhaveaneffectuponthesignofthealpha,ifthefund'srisklevelisnotconstant.Fluctuationmayoccurduetoshiftsassociatedwithmarkettiming,orbecauseoftheoptionnatureofleveredsecurities.TheJensenalphameasuremaybepositiveornegativedependingonthecovariancebetweenchangesinitsriskpostureandthereturnsonthefactors.Fabozzi,Francis,andLee(1980) -H'* foundthattheuseofmonthlyreturnsresultsinthebetacoefficientofthefundremainingfairlyconstant.Thus,sincethisdataconsistsofannualizedmonthlyreturns,therisklevelofthefundunderconsiderationshouldremainfairlyconstant.Thisconstantrisklevelmayalleviatesuchproblemsasarecommonwithmarkettimingtechniquesthatmaycauseshiftingintherisklevelofthefund.However,consideringthepossibilityofanonconstantrisklevel,theJensenalpha(theintercept)isdividedbytheintercept'sstandarddeviationtoadjustforthisrisk.̜xXxX+\5xTreynorIndex#x\5g##xX+Xxg# t    TheTreynorIndexcomputedas̜mA5%!`|+ `E+ ,Qftm ~ $~ $$$ Ӏ $$~ $~ $ _whereR=averagereturnsonthestock(portfolio)  Ѐ_  Rf=averageriskfreerate  0  =stocks(portfolios)beta.p$$ xXxX+^5xSharpeIndex#x^5j##xX+Xxj# H   TheSharpeIndexiscomputedas̜mA{?|5%!`|{ `E{.! !f vm ) $) $$$  $$) $) $ Ӏ_whereR=averagereturnsonthestock(portfolio) %Z& Ѐ_  Rf=averageriskfreerate '2!( 0  %=standarddeviationofthestocks(portfolios)returns.(")$$  xXxX+ ,&- KO  AngChuaModel#xX+XxMm# d   AngChuauseamodifiedcharacteristicequationthatrequiresinformationaboutthemarket ( portfolio,theriskfreeasset,andinformationbytheinvestoraboutthereturnandsystematicskewnessofazerosystematicriskandnonzeroinvestmentopportunitytodevelopanExcessReturnIndex.SimilartoJensensmodeltheexcessreturnissimplytheinterceptofthecharacteristicequation.AngChuascharacteristiclineisthefollowing:mA5%!`|gC"" `Eg""U# m $$     $$whereRi=actualreturnoffund7=i7= ` 0  R򛀜f=riskfreereturnononemonthTreasuryBillL$$ 0  ERi󛀜=excessreturn8$$ 0  C1i󛀜=regressioncoefficientforthelineartermforfund(8=i8=9=I9=)$$$ 0  Rm=returnonvalueweightedNewYorkStockIndexorReturnonDowJones  (#  @(#  @(# $$ Ѐ   ` BondAverage(dependingontypeoffund)  0  C2i󛀜=regressioncoefficientforthequadratictermforfund(:=I:=;=i;=).$$ ̜TodeveloptheAngChuacharacteristicline,thebetaforeachcompanyforeachtimeperiodwascalculatedasfollows̜eA-` 0  `Eu!<q d tp!p! e $$   $$Ameasureofskewnessforazerobetaportfolioiscalculated̜eA-` 0  `E(.d#f(p!e $$  $$where  Rzit󛀜=Blackszerobetaportfolioreturnforfund<=I<===i==attimet l-'* Ѐ_0  Rzi=themeanofthezerobetaportfolioforfund>=I>=?=i?=returnD/(,$$ oOa  0  Rmt=marketreturnattimetd$$ Ѐ_0  Rm=meanmarketreturn.<$$ ;=oOa  ;=̜Nexttheskewnessofeachfundforeachtimeperiodiscalculated̜eA-` 0  `Ec dLb p!me $$    $$Thebenchmarkbecomes̜eA-` 0  `ES<kd tLp! eߛ $$   $$whereRRit=theexpectedrateofreturnforfund(i). r Theexcessreturnforfund(i)iscalculatedmA5 %!`|P{d `E{d_ 26 | m %$$%$$$$ %$$%$$%$$%$$ %$$%$$%$$%$$ $$%$$%$$where  ERit=ExcessReturn. Z$ Wenowhavethecharacteristicline̜eA- ` 0  `E"dU"p! eߛ $$   $$Fromthecharacteristicline,theExcessReturnIndexisdeveloped̛jA2 "`t  `E ( =  (p! j $$    $$whereY0= ` themeanof(RItRft) +V%4 0  Y1= ` themeanof(RmtRft),B&5$$ 0  Y2= ` themeanof(RmtRm)2-.'6$$ 0  C1i󛀜=regressioncoefficientfromthecharacteristicequationforfund(i)d$$ 0  C2i󛀜=regressioncoefficientforthequadratictermfromthecharacteristicP$$    ` equationforfund(i).?~  NinetyoneY1andY2areobtained.TheirsignificancewillbeexaminedwithaT-test.The91fundswillberankedbytheY1andY2toseeiftheresultsdifferfromtheothermodels.xXxX+ PrakashBearModel  :  #xX+Xx#  ThePrakashBearModelisbaseduponathreemomentCAPMthatisnot t  Ѐpredicatedontheexistenceofazerosystematicriskportfolio.ThePrakashBear̀threemomentcompositemeasurecanbewritteninexpostformas : $  mA5 %!`| `E( ~ 2 hZ!9 m $$     $$ where  @@Rjt=returnoffundjattimet.    󀀀 ` N x Z1jt=Covariance(Rjt,Rmt)=1/n&𛀜(RjtRj)*(Rmt󀀄Rm). d Ѐt=1̀N@=@=񀀛̜Z2jt=CoSkewness=1/n&(RjtRj)*(Rmt󀀄Rm)2. ( Ѐt=1Rmt=marketreturnattimet. d! Є̜Rj=meanreturnoffundj. <#  ЄRm=meanreturnoffundm. %" М@0=interceptregressioncoefficient. &# М1=regressioncoefficientforcovariance. '' $ М2=regressioncoefficientforcoskewness. N(!% N=numberofmonthsintheobservationperiod.̜ejt=errortermwhichiswidesensestationary. *8$' М̛ ,&) @Iftheinterceptisstatisticallypositivethefundhasearnedexcessreturns.Thefundsperformancecanberankedby=1/2.Ninetyone0,1,2areobtained.Theirstatisticalsignificanceare < examinedbyaTtest.The91fundswillberankedbytoseeiftheresultsdifferfromothermodels.    xXxX+StephenProffittModel#xX+XxB#   М  StephenProffittextendedtheworkofPrakashBear(1986)developingageneralized  & performancemodelthatallowsformultiplemomentsofutility.TheStephenProffittgeneralizedperformancemodelcanbewritteninexpostformas:eA- ` 0  `EM dF p!l e $$    $$where:̜Rjt=returnoffundjattimet. $ ЀNZnjt=1/n&𛀜(RjtRj)*(Rmt󀀄Rm)n.  Ѐt=1n=1,2,3forcovariance,coskewness,andcokurtosisrespectively.Rmt=marketreturnattimet. \ Є̜Rj=meanreturnoffundj. 4 Є̜Rm=meanreturnoffundm. p   М0=interceptregressioncoefficient. \!! М1=regressioncoefficientforcovariance. H"" М2=regressioncoefficientforcoskewness. 4## М3=regressioncoefficientforcokurtosis.  $$ N=numberofmonthsintheobservationperiod.̜ejt=errortermwhichiswidesensestationary. %& Iftheinterceptisstatisticallypositivethefundhasearnedexcessreturns.Thefundsperformancecanberankedby=2/3.Ninetyone0,1,2,and3areobtained.Theirstatisticalsignificanceare *0$+ examinedbyaTtest.The91fundswillberankedbytoseeiftheresultsdifferfromothermodels. l,&- ЇxXxX+  EMPIRICALRESULTS #xX+Xxy#ԛ d   Table1(AC)givestheriskadjustedrankingofthefundsbytypeandfundnumberforthemodels.TherankingsarealmostthesameforthebondfundsunderallthemodelsexcepttheAngChuamodel,thePrakashBearmodel,andtheStephenProffittmodelwhichtakesintoaccounttheϜskewnessandkurtosisofthereturns.Inallthemodelswiththeexceptionofthehighermomentmodelsthebondfundsinterceptsarenotstatisticallydifferentfromzeroand,therefore,rankingthefundsbyriskisamuteexercise.Howeverwhenskewnessisincludedinthemodel,threeofthefundsinthePrakashBearmodelhadstatisticallypositivesignificantinterceptsatthe95%confidencelevel.WhenkurtosisisincludedintheStephenProffittmodel,allofthebondfundshavestatisticallypositiveinterceptsatthe95%confidencelevel.Insummary,thePrakashBearmodelfound43%andtheStephenProffittmodelfound100%statisticallysignificantinterceptsinthebondfunds.@BB8InsertTable1(A)AboutHere  Ofthebalancedfunds,100%ofthefundsdifferinrankingpositionwhenallmodelsareconsidered.Whenthehighermomentmodelsareexcludedthan53%ofthefundsdifferinorderofrankingamongthefourmodels.Thesignificantinterceptranksinthesameorderundermeanvariancemodels.TheAngChuamodelhasthesamesignificantintercept(fund#46)thatwassignificantintheϜmeanvariancemodels.ThePrakashBearfoundallbuttwofundsandtheStephenProffittmodelfoundallbutonefundhadstatisticallysignificantpositiveintercepts.Insummary,boththeAPTandCAPMmodelfound7%,theAngChuamodelfound7%,thePrakashBearmodelfound87%andtheϜStephenProffittmodelfound93%statisticallysignificantinterceptsinthebalancedfunds. @``9InsertTable1(B)AboutHere +0%(   Ofthestockandspecialtyfunds,100%ofthefundsdifferinrankingpositionwhenallseven modelsareconsidered.OfthesignificantinterceptsundertheAPTmodel71.5%variesfromtheCAPMmodel.TheAPTmodelhad13significantinterceptswith10ofthemnegative.TheCAPMmodelhad15significantinterceptswitheightofthemnegative.TheAngChuamodelhasthreestatisticallysignificantnegativeand32statisticallysignificantpositiveintercepts,thePrakashBearmodelhas63staticallysignificantpositiveintercepts,andtheStephenProffittmodelhas67statisticallysignificantpositiveintercepts.Insummary,theAPTmodelfound19%,theCAPMmodelfound22%,theAngChuafound51%,thePrakashBearmodelfound91%,andtheStephenProffittmodelfound97%ofthestockfundswithstatisticallysignificantintercepts.@9InsertTable1AboutHere  Thehighermomentmodelshadmoresignificantinterceptsandtherankingofthefundsvariedsignificantlyfromtheotherfourmodels.Inaddition,thesignificantinterceptsweremostlypositive.ϜTheconclusionseemstobethatskewnessandkurtosisarefactorsthatmustbeconsideredwhendevelopingperformancemeasures.Inaddition,whenrankingthefunds,themodelusedbecomesofincreasingimportancewhichcanbeevidencedbythevariationinrankingorderandthenumberofstatisticallysignificantinterceptsastheobjective(type)offundchangestoastockandspecialtyfund.  The"t"valueforthetwotail5%levelofsignificanceat81degreesoffreedomis1.994.Thus,consideringallfunds,itisexpectedthat4.6observationswillhavea"t"valuegreaterthan1.994.IntheAPTmodel,11fundshavesignificantinterceptswhichisgreaterthanrandomchance.Atthis5%levelofsignificance,theJensenCAPMmodelhadsevenfundswithsignificantintercepts,theAng Chuamodelhad27fundswithsignificantintercepts,thePrakashBearmodelhad77fundswith D+$( significantintercepts,andtheStephenProffittmodelhad87fundswithsignificantintercepts;also,greaterthanrandomchance.Fortheotherlevelsofsignificance,similarresultsarefound.   Althoughthe"t"testmeasuresthesignificanceoftheindividualpointestimate,itdoesnotsignifythe"goodnessoffit"forthemodelnortheexistenceofalinearrelationshipbetweentheindependentanddependentvariable.The"goodnessoffit"testresultsweremuchhigherforthestockandspecialtyfundsthanforthebondfunds.Tovalidatethemodelnotonlythe"adjustedR2"butalso 8  the"F"testfortheindividualfundswereconsidered.  TheFstatistics,withK1andNKdegreesoffreedom,allowthehypothesisthatnoneoftheindependentvariableshelpstoexplainthevariationofthedependentvariableaboutitsmeantobetested.Inotherwords,theFstatisticteststhejointhypothesisthat2=3=...=K=0.Ifthenull  hypothesisistrue,thentheR2andFwouldbecloseto0. p   Ofthe91mutualfundsinthefivefactorAPTmodel,90rejectthenullhypothesisatthe95%confidencelevel,and88didlikewiseatthe99%confidencelevel.Similarresultsappearintheothermodels.Therefore,theFtestvalidatesthefund'sR2. \   ӄInsertTable2(AC)AboutHere    Finally,theFriedmanprocedureisappliedtothethreecategoriesoffundstotestfordifferencesinthelocationoftherankedfundsbythesevenmodelswithintheirobjectivecategory.ThedistributionoftheFriedmanstatisticapproximatelyfollowsachisquaredistributionwithk1degreesoffreedom.Thisapproximationworkswellasboththeblocksandfactorlevelsexceedfive.TheFriedmantestprocedure,liketheKruskalWallistestprocedure,istorejectthenullhypothesisifthecomputedFriedmanstatisticliesintherighttailofthechisquarecurve;thatis,rejectthenullhypothesisiftheFriedmanstatisticisgreaterthanthechisquarestatisticassociatedwiththerighttail -&* areaofandk1degreesoffreedom.AsTable2(AC)indicates,inallcases,thenullhypothesiswas d rejectedatasignificancelevelof.025withtheappropriatedegreesoffreedom.Thefundswererankeddifferentlyunderthevariousmodels.Thisresultfurthersubstantiatestheimportanceofthemodelchosentoevaluatetheperformanceoffunds.@D xXxX+CONCLUSION   ` #xX+Xx}#  Thestudysresultsonmutualfundperformancereportedabovefurthervalidatepreviouswork(Jensen,1969;Jensen,1968;Sharpe,1966;McDonald,1974;Fama,1972;Friend&Blume,1970)whenmeanvariancemodelsareused.Itsuggeststhatfundmanagers,despitefundtype,are,onthe t   average ,notabletopredictsecuritypriceswellenoughto outperform abuyandholdpolicywhen L themeanvariancemodelsareusedtomeasureperformance.However,whenskewnessisconsidered,Ϝ36ofthe91fundsintheAngChuamodel,79ofthe91fundsinthePrakashBearmodel,and88xXxX+ԛ#xX+Xx#of  the91fundsintheStephenProffittmodelhadsignificantintercepts.Inaddition,alloftheseinterceptsexceptforfourwerepositivewhichsuggeststhatfundmanagers,despitefundtype,are,onthe average ,abletopredictsecuritypriceswellenoughto outperform abuyandholdpolicywhen x thehighermomentmodelsareusedtomeasureperformance.Theseresultleadstotheconsiderationthattheuseofamodelincorporatinghighermomentsmaybeabettermeasureofperformancethanthemoretraditionalmeanvariancebasedmodels.  Unlikepreviousempiricalwork,theoverwhelmingevidencesuggeststhatfundmanagersdo performaswellas themarket.Inaddition,inrespectofthosefundswithexcessreturns,fourunder (!$ theAPTJensenmodel,eightundertheCAPMJensenmodel,32xXxX+ԛ#xX+Xx#undertheAngChuamodel,79under )#& thePrakashBearmodel,and88undertheStephenProffittmodelhavepositiveexcessreturns.As $,%( theperformancemodelsincludehighermoments,thestatisticalsignificanceofapositiveinterceptbecomesmorefrequentandtheabilityoffundmangerstoearnexcessreturnsisrevealed.Therefore,somefundsdooutperformthemarket.  ConsiderabledifferencesexistbetweentheperformancemeasuresyieldedbythetraditionalCAPMJensenmodelandthoseproducedbytheAPTJensenlikemodel.TheCAPMmodelhasmorepositiveinterceptsthantheAPTmodel,whichresultsinadifferenceinperformanceranking.ThehighermomentmodelshavemoresignificantinterceptsthaneithertheCAPMJensenortheAPTJensenlikemodel.Inaddition,thehighermomentmodelshavemorepositiveinterceptsthaneithertheCAPMJensenortheAPTJensenlikemodel.Ifaportfoliomanagerispaidaccordingtohisperformanceranking,thereisagreatinterestinwhichmeasureisused.Thesedifferencesemphasizetheimportanceofknowingtheappropriatemodelforriskandexpectedreturn.  InrankingthefundsbytheriskadjustedJensenalpha,themodelusedbecomesincreasinglyimportantastheobjective(type)ofthefundchanges.WhenthemeanvariancemodelsortheAPTmodelisused,virtuallynodifferenceexistsinrankingbetweenthemodelsforbonds.However,thehighermomentmodelsdohavemorefundswithsignificantinterceptsthanthemeanvarianceorAPTmodels.Inthebalancedfunds,100%ofthefundsvaryinorderofrankingwhenallsevenmodelsareconsidered.Evenwhenthehighermomentmodelsareexcluded,56%ofthefundsstillvaryintheorderofrankingbetweentheremainingfourmodels.Stockandspecialtyfundsvaryintheorderofrankingby100%.Ifthesignificantinterceptsareonlyconsidered,thenonlythestockandspecialtyfundsvaryinrankingwhenthehighermomentmodelsareexcludedfromtheanalysis.Theirvariance inorderofrankingis71.5%forthesignificantintercepts.However,whenoneallowsformodels D+$( incorporatingskewnessandkurtosiseventhefundswithsignificantinterceptsvaryinorderofrankingmuchmorethanwhenthemeanvariancemodelsortheAPTmodelareusedsolely.   Inadditiontotheaboveconclusions,apuzzleappearsinthestudysresults.Examinationofthestockandspecialtyfundsusingthe"goodnessoffit"testontheperformancemodelsshowsthat,ontheaverage,theperformancemodelsexplainsalargeportionofafundsreturn.Theresultssuggestthatthemodelsworkwellforstockandspecialtyfunds.However,inexaminingthebondfundsforthesame"goodnessoffit"test,theAPTandCAPMperformancemodelsexplainlessthan25%ofafund'sreturnwhiletheStephenProffittmodelwhichincludeskurtosisoftheprobabilitydistributionexplainsmorethan41%andontheaverage58.5%ofafundsreturn.  Furtherresearchisnecessaryintowhyper