S U P E R V I S O R : M I C H A E L C H R I S T E N S E N P ERFORMANCE EVALUAT ION O F 6 0 DAN I SH

MUTUAL FUNDS

MICHAELVILLADSEN(286274)

U13BSC

AarhusSchoolofBusinessandSocialSciences

Spring11

AbstractIn2011themarketvalueoftheDanishmutualfundsexceededDKK1200billion,investedby

more than 835.000 people. Therefore, the performance by the mutual funds is of great

interest toa largeproportionof theDanishpopulation, andalongwith thehugeamountof

moneymanagedbytheDanishmutualfunds,itisveryimportanttoshedlightonthereturns

thatthemutualfundsareabletoobtain.

Thepurposeofthisthesisisthereforetoprovideaperformanceanalysisof60Danishmutual

funds in the period from 2001‐2009. The analysis will cover three investment categories:

Danishstocks,Europeanstocks,andGlobalstocks.

Thetheoreticalframeworkofthethesisisbuildaroundthetheoryofefficientmarketsandthe

Capital Asset Pricing Model. Consequently, the performance measurements used in the

analysiswillfollowthesetheoriesaswell.Inordertoanalysetheperformanceofthemutual

funds,relevantbenchmarkshavebeenchosen,andthepurposeofthethesisistoseewhether

themutual funds are able to outperform those benchmarks or not. The analysiswill begin

withtheJensenindex,howeverthisisasingleindexmodel,andthereforeitmightnotbeable

toexplainallofthevariationinthereturnsofthemutualfunds.Therefore,theJensenindexis

followedbyamulti‐indexmodel,where4 to7benchmarksare included,dependenton the

investment category of the mutual funds. However, the Jensen index and the multi‐index

modelonlyfocusesontheselectionabilityofthefunds,andthereforeitisdesiredtoseparate

theselectionandtimingabilitiesofthemutualfundsusingtheTreynor&Mazuymodel.

The results from the analysiswere that in the Jensen index, 23 funds underperformed the

index,31fundsperformedneutrally,and6fundsoutperformedtheindex.Inthemulti‐index

model,thesenumberschangedto31,27,and2,respectively,therebyshowinganevenweaker

performancebythemutualfunds.

Takinganaverageofthesetwoanalyseswouldrevealthatclosetohalfof themutual funds

haveperformedsobadthattheyarenotabletocovertheirownexpenses.Thisshouldraise

someconcernregardingtheworkdonebytheDanishmutualfunds.

However,positiveaspectswerealsofoundintheanalysis,sincetwofundsshowedsignificant

positiveperformanceinbothofthemodels.Thosefundsbeing“SEBinvestDanskeaktier”and

“Valueinvest Danmark, Blue Chip” showing a monthly over performance of 0,3271% and

0,3376%, respectively. Itmust thereforebe concluded that these fundshavedone their job

reallywell.

Finally,theTreynorandMazuymodelwasusedtoseparatetheselectionandtimingabilities

ofthemutualfunds.Surprisingly,itwasdiscovedupthat8mutualfundsinvestinginDanish

stocks showed significant timingabilities.However, for the remainingmutual funds, timing

abilities were not present. None of the funds showing timing abilities showed significant

positive selection abilities, and the result of that was that the α values where decreased

comparedtotheJensenindex. Itshouldfinallybenoted,thatnoneofthetwopreviousbest

performingfundsshowedsignificanttimingabilities.

The overall conclusion on the thesis is that one should be careful when selecting mutual

funds. The reason of that is that the returns obtained by the mutual funds compared to

relevantbenchmarksdifferalot,andonewouldthereforeearnsignificantlymoremoneyby

choosingthebestfunds.Finally,itshouldbementionedthathighreturnsofamutualfundin

one period does not guarantee high returns in the following period, and therefore the

selectionofmutualfundsshouldnotsolelybebasedonpreviousreturns.

Tableofcontents

1.INTRODUCTION 1

1.1.PROBLEMSTATEMENT 21.2.DELIMITATION 31.3.THEORETICALFRAMEWORK 3

2.INVESTMENTFUNDSINDENMARK 4

2.1GENERALLYABOUTTHEINVESTMENTFUNDS 42.2LEGISLATION 52.3COSTS 6

3.THEORY 8

3.1EFFICIENTMARKETS 83.2CAPITALASSETSPRICINGMODEL(CAPM) 103.2.1CRITIQUEOFTHECAPITALASSETPRICINGMODEL(CAPM) 123.3RISKMEASUREMENTS 143.3.1SYSTEMATICRISK 143.3.2UNSYSTEMATICRISK 153.4PERFORMANCEMEASUREMENTS 163.4.1JENSEN’SALPHA 183.4.2CRITIQUEOFJENSEN’SALPHA 193.4.3MULTI‐INDEXMODEL 213.4.5MARKETTIMING 23

4.DATA 28

4.1CHOICEOFMUTUALFUNDS 284.2CHOICEOFBENCHMARKS 294.2.1INDEXMETHODOLOGY 304.2.2BENCHMARKFORTHEJENSENINDEX 314.2.3BENCHMARKFORTHEMULTI‐INDEXMODEL 344.3CHOICEOFRISKFREERATE 374.4SURVIVORSHIPBIAS 374.5ASSUMPTIONSFORTHETESTS 39

5.PERFORMANCEANALYSIS 45

5.1JENSEN’SALPHA 455.1.1DANISHSTOCKS 465.1.2EUROPEANSTOCKS 475.1.3GLOBALSTOCKS 485.1.4CONCLUSIONONJENSEN’SALPHA 505.2MULTI­INDEXMODEL 515.2.1DANISHSTOCKS 515.2.2EUROPEANSTOCKS 525.2.3GLOBALSTOCKS 545.2.4CONCLUSIONONTHEMULTI‐INDEXMODEL 565.3MARKETTIMING 575.3.1DANISHSTOCKS 575.3.2EUROPEANANDGLOBALSTOCKS 585.3.2CONCLUSIONONTHETREYNORANDMAZUYMODEL 595.4CONCLUSIONONTHEPERFORMANCEANALYSIS 60

6.DISCUSSIONANDREFLECTIONS 61

7.CONCLUSION 66

8.BIBLIOGRAPHY 68

APPENDIX1:JENSEN’SALPHA 70

DANISHSTOCKS 70RESULTS 70ASSUMPTIONS 70EUROPEANSTOCKS 71RESULTS 71ASSUMPTIONS 71GLOBALSTOCKS 72RESULTS 72ASSUMPTIONS 72

APPENDIX2:MULTI­INDEXMODEL 73

DANISHSTOCKS 73

SIGNIFICANTBENCHMARKS 73ASSUMPTIONS 73RESULTS 74EUROPEANSTOCKS 75SIGNIFICANTBENCHMARKS 75ASSUMPTIONS 75RESULTS 76GLOBALSTOCKS 77SIGNIFICANTBENCHMARKS 77ASSUMPTIONS 77RESULTS 78

APPENDIX3:MARKETTIMING 79

DANISHSTOCKS 79RESULTS 79ASSUMPTIONS 79EUROPEANSTOCKS 80RESULTS 80ASSUMPTIONS 80GLOBALSTOCKS 81RESULTS 81ASSUMPTIONS 81

APPENDIX4:BENCHMARKSOFTHEINVESTMENTFUNDS 82

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1.IntroductionTheamountofmoneyinvestedintheDanishmutualfundshasexperiencedalargeincreasein

the period from 2001 to 2010, increasing fromDKK 250 billion to DKK 1.200 billion. The

majordevelopmentintheamountofmoneyinvestedputsalargefocusonthereturnsthatthe

mutualfundsareabletoobtain.

Asawiseinvestor,oneshouldbeinterestedinselectingthebestmutualfundstotakecareof

one’smoney.However,thenumberofmutualfundshasalsoincreasedalongwiththeamount

ofmoneyinvested,causingittobecomeevenmoredifficultforaninvestortodecidewhereto

placehismoney.Inordertohelpdecideuponwhichmutualfundstochoosemanydifferent

performance analyses of the mutual funds have been made, some more valid than other.

However, as a general rule proper performance evaluation ofmutual funds should rest on

risk‐adjusted returns, as the investorsmust be considered risk‐averse, therebywanting to

obtainthelargestpossiblereturnwiththesmallestamountofrisk.Inaddition,investorsmust

alsobeinterestedinthereturnthemutualfundhasobtainedcomparedtoarelevantmarket

indexfollowingthereasoning,thatoneshouldnotbesatisfiedwithobtaininga10%returnin

amutualfundifthereturnofthemarketwithasimilarriskhasbeen20%.

Nevertheless, the mutual funds are facing a hard time, following the efficient market

hypothesisbyFama(1970).Theefficientmarkethypothesisstatesthatifamarketisefficient

in it strongest form, then the stock prices would follow a random walk, thereby making

predictionsof futurestockpricesworthless. Ifmarketsareefficient, thetheorystatesthata

passive investment strategy would be the best choice. Still, most mutual funds choose an

activeinvestmentstrategyinthebeliefthattheyareabletomakevaluablepredictionsabout

futurestockpricesenablingthemtooutperformtheirbenchmark.

However, Christensen (2001) argues thatmutual funds can use different strategies, which

maketheirinvestmentslookbettercomparedtotheirbenchmarks.Oneofthembeingthatthe

mutual fund uses a benchmark where dividends are excluded and afterwards including

dividendsinthecalculationoftheirownreturn.AccordingtoChristensen(2001)thiswould

givethemutualfundsaheadstartof2‐3%comparedtotheirbenchmark.

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Another interesting aspects in performance analysis of mutual funds is performance

persistency. Christensen (2004) analysed whether performance persistency exists on the

market forDanishmutual funds.Hereachedtheconclusionthatperformancepersistencyis

non‐existentontheDanishmarket,meaningthatonecannotpredictwhichmutualfundswill

perform thebest in the followingperiodbasedonhow themutual fundsperformed in the

previousperiod.

Consequently, it could be argued that the previous returns are of no relevance to new

investors,sincetheonlyonesbenefitingfromthosearetheinvestorsofthattime.However,

many private investors are still interested in the how mutual funds have performed

previously,sincegoodpreviousperformancecangiveinvestorssomekindof“false”security,

even though this is conflicting with the results reached in the literature regarding

performancepersistency.

Thepurposeofthisthesisistofindthebestperformingmutualfundsintheperiod2001to

2010inthethreemostcommoninvestmentcategoriesfortheDanishmutualfunds,namely

Danish stocks, European stocks, and Global stocks. The evaluation will be based on risk

adjustedreturnsandrelevantbenchmarksincludingdividends.

Thethesiswillshowaninformativepictureoftheperformanceofthemutualfunds,whichcan

beusedinevaluationofwhethertheirobtainedresultsaresatisfactoryornot.Ifonebeliefs

that the performance ofDanishmutual funds is persistent one can also use the analysis to

choosewheretoplaceyourfutureinvestment.

1.1.Problemstatement

Basedontheintroduction,theaimofthethesisistoanalyse60Danishequitymutualfunds

investing in Danish stocks, European stocks, or Global stocks. The period of analysis is 10

yearsrangingfrom2001‐2010.

Thespecificquestions,whichareanalysed,areasfollows:

‐ HowhastheDanishmutualfundsperformedcomparedtorelevantpassivebenchmarks?

‐ DoestheDanishmutualfundspossesstimingabilities?

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1.2.Delimitation

Inordertogetamorethoroughanalysis,certainlimitationshavebeenmadeintheselection

ofthemutualfundsfortheanalysis.

Firstofall,ithasbeendecidedtokeepthefocusinthisthesisonDanishequitymutualfunds.

Thereasonforthischoiceis,thatismustbeexpectedthatthestockmarketsaremorevolatile

compared to the bonds market, and therefore it is found more interesting to analyse the

mutualfundsinvestinginstocks.

Inaddition,ithasbeendecidedtolimittheinvestmentcategoriestoonlythree;thosebeing;

Danishstocks,Europeanstocks,andGlobalstocks.Thereasonforthesespecificchoicesisthat

these three investment categories are themost common for the Danishmutual funds, and

thereforerelativelylargesamplesofmutualfundscanbegatheredineachofthecategories.

The classification of the mutual funds in the three categories will follow the classification

madebytheFederationofDanishInvestmentAssociates(IFR).

Intermsoftheselectionofthemutualfundsforthethesis,ithasbeendecidedonlytoanalyse

mutualfunds,whichhavebeenrunningfortheentireperiodofanalysisfrom2001to2010.

However, this raises the problemof survivorship bias,whichwill be discussed later in the

thesis.Thoughitshouldbenoted,thatthethesishasbeenlimited,sothatsurvivorshipbiasis

notdealtwithintheanalysis.

1.3.Theoreticalframework

The theoriesandanalyses chosen for the thesisareallbuildupon theCapitalAssetPricing

Model (CAPM). Inorder toanalyse theperformanceof theDanishmutual funds, the Jensen

index will be used as a single index model, followed by a multi index model. In order to

separate the selection and timing abilities of themutual funds, the quadratic Treynor and

Mazuymodelwillbeusedaswell.A thoroughdiscussionandargumentation for theuseof

eachofthemodelswillbeincludedinthetheorysection.

Finally, the results thatwill be reached in the analysiswill be compared to similar results

obtainedintheliterature.

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2.InvestmentfundsinDenmark

Thissectionisadescriptivesection,whichwillgiveabriefintroductiontotheDanishmutual

funds. It therefore serves the purpose of providing the reader with some very basic

knowledgeabouttheDanishmutualfunds.

2.1Generallyabouttheinvestmentfunds

TheFederationofDanishInvestmentAssociates(IFR)has30mutualfundsthataremembers

ofthefederation.InbroadtermthismeansthattheDanishinvestorhas30differentmutual

fundstochooseamong.Allofthesemutualfundsthenhavenumerousofdifferentinvestment

portfolios,whichaninvestorcouldchoosetoplacehis/hermoneyin.

TheamountofmoneyinvestedintheDanishmutualfundsisalsoofinterest.

Christensen(2005)concludesthattherehasbeenatremendousincreaseinthemarketvalue

ofthemutualfundsworldwide,butforDenmarkthedevelopmenthasbeenmuchbiggerthan

fortheEuropeanUnionasawhole.Heconcludes,thatthemarketvalueoftheDanishmutual

funds increased fromUSD3 billion in 1992 toUSD57 billion in 2002. This amounts to an

annual increase of 34%, which can be compared to an annual increase of 18% for the

EuropeanUnion.

However,thesenumbersareratherold,andthereforethebelowdiagramhasbeenmadeto

show how themarket value of the Danishmutual funds has developed over the period of

analysisfrom2001to2010.

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Source:StatisticsDenmark

The diagram shows that themarket value of the Danishmutual funds has increased from

approximately DKK 250 billion in 2001 to approximately DKK 1200 billion in 2011. This

amountstoanannualincreaseofapproximately17%,therebyshowing,thatthedevelopment

hasslowedabitdown,comparedtothepreviousperioddescribed.

The diagram shows a steady development over the period as a whole, however a major

decrease occurred from 2008 to 2009. This was probably caused by the decreasing stock

pricesduringthefinancialcrisis.

2.2Legislation

Before investing your money in a mutual fund, it is a good idea to get familiar with the

legislationonthearea.Rememberthatmostofthelegislationismadeinordertoprotectthe

consumer.Thereforethissectionwillgiveabriefdescriptiononsomeofthemainlegislation

onthemutualfundswhosemaininvestmentcategoryisstocks.

Firstofall, “legislationonmutual funds”(lovominvesteringsforeninger)stipulatesruleson

howmutualfundsshouldplacetheirinvestments,sothatproperriskdispersionisobtained.

The law requires that no more than 10% of the capital can be invested in unquoted

companies.Inaddition,asinglestockmustnotwaymorethan5%oftheentireinvestment,

andfinally,theportfolioofthemutualfundsshouldcontainatleast16differentstocks.

Usuallythemutualfundsholdfrom30to250securities(Christensen,2005).

0

200

400

600

800

1000

1200

1400

2001 2002 2003 2004 2005 2006 2007 2008 2009 2010 2011

DKKinbillions

Year

MarketvalueofDanishinvestmentfunds

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In addition, the legislation states that the mutual funds should invest at least 75% of the

investmentinsecuritieswithintheirmaininvestmentcategory.However,thisalsomeansthat

themutual funds have the possibility to invest up to 25%of their investment in securities

outsidetheirmaininvestmentcategory.

2.3Costs

EachyeartheFederationofDanishInvestmentAssociates(IFR)publishesofcostanalysisof

theDanishmutualfunds.

The analysis states that 835.000 Danes own shares in themutual funds,meaning that the

costsofthefundsarerelevanttoalargeproportionoftheDanishpopulation.

Thegeneralresultsshowthattheadministrationcosts increasedforthestockdepartments,

whereas they decreased in the foreign bonds departments. In total the analysis showed a

decreaseintheaveragecostsforallfunds,decreasingfrom1,05%in2008to1,02in2009.

In termsof thisspecificanalysis, itwouldbe interesting toseewhat thecostsare for the3

investmentcategoriesfortheanalysis:

Thetablebelowshowsthis,basedonthereportbyIFR(Investeringsforeningsrådet,2010):

The most interesting thing from the table is probably the large spread in annual

administration costs for the funds investing in global stocks, showing a difference of 4%‐

pointsbetweenthecheapestandthemostexpensivefund.

One should keep inmind that a high cost of investing in a fund does not guarantee better

returns.Therefore,oneshouldalsokeepthecostsofinvestinginaspecificfundinmindwhen

choosingamutualfund.Whencalculatingthereturnsofthemutualfunds,thecostshavebeen

deducted,andthereforethehigherthecosts,themoredifficultitbecomestoout‐performthe

benchmarkintheanalysis.

Theadministrationcostsdoesnotcoverallthecostsaninvestorhastopaythemutualfund,

and therefore the ÅOP (Annual total cost in percent) has been included as well. From the

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numbers it is seen that among the three, investments in Danish stocks are the cheapest,

whereasinvestmentsinglobalstocksarethemostexpensive.

It therefore implies that if one decides to invest in a category,which has higher costs, one

shouldalsoexpectthatthesewillgiveahigherreturn(Investeringsforeningsrådet,2010)

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3.TheoryThis section will cover the theory, which will be used in the analysis. It is therefore an

extensionofthetheoreticalframework,explainingdifferenttheoriesandwhytheyhavebeen

chosen. The theory will start by going into the theory of the efficient market hypothesis

followed by a discussion of the Capital Asset PricingModel. The reason of that is that the

performancemeasurementsused intheanalysis isbuilduponthosetheories,andtherefore

theylaythefoundationforthelaterexplanationofthetheoriesusedintheanalysis.

3.1Efficientmarkets

Anefficientmarket isdefinedasamarketwhereall information iscontained inthecurrent

stock price. Therefore, if a market is efficient, it would not be possible to find misvalued

stocks. The reasoning behind this is supply and demand. If a stock is overvalued, then

investorswillsellthestock,whereasinvestorswillpurchaseundervaluedstocks.Therefore,

supply and demandwill quickly adjust the price to a correct level, thereby eliminating the

opportunityofearningsuperiorprofitsoveralongperiodoftime.Inotherwordsthiswould

alsomean,thatnotevenareallyhardworkingmanwouldbeabletoearnsuperiorprofits.If

thisisthecase,thentheefficientmarkethypothesisissaidtobefulfilled(Brealeyetal.,2007).

In1970EugeneF.Famafoundtheneedtofurtherdeveloptheefficientmarkethypothesis.

Hearguedthatwhenanefficientmarketisdefinedasamarketthatfullyreflectallavailable

information, then the null hypothesis would be rather extreme, and therefore one cannot

expectittobeliterallytrue.Therefore,hedividedtheefficientmarkethypothesisintothree

subcategories,whichcouldindividuallybetested.

Weakformefficiency:Thistypeofefficiencyischaracterizedbythefactthatpricesreflectall

theinformation,whichiscontainedinthehistoricprices.Changesinthestockpricesaresaid

to follow a random walk, and therefore analysis of patterns in previous stock prices are

valuelesswhentryingtopredictfuturestockprices.

Semistrong form efficiency: This type of efficiency is similar to the weak form efficiency,

however this also includes all public available information. This means, that one simply

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cannotearnsuperiorprofitsjustbyreadingthefinancialpress,studyingfinancialstatements,

etc. In other words this implies, that in the moment new information about a company

becomespublic,thenthestockpricewillimmediatelyadjusttothat.

Strong formefficiency:The last formofefficiency is thestrong formefficiency,whereprices

includeallavailableinformation.Thismeansthatinformationsuchas“insider”informationis

alsocontainedinthestockprice,andtherefore,itisabsolutelyimpossibleforaninvestorto

earnsuperiorprofits.

The advantageof dividing the efficientmarkethypothesis into threewas, that if itwasnot

fulfilled, then one would be able to pinpoint exactly at which level of information the

hypothesisbreaksdown.

In Fama’s analysis he found that there was no important evidence against the weak and

semistrongformefficiency,andthattherewasonlylimitedevidenceagainstthestrongform

efficiency.Whenrelating this to theperformanceofmutual funds, thiswouldmeanthathis

analysispoints inthedirection, thatmutual fundsshouldnotbeabletooutperformpassive

marketindexes.Thesimplereasoningbehindthatis,thatifthestockmarketisfullyefficient,

noanalysisofanykindcouldhelppredictthefuturestockprices,therebyenablingthespecific

fundtoearnsuperiorprofitsoveralongerperiodoftime(Fama,1970).

In1991EugeneF. Famaconducteda similar analysisofmarket efficiency,however, in this

analysishenamedthethreetypesofmarketefficiencydifferently.Theywerecalled:

1.Testforreturnpredictability

2.Eventstudies

3.Testsforprivateinformation

Thesecondandthethirdcategoryhavethesamecoverageaspreviously,but justwithnew

names. Themain difference lies in the first category, which besides the forecasting power

basedonpastreturnsnowalso includethingsas forecastsbasedonvariables likedividend

yields,interestrates,seasonality,etc.

TheresultfromFama’sanalysiswas,thatheconcludedthatreturnswerepredictablebased

uponpastreturns,dividendyields,etc(Fama,1991).

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Toconcludeonthemarketefficiency,onecansaythatifyoubelieveinmarketefficiencyinit

strongest form, then your best choice would simply be to choose a passive investment

strategy. The reasonof that is, that if themarket is efficient in its strongest form, then the

stockpriceswill followarandomwalk.Therefore, thebest thingyoucandoasan investor

wouldbetominimizeyourtradingcostsandinformationseekingcosts.

However,itdoesnotseemasifmutualfundsbelieveinmarketefficiencyinitsstrongestform,

since many of them choose an active investment strategy. Hence, they must believe that

through different kinds of analysis, they would be able to outperform the market. This is

identical totheresultsFamareached inhissecondanalysisofmarketefficiencyfrom1991,

whereheconcludesthatitwaspossibletopredictfuturestockreturns.

Inrelationtothisthesis,thisindicatesthatthereisapossibilitythatthemutualfundsareable

tooutperformthemarket.

3.2CapitalAssetsPricingModel(CAPM)

CAPM was among others developed by William F. Sharpe in 1964, and together with the

efficientmarkethypothesis,thecapitalassetpricingmodelformsthetheoreticalframework

formanyfinancialanalyses.

Asthenamesuggests,itisamodel,whichisusedtodeterminethepriceofcapitalassets.

Basically,themodelisbuildupontwothings;firstly,ariskpremiumbaseduponbetaandthe

marketriskpremiumandsecondly,therisk‐freerate(Brealeyetal.,2007).

CAPMisbaseduponthesecuritymarketline(SML),shownbelow:

SML: Riskpremiumoninvestment=beta*expectedmarketriskpremium.

Thismeans that according to SML the risk premium youwould earn on our investment is

equaltotheexpectedmarketriskpremiummultipliedbythesystematicriskyouarewilling

totake.

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The CAPM model works with returns instead of risk premiums, and therefore the CAPM

modellookslikethis:

CAPM: Expectedreturnonstock=Risk‐freeinterestrate+(beta*marketriskpremium)

r=rf+β(rm–rf)

Where:

r=Returnoninvestment

rf=Riskfreerate

β=Measureofthesystematicrisk

rm=Returnfromholdingmarketportfolio

BelowtheCAPMmodelhasbeenillustrated.

When looking at the abovediagram,one can see, that if you arenotwilling to takeon any

systematicrisk,thenyouwouldearntherisk‐freerate.Thiswouldbeequivalenttoinvesting

inrisk‐freetreasurebills,sincethisisthelessriskyassetyoucanhold.

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Ontheotherhand,ifaninvestorinvestedeverythinginthemarketportfolio,thenhewould

haveabetaof1.AccordingtotheCAPMmodel,hewouldthenearnthemarketreturn.

TheessenceoftheCAPMisthatyouarerewardedbyareturnontheSMLlinedependonthe

risk(β)youarewillingtotake(Brealeyetal.,2007).

3.2.1CritiqueoftheCapitalAssetPricingModel(CAPM)

The CAPM has received a lot of critique for being based upon assumptions, which are so

theoretical,thattheyareunrealisticinapracticalsense.

TheCAPMassumptionsareasfollows(Mullins,1982)and(Jensen,1967):

1. Thesecuritymarketiscompetitiveandefficient.

2. Investorsarerationalandrisk‐averse,andthereforewanttomaximizereturnsbasedon

therisktheyaretaking.

3. Themarketisfrictionless,andthereforeitdoesnotincludetransactioncosts,taxes,and

restrictionsonborrowingandshortselling.

4. Allinvestorscanborrowandlendunlimitedamountsattheriskfreerate.

5. Investors agree on common expectations about performance and risk of all securities

(thereforeallinvestorshavethesameprobabilityofgettingcertainfuturereturns).

6. Allinvestorshavethesametimehorizonfortheirinvestments.

7. Thepriceofsecuritiesisnotaffectedbysalesandpurchasesofindividualinvestors.

8. Allinvestorshavethesameopportunitiesofinvestingindifferentstock.Inotherswords,

allinvestorshavethepossibilitytopurchaseeachandeverysecurityinthemarket.This

wouldalsorequirethesecuritiestobe infinitelydivisible,sothatsothatevery investor

hasthesameinvestmentopportunitiesindependentoftheparticularinvestor’sfortune.

When reading through theseassumptionsofCAPM,one caneasily see, that theyare rather

theoretical, and therefore, it is impossible to fully fulfil theseassumptions in reality. Just to

discuss a few, one can mention that it is doubtful whether all investors have the same

investmentopportunitiesindependentlyoffortune.Thereasonofthatis,thatsomesecurities

areveryexpensiveandnotdividable,meaningthatitrequiresaratherlargefortunetoinvest

in those securities. Thiswill limit the opportunities for the lesswealthy investors. If a less

wealthy investorwantstheopportunityof investing inallsecurities,hewouldhavetodo it

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throughamutual fund.Therebyhewouldhavetopaysomecosts for investingthroughthe

fund.

Even foran investornotusingamutual fund, it isverydoubtful thathehasno transaction

costs. The reason of that is, that for private investors there is some sort of brokerage on

literallyeverytransaction.

InspiteofthecritiqueanddoubtfulassumptionsregardingCAPM,onemightwonderwhether

themodel isworthanything inreal life.Thesimpleanswer isyes. It isnotaperfectmodel,

howeveritisveryuseful.

In1982DavidW.Mullins,Jr.conductedananalysisofwhethertheCAPMworks.Theresults

fromhisanalysiswerethatCAPMwasnotperfect,however itwasaveryusefuladditionto

otheranalytical toolkits.He furtherargued thateven though theassumptionsofCAPMare

very theoretical and unrealistic, it is often necessary to simplify reality in that manner in

ordertodevelopusefulmodels.Inadditionhearguesthattheunderlyingassumptionsofthe

modelarenotnecessarilythemostimportantthings,andthatimportanceshouldbeattached

tothevalidityandusefulnessofthemodel’sprescription.

ResultsfromdifferentanalysesofCAPMhaveshownthatasameasureofrisk,betaseemsto

be related to past returns. In addition, the relationship between past returns and beta has

shown to be linear, and that the relationship between the two is positively sloped. Finally

research has shown, that CAPM overestimates returns for low beta securities and

underestimatereturns forhighbetasecurities.Thiswill cause that theempiricalSML tobe

lesssteeplyslopedthanthetheoreticalSML(Mullins,1982).

To briefly sumup on the empirical tests of CAPM, one can say that they do not absolutely

validatethemodel,howevertheysupportsomeofthemainimplicationsofCAPM.Firstofall,

theysaythatthesystematicrisk(beta)appearstoberelatedtopastreturns,secondly, that

thereisapositiverelationsbetweenriskandreturn,andfinally,thattherelationshipbetween

riskandreturnappearstobelinear.

Theperformancemeasurements thatwill beused in this thesis arebaseduponCAPM, and

thereforethevalidityofCAPMofimportanceforthisthesis.

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3.3Riskmeasurements

Infinanceyouoftendistinguishbetweentwodifferenttypesofrisk,andthereforethesewill

brieflybeexplainedinordertoavoidconfusion.Theperformancemeasurementsusedinthis

analysis only focuses on the systematic risk, and therefore thedistinctionbetween them is

important.

3.3.1Systematicrisk

Thesystematicriskistherisk,whichliesinthemarket.Therefore,thisriskcannotbeavoided

regardless of how diversified your portfolio is. In other words it can be said that the

systematicriskisamacroeconomicrisk,simplybecausethatitisariskwhichtosomeextent

will affect the entire market. An example of this systematic risk, which affects the entire

market, could be the latest financial crisis, or simply just fluctuations of the market. The

degreetohowmuchasecurityisaffectedbythesystematicriskdependsonitsbeta(whichis

thenotationforsystematicrisk)(Brealey,2007).

Securitieswithlowbetaswillbelessaffectedbymacroeconomicchangesthansecuritieswith

highbetas.

Inotherwords,thebetaisasimplemeasurementoftheinvestmentsriskcomparedtotherisk

ofthemarket.Betaiscalculatedinthefollowingway:

€

β j =cov( ˜ R J , ˜ R M )σ 2 ˜ R M

(Jensen,1967)

Where

€

cov( ˜ R J , ˜ R M ) isthecovariancebetweenthereturnsoftheinvestmentandthereturnsof

thebenchmark,and

€

σ 2 ˜ R M isthevarianceofthereturnsofthebenchmark.

In terms of this thesis, betawill be found using regression analysis. In regression analysis,

beta is the slope parameter of the regression line. In more statistical terms, the slope

parameter(beta)isfoundusingthefollowingformula:

Page15

€

ˆ β =(xi − x

i=1

n

∑ )(yi − y )

(xi − x )2

i=1

n

∑ (Wooldridge,2008)

Whenthinkingaboutthecalculationmethodsofbetashownabove,onewillalsorealisethat

theyarebuilduponpastreturns.Whetherthisisvalidornotcanbediscussed,howeverIwill

juststicktotheconclusionreachedbyDavidW.Mullins,Jr.Thiswasdiscussedintheprevious

section,andtheconclusionwasthatbetaseemedtoberelatedtopastreturns.

3.3.2Unsystematicrisk

Where the systematic risk couldbeexplainedas themacroeconomic risk, theunsystematic

riskcanbeexplainedasamicroeconomicrisk.

Thismeansthattheunsystematicriskisarisk,whichisparticulartoacertaincompanyand

perhaps its closest competitors. Therefore, the unsystematic risk is a risk, which can be

avoidedbyholdingadiversifiedportfolio.Thefirstreasonofthatis,thatifyouhavealarge

portfolio of stock, the total return of your portfolio will only be slightly affected by

fluctuationsinthereturnofasinglestock.Inotherwords,onestockonlyhasalittleeffecton

theentireportfolio.Itisalsorequiredthatyourportfolioshouldbediversified,meaningthat

youholdsecurities inmanydifferent industries, countries, etc. If youwant toeliminate the

unsystematic risk of your portfolio you are not necessarily looking for securities with the

lowestunsystematicrisk.Theimportantthinginordertoeliminatesystematicriskisthatyou

hold securities in your portfolio that has unsystematic risk that outweighs each other

(Brealey,2007).Anexampleofthiscouldbetwostockswhosestockpricesarehighlyaffected

bythegasolineprices.Ifoneofthesecompanieshighlybenefitfromhighgasolineprices,and

theotherloosesonhighgasolineprices,thentheymustindividuallyhavehighunsystematic

risk,howeverbyholdingbothof thestocks inyourportfolio,mostof theunsystematicrisk

willoutweigheachother.

Thediagrambelowshowstherelationshipbetweensystematicriskandunsystematicriskina

portfolio.

Page16

Source:(Brealey,2007)

Itisimportanttokeepinmindthatmostfinancialtoolsonlyworkwithsystematicrisk.The

reason of that simply is that unsystematic risk can be avoided by holding a diversified

portfolio,andthereforecleverinvestorswillhaveasystematicriskcloseto0.Intermsofthis

thesis, itwillbeassumed that themutual fundsholddiversifiedportfolios,whicheliminate

unsystematicrisk.Therefore,thisassignmentwillonlydealwiththesystematicrisk.

3.4Performancemeasurements

When analysing the performance of Danish mutual funds, there are several performance

measurements to choose between. The Federation of Danish Investment Associates

(Investeringsforeningsrådet) has since the year 2000used the Sharpe ratio to evaluate the

performanceoftheDanishmutualfunds.

The Sharpe ratio is a risk adjustedperformancemeasurement basedon the capitalmarket

line (CML). Thus, the Sharpe ratio uses the total risk of the portfolio instead of only the

systematicrisk(Christensen,2003).

TheSharperatioiscalculatedusingthefollowingformula:

Sharpe=(returnonportfolio–riskfreerate)/standarddeviationonportfolioreturn

Page17

As discussed previously, it must be assumed that mutual funds hold diversified portfolio,

which eliminates the unsystematic risk, and therefore the Sharpe ratio is not the best

performancemeasurementinthiscase.

Instead, thisanalysiswilluse the Jensen index/Jensen’sAlpha tocalculate therisk‐adjusted

returnsofthemutualfunds,andthereareseveralreasonsofthat.

Firstofall,theinterpretationofJensen’salphaiseasier,asitshowsthepercentageexcessrisk

premiumthatamutualfundhasearnedcomparedtoitsbenchmark.HenceaJensenindexof

0,15%simplyjustmeansthatthemutualfundhasearnedanexcessreturnof0,15%pertime

unitcomparedtoitsbenchmark.Ontheotherhand,theSharperatiogivesyouriskpremium

per risk unit, and therefore it can be difficult for private investors to interpret whether a

Sharperatioof0,5isgoodorbad.

Secondly,Jensen’salphaiscalculatedusingregressionanalysis,meaningthatitcandirectlybe

seen from the output whether the results are statistically significant or not. If we again

compare to theSharperatio,wecannotdirectly interpretwhetheraSharperatioof0,25 is

significantlybetterthanoneof0,3withoutdoingadditionalcalculations.

Thirdly, Jensen’s alphamakesa relative comparison to abenchmarkwithin the calculation.

UsingtheSharperatio,onewouldhavetocalculatetheSharperatioforthebenchmark,and

thencompareittothatofthemutualfund.

Finally,Jensen’salphacanbeusedonbothefficientandinefficientportfolios,sinceitisbased

onthesecuritymarketline.Inotherwords,itcanbeusedforportfoliosthatlieonthesecurity

market line(theefficientones)andforportfoliosthatdonot lieonthesecuritymarket line

(theinefficientones).IfweagaincomparetotheSharperatio,thenthisisbasedonthecapital

marketline,andthereforeitcanonlybeusedforefficientportfolios(Christensen,2003).

In the analysis by Christensen (2003), he further concludes that the tradition in academic

literatureistouseJensen’salphaandthatmorethan100Americanresearchstudiesarebuild

uponthismethod.

Basedon theabove‐mentionedarguments, Jensen’sAlphawillbeused for theperformance

evaluationinthisthesis.

Page18

3.4.1Jensen’sAlpha

Jensen’s alphaor Jensen’s index is a riskadjustedperformancemeasurementdevelopedby

MichaelC.Jensenin1967.

ThemodelisbuilduponCAPM,andthereforeitassumesthatportfoliosarewelldiversified,

so that unsystematic risk is eliminated. Since the model is based upon CAPM, the same

assumptionsapplyforJensen’salphaasforCAPM(Jensen,1967).

Jensen’s alpha is calculated as the constant in the following regressionmodel (Christensen,

2003):

€

rpt − rft =α + β(rmt − rft ) + et

Isolatingalpha,weobtain:

€

α = (rpt − rft ) − β(rmt − rft ) + et

Where:

€

rpt :Returnoftheportfolioofthemutualfundinperiodt

€

rmt :Returnofthebenchmarkinperiodt

€

rft :Riskfreerateinperiodt

€

β :Estimateofthesystematicriskofthemutualfund

€

et :Errorterminperiodt

€

α :Performancemeasurementforthemutualfund.

In more technical terms, Jensen’s alpha is estimated using regression analysis in Eviews,

holdingtheriskpremiumofthemutualfundasthedependentvariableandtheriskpremium

of the benchmark as the independent variable. Jensen’s alpha shows up as the constant or

interceptinthemodel.

IftheestimateofJensen’salphashowsuptobepositiveandsignificantlydifferentfrom0,itis

concluded that the mutual fund has outperformed the benchmark. Oppositely, if it is

significantly negative, the benchmark has outperformed themutual fund. If Jensen’s alpha

shows up to be insignificant, it is concluded that themutual fund has performed neutrally

comparedtotheirbenchmark.

Instatisticalterms,thehypothesesthatarebeingtestedlookslikethis:

Page19

€

H 0 :α = 0H1 :α ≠ 0

WhentestingthesignificanceofJensen’salpha,thep‐valueofthecoefficientwillbeused.The

significanceofJensen’salphawillbothbeevaluatedata5%anda10%significancelevel. If

thep‐valueisbelow10%,thenJensen’salphaissignificantata10%level,andif it isbelow

5%itissignificantata5%level.

AgraphicalillustrationofJensen’salphaisshownbelow:

The above diagram shows that Jensen’s alpha is the vertical distance from the SML to the

returnofthefund.(Pleasenotethatthelineforthemutualfundscouldhaveadifferentslope

than the SML line, and therefore the distance from SML to the return of themutual funds

shouldbemeasuredintheinterceptofthey‐axis).

3.4.2CritiqueofJensen’salpha

Asmanyother financialmodels, Jensen’s alphahas received somecriticism.Firstof all, the

modelisbuilduponCAPM,andthereforeithasreceivedthesamecriticismasCAPM,which

haspreviouslybeendiscussed.Themainpointof thatwas thatCAPMsimplifies realities in

suchamanorthatsomearguethattheassumptionsareimpossibletofulfilinreality.

Page20

Roll (1977) further discusses the CAPM and the implications it has on performance

measurementsusingmodelsbuilduponCAPM.Mostofhisargumentationisconnectedtothe

choiceofbenchmark.

Firstofall,hearguesthatthechoiceofbenchmarkaffectsthesizeofthebetafortheportfolio.

In otherwords, thismeans that aportfoliowill havedifferent valuesof betadependenton

which benchmark is used as proxy for themarket portfolio. The reason of that is, that the

varianceofthemarkedportfolioisusedinthecalculationofthebetaofamutualfund.This

also implies that two investorswithexactly thesameriskprofilecanhavedifferentbetas if

theyusedifferentproxiesforthemarkedportfolio.

Thisthereforeimpliesthatbetaisnotanunambiguousmeasurementofthesystematicriskof

theportfolio.

Roll(1977)alsodiscussesthattherearetwodifficultieswhenusingaproxyforthemarked

portfolio.Firstofall, theproxycouldbeefficienteventhoughthe truemarket is inefficient,

therebyshowingan incorrectpictureof the truemarket.Secondly,heargues that it isvery

likely that there will be a high correlation between reasonable proxies for the marked

portfolio, and therefore it could seem as if the choice of either one of them is of less

importance.However, this is not the case according tohis analysis, becausehe argues that

eventhoughtheproxieshavehighcorrelation,theycanresultinquitedifferentinferencesin

theactualperformanceanalysis.

Theconclusiontothismustthereforebethatthechoiceofbenchmarkisofcrucialimportance

fortheresultsoftheperformanceanalysis.

TheimportanceofcorrectuseofbenchmarkswasshownbyIppolito(1989).Inthisanalysis,

Ippolito reached the opposite conclusion of most other analyses, namely that the mutual

funds had been able to outperform themarket. However, Elton et al. (1993) reviewed the

analysis, and reached the conclusion that Ippolito’s resultswere causedby incorrectuseof

benchmarks. Itshowedup,thatmanyofthemutual fundsIppolitohadanalysedinvestedin

“small” American stocks,whichwere not included in the S&P 500‐indexwhich he used as

benchmark.The“small”stockshadperformedbetterthantheS&P500‐indexintheperiodof

analysis,andthereforetheanalysisbyIppolitoshowedsuperiorperformancebythemutual

Page21

funds. However Elton et al. (1993) redid the analysis using appropriate benchmarks and

reached the same conclusionasother analysis, namely that themutual fundshadnotbeen

abletooutperformthemarket.

The example above shows, that reasonable use of benchmarks is crucial in order to reach

valid conclusions in the analysis. Therefore one should act carefully when choosing the

benchmarksforananalysis.

Inthediscussionofmarketefficiency,Roll(1977)alsodiscussesthatiftheproxyusedforthe

market portfolio is efficient, then none of the estimated Jensen’s alpha can be significantly

positive. The simple argumentation behind this is that if the marked is efficient, then all

informationisreflectedinthestockprice.Thus,itwouldbeimpossibleforamutualfundto

outperformtheindex,andthereforetheyshouldshowuptohaveinsignificantJensen’salpha.

As previously discussed, Fama (1991) showed that the stockmarkets where not perfectly

efficient,sothereforeitshould,atleastaccordingtoFama,bepossibleforthemutualfundsto

outperformthebenchmarks. Inrelationto this,Roll (1977)discusses that if theproxyused

forthemarketportfolioisnotefficient,thenhowcanitbejustifiedtouseitasabenchmarkin

aperformanceanalysis,whentheperformanceanalysis isbuilduponCAPMwhichrequires

efficientmarkets.

Itshouldfinallybenoted,thatthemulti‐indexmodelisbasedontheexactsametheoryasthe

Jensenindex,andthereforethecritiqueappliesforthemulti‐indexmodelaswell.

3.4.3Multi‐indexmodel

OnedrawbackofusingtheJensenindex,asdescribedabove,isthatitisasingleindexmodel.

It therefore requires that the investment objective of the mutual funds is well‐defined,

meaningthatitcanbeexplainedbasedononlyonebenchmark.

DanishlegislationrequiresthatDanishmutualfundsinvestatleast75%oftheirinvestment

withintheirmaininvestmentcategory.Thismeans,thattheDanishmutualfundsareallowed

toinvestupto25%ofitsvalueinanyotherinvestment.These25%canbeinvestedinboth

stocksaswellasbonds,howevermostmutualfundschoosetoinvestsolelyineitherstocksor

bonds.Thereasonofthatis,thatiftheychooseacombination,thefundwillbeclassifiedasa

mixedfund,whichchangesthetaxationofthefund(Christensen,2003).

Page22

The 25%‐rule could indicate that the performance of Danish mutual funds cannot be

evaluatedbasedononebenchmarkonly,andthereforeamulti‐indexmodelwillbeincluded

inordertoseewhetherthisisbetteratexplainingthereturnsofthemutualfunds.

Themulti‐indexmodelwillboth includestock indexesaswellasbonds indexes inorder to

coverallthepossibilitiesthemutualfundshave.

Differentmulti‐indexmodelswillbeuseddependentonwhatthemaininvestmentcategoryof

the investment funds. Therefore, the specificmodelswill be show in the section “choice of

benchmark”.

The purpose of using a multi‐index model is to increase the amount of variation in the

dependent variables explainedby independent variables in themodel. In order to evaluate

uponwhetherithasthedesiredeffect,oneshouldlookatadj.R2.Iftheadj.R2increaseswhen

includingotherbenchmarks,thismeans,thatthenewbenchmarkscapturesomething,which

thesinglebenchmarkwasnotabletocapturebefore.Thereasonofusingtheadj.R2insteadof

just the regular R2 simply is that the adj. R2 allows for comparisons betweenmodelswith

different numbers of independent variables. Adj. R2 has been corrected to allow this

comparison.

However, it is a requirement that the other benchmarks included in themulti‐indexmodel

havebeta‐coefficientsthataresignificantlydifferentfromzero.Ifthisisnotthecase,thenit

means that theyare insignificant, and therefore theydonothavea significanteffecton the

model,andshouldberemovedfromthemodel.

Thus,thereisachancethatthemulti‐indexmodelwillshowupwithexactlythesameresult

as thesingle index‐model.Thiswouldoccur ifall theadditionalbenchmarksshowup tobe

insignificant. If benchmarks show up to be insignificant, they should be removed from the

model onebyone, startingwith theonewith thehighestp‐value.Each timea variablehas

beenremoved,anewregressionmodelshouldbeestimated,andexactlythesameevaluation

shouldtakeplaceuntilthemodelconsistsofonlysignificantvariables.

Themulti‐indexmodelwillbeestimatedusingregressionanalysis,similarlytotheestimation

ofJensen’salphainthesingleindexmodel.Itwillbedonebyregressingtheriskpremiumof

the fundas thedependentvariableholdingtheriskpremiumof thedifferentbenchmarkas

theindependentvariables.

Page23

3.4.5Markettiming

TheJensenindexandthemulti‐indexmodelisonlyconcernedwiththeselectionabilityofthe

mutual funds to earn superior return. However, superior returns can also be obtained by

timingthemarket,andthereforeTreynor&Mazuy(1966)developedamodel,whichcanbe

used toanalyseboth theselectionand timingabilitiesofamutual fund. Inotherwords the

performanceanalysisisdividedintotwopieces,oneanalysingthemutualfundsabilitytofind

undervalued stocks (selectivity) and one analysing the mutual funds ability to predict the

directionthemarketwillbemovingin(timing).Thiswouldallowonetoseespecificallywhich

ofthesetwothingscausesagivenperformancebyaninvestmentfund.

Themodelwritesasfollows:

€

rpt − rft =α + β(rmt − rft ) + γ(rmt − rft )2 + et

Themodel is similar to the Jensen index, however in this themodel by Treynor &Mazuy

(1966)aquadratictermhasbeenincluded.

Inthemodel,αrepresentstheselectionabilityofthefund,whereasγrepresentsthetiming

abilityofthefund.TreynorandMazuy(1966)arguethatifγshowsupsignificantandpositive

intheregressionmodel,thisindicatesthatthemutualfundpossessedtheabilitytotimethe

market.

Thereasoningbehindthemodelisasfollows:

Ifmutual funds posses the ability to time themarket, they should adjust the beta of their

portfolio upwardswhen themarket is rising,whereas they should adjust the beta of their

portfolio downwards when the market is falling. The thoughts behind this is that mutual

funds should maximize their returns by having high betas when the market is moving

upwards,whereastheyshouldminimizetheirloosesbyhavinglowbetaswhenthemarketis

moving downwards. This would therefore also require the mutual funds to adjust their

systematicrisk,correspondingtotheirexpectationsaboutthemarket.

TreynorandMazuy(1966)usedthefollowingthreefigurestoexplainthedifferentscenarios

in the market timing analysis. The models are based on a “characteristic line”, which is

Page24

obtainedbyplottingthereturnofthemarketonthex‐axis,holdingthereturnofthemutual

fundonthey‐axis.

Thediagrambelowillustratesamutual fundholding itsbetaconstant, therebynotworking

withmarkettimingbyadjustingbetatotheexpectationsofthemarket.

Thenextdiagramillustratesthemutualfundthatperfectlytimesthemarketineachsituation.

Thismutualfund(blackline)willholdaportfoliowithahighbetawhenthemarketyieldsa

highreturn,whereasitwouldholdaportfoliowithalowbetawhenthereturnofthemarket

isbad.Notethattheredlineisfortheinvestmentfund,whichhastimedthemarketincorrect,

byhavinglowbetainincreasingperiods‐,andhighbetaindecreasingperiods.

Page25

However, the illustration above is rather unrealistic, since it requires that themutual fund

perfectly times fluctuations in the market in each and every situation. Therefore, a more

realistic situation would be that the fund is able to time the market more often than the

average.Thiswouldcausetheillustrationtolooklikethis:

Inthesituationwherethemutualfundisbettertothetimethemarketthantheaverage,the

characteristiclinewouldhaveaconvexshapecausedbythesquaredterminthemodel.This

wouldimplythatthemutualfundgraduallyadjuststhebetaoftheportfoliosothatitmatches

theexpectationsofthemarket.Thelineshowsthatthegradualadjustmentcausestheslopeof

thelinetobecomesteeperandsteeper,thehigherthereturnofthemarketis.

Page26

Basedonthemodelabove,italsomakessensethatthequadratictermisusedtocapturethe

effectofmarkettiming.

Wooldridge(2008)arguesabout theeffectsofusingaquadratic terminamodel.Themain

point is that a quadratic term can be used to capture decreasing or increasing marginal

effects, and in this case, market timing of the mutual funds must be characterised as an

increasingmarginaleffect.

Inotherwordsthismeans,thattheregularregressionmodelassumeslinearity,andtherefore,

ifitshowsupthatthemodelisnotlinear,thelinearregressionwillnotbeabletoexplainthat.

This iswhere thequadratic termcomes inhandy,because it allows themodel to takeona

shape, which is not linear, and thereby the model with the quadratic term might help in

explainingsomething,whichtheregularmodelwasnotabletoexplain.

In termsofmarket timing for themutual funds, thiswouldalso imply, that if thequadratic

termshowsup tobepositiveand significant, theremustbeamarginal effect in themodel,

whichmeansthatthemutualfundhasadjustedthebetaoftheportfoliocorrespondingtothe

developmentinthemarket.

TheestimationoftheregressionmodelwillbedoneinEviews,usingthesamedatasetasfor

the Jensen index. However, for the estimation of the model, the Newey West

heteroskedasticityandautocorrelationconsistent (HAC)standarderrorswillbeusedonall

regressions,followingChristensen(2003),wherehearguesthatthequadratictermimposesa

heteroscedasticitytypeofproblemintothemodel,andthereforeit isparticularlyimportant

tousetheheteroscedasticityconsistentstandarderrors.

Thus, the only assumption that will be discussed for this model is the assumption about

normalityintheerrors.

Grant(1977)arguedthatthepresenceofmarkettimingwouldcausetheestimateofJensen’s

alpha to be downward biased, meaning that the estimate of Jensen’s alpha would be

underestimatedcomparedtotheactualperformanceofthemutualfund.

In other words, this means that in the presence of market timing, the value of α will be

underestimatedcomparedtotheJensenregressionbecauseJensenalphadoesnotaccountfor

themutual funds’abilitiestotimethemarket.Thecharacteristic linewouldbe linear inthe

Page27

Jensenindex,whereasinthepresenceofmarkettimingitwouldbeconcavelyshapedinthe

TreynorandMazuyregression,therebypushingtheinterceptfurtherdowntheY‐axisinthe

TreynorandMazuyregressioncomparedtotheJensenindex.

Finally, itshouldbenoted,thatit is likelythattheestimateofαwillchangewhenincluding

the quadratic term in the model, independently of whether the quadratic term shows up

significantornot.Itshouldthereforebementioned,thataccordingtoregularregressionrules,

thequadratic termshouldbe removed fromthemodel, if it showsup insignificant, thereby

leavingbehindtheJensenregression.

Thereasonofthatis,thatifavariableshowsupinsignificant,itdoesnothaveanysignificant

effectonthemodel.Therefore, itwouldonlycausedisturbanceoftheothervariablesinthe

model,ifthequadratictermisleftinthemodel.

Page28

4.DataThechoiceofdataisofgreatimportancefortheanalysis.Ifwrongchoicesaremade,itisvery

likely that the analysis will be biased, and therefore the results and conclusions will be

misleading.Thus,thissectionwillcoverthethoughtsbehindthechoicesthathavebeenmade

whenchoosingthedatafortheanalysis.Inaddition,certainassumptionsarerequiredtobe

fulfilled in order for the results of the analysis to be valid. Therefore, this sectionwill also

coveratheoreticaldiscussionoftheassumptionsandhowtheywillbetestedintheanalysis.

4.1Choiceofmutualfunds

IthasbeendecidedthatthisanalysisshouldcoverDanishmutualfundsinvestinginstocksin

thefollowingthreeareas:

1. Danishstocks

2. Europeanstocks

3. Globalstocks

Thesethreeareasofinvestmenthavebeenchosen,sincetheyarethemostcommonareasof

investmentfortheDanishmutualfunds.Inotherwords,thesethreecategoriesaretheones

thatcangatherthelargestsamplesizes.

Whenselectingthespecificfundsfortheanalysis,certaincriteriahastobemet:

• Themutualfundshouldexistfortheentireperiodfrom2001‐2010

• Themutualfundshouldonlyinvestinstocks

• Themutualfundshouldfitintooneofthethreecategories(inordertodecide

upon that, the categorization made by IFR has been used, as well as

descriptionsonthehomepagesoftheindividualfunds).

Intotal,60mutualfundshavebeenabletomeettherequirementsforthisanalysis.Theyare

dividedintothethreecategoriesasfollows:

1. Danishstocks(16mutualfunds)

2. Europeanstocks(21mutualfunds)

3. Globalstocks(23mutualfunds)

Page29

Thereturnsof themutual fundsaredeterminedas logreturnsusingmonthlyobservations,

whichtotallyamountsto120observationsforeachfund.

4.2Choiceofbenchmarks

The purpose of a benchmark is to show the general development in specific area of

investment. A benchmark should therefore be an expression of “best practice”. In other

words,thismeansthatthebenchmarkyouchooseshouldbeconsistentwithyourinvestment

strategy.This is themost important criterion for choosingabenchmark,however thingsas

investmenthorizon,restrictions,etc.,shouldalsobetakenintoconsideration.Inotherwords,

thismeans that the perfect benchmark should perfectlymatch your investment criteria. In

realityit isoftendifficulttofindabenchmarkthatperfectlymatchesyourinvestmentonall

criteria.Therefore,apublicindexisoftenchosenasbenchmark,becauseitiseasytoaccess,

stilloneshould thoughkeep inmindthat thiswillonlybeanapproximation to theoptimal

benchmark(Christensen,2001)

Whenchoosingabenchmarktherearecertainpitfallsoneshouldbeawareof.

Firstofall,itisimportantthatyouhaveconsistencybetweenwhetherdividendsareincluded

in your calculations or not. The reason of that is that when calculating the returns of the

mutualfundsyoucorrectfordividend.Ifyouthenchooseabenchmarkwheredividendshave

been excluded, your investment portfolio will already have a head start of 2‐3%‐point

(Christensen,2001).

Somemutual fundsstill chooseabenchmarkwheredividendsarenot included.Thereason

fordoingsoisthatitbecomeseasierforthemutualfundtooutperformtheirbenchmark.One

shouldkeepinmind,thatwhencalculatingthereturnofthemutualfund,costsofrunningthe

mutualfundhasbeendeductedfromthereturn,andthereforeitisdifficultforamutualfund

tooutperformabenchmark,whichiscost less.Stilloneshouldbeawareofthefactthatthe

correctcomparisonistoincludedividendsinboththecalculationofthereturnofthefund,as

wellasforthereturnofthebenchmark(Christensen,2001).

Whencalculatingthereturnsofthemutualfundsforthisanalysis,correctionsfordividends

havebeenmade.Therefore,thebenchmarksfortheanalysisshouldincludedividendsaswell.

Page30

Therearedifferentsuppliersofthesebenchmarks,andoneofthemajoronesisMSCIBarra,

which is a company owned by Morgan Stanley. They supply many different relevant

benchmarks,whicharecommonlyusedamongtheDanishmutualfunds.However,beforeone

isabletochooseappropriatebenchmarks,oneshouldbefamiliarwiththemethodologyused

intheindexes.

4.2.1Indexmethodology

Inordertochoosethemostappropriatebenchmark,oneshouldbeawareofthemethodology

usedintheindexes.Therefore,thissectionwillgiveabriefdescriptionofthat.

TheNASDAQOMXgrouppublishesindexes,whicharerelevanttothisanalysis,andtherefore

abriefdescriptionoftheirmethodologywillbegivenhere.

Firstofall,theyhavethreedifferent“measurement”typesusedfortheindexes,whichcanbe

combinedinanyway.

Grossindex(GI):Asthenamesuggest,thisisagrossindex,andthereforedividendshavebeen

includedintheindex.Theindexassumesthatthedividendsarereinvestedthedayafterthey

havebeenpaidout.

Price index (PI): This index is a price index, and therefore the index does not include

dividends. The index only focuses on movements in the stock prices, and therefore the

differencebetweenthegrossindexandthepriceindexsimplyisattributabletothedividends.

Cappedindex(Cap):Acappedindexmeansthattheindexhasbeencorrected,sotheweightof

asinglestockhasanupperlimitintheindex.Incaseastockexceedsthemaximumweight,the

stockisweightedbythemaximumweight.

Toconcludeontheabovedescriptions,itisfirstofalldesiredthatdividendsareincludedin

thebenchmarks,basedonthepreviousdiscussion.Therefore,thechosenbenchmarkshould

beagrossindex(GI).Inadditionitisalsodesiredthattheindexiscapped,sothatnosingle

stockhastolargeaneffect.Thisisinaccordancewiththelegislationregardingtheinvestment

funds,whichrequiresthatasinglestockcannotwaymorethan5%intheportfolio.Therefore,

acappedindexismorelikelytoshow“bestpractice”.

Page31

Themainconclusionisthereforethatthechosenbenchmarksshouldbeacappedgrossindex.

Onefinalthingthatwouldbedesirableisthattheindexisadjustedforfreeflow.Afreeflow

adjustment means that index has been corrected, so that only the share capital, which is

availableonthemarket,isincludedintheindex.

4.2.2BenchmarkfortheJensenindex

This section will cover a discussion of why certain benchmarks have been chosen. When

choosingthebenchmarkforthemutualfundsinvestinginDanishstocks,severalothergood

alternativesexist.Therefore, adiscussionof thealternativebenchmarkwill be included for

theDanishstocks.However, fortheotherinvestmentcategoriesthemaindiscussionwillbe

regardingthebenchmark,whichhasbeenchosen.

The discussion of the benchmarks will be based upon what the different funds use as

benchmarks.Appendix4showsanoverviewofthebenchmarksusedbythedifferentmutual

funds.

Thedescriptionof thedifferent indexes isbasedonexplanationsfoundonthehomepageof

thesuppliersoftherespectiveindexes.

Danishstocks

Choice:

OMXCopenhagenCap_GI

This index is a total index, which includes all of the stocks that are registered on the

Copenhagen Stock Exchange, and therefore the purpose of the index is to show the actual

conditionaswellaschangesofthemarket.Theindexisbothagrossindexaswellasacapped

indexhowever it hasnot been adjusted for free flow.Despite this fact, this indexhasbeen

chosenasbestpracticefollowingthediscussionofthealternativesbelow.

ThedatafortheindexhasbeencollectedfromCopenhagenStockExchange,andthereforethe

indexismeasuredinDKK.

Alternatives:

OMXCopenhagen20

This index includes the 20 most traded stocks on the Copenhagen Stock Exchange. The

advantage of this index is that the stocks that are included are the most liquid stocks.

Page32

However,itonlyincludes20differentstocks(TheOMXC20stocks),andthereforeithasbeen

concluded that this is not enough to cover the possible investments made by the mutual

funds.Inaddition,theindexisapriceindex,meaningthatdividendshavenotbeenincluded

intheindex.Finally,theindexisneithercapped,meaningthatcompanieslikeNovoNordisk

andA.P.MøllerMærskwillhavealargeweightintheindex.Basedonthis,theindexhasbeen

foundinappropriatefortheanalysis.

OMXCopenhagenBenchmarkCap_GI

This index includes between 50 and 80 of the largest and most frequently traded stocks

representingmost of the different sectors on theDanish stockmarket. The index is both a

cappedindexaswellasagrossindex,andinadditionithasalsobeenadjustedforfreeflow.

Therefore, this would be themost appropriate benchmark to use for the analysis, since it

fulfilsalltherequirementspreviouslystated.However,itonlydatesbacktoOctober242005,

andthereforeitcannotbeusedforthisanalysis.Finally,itshouldbenoted,thatmanyofthe

mutualfundsincludedintheanalysisusesthisindexasbenchmarkontheirportfolios.

MSCIDenmark

ThefinalalternativecouldbetousetheMSCIDenmarkindex.Theadvantageofthisindexis

thatitcanbechosenasbothcapped,grossindexandfreeflowadjusted.Therefore,theindex

couldfulfiltherequirements,whichwerepreviouslystatedforthebenchmarks.However,the

disadvantageisthattheindexismeasureinUSD,sothereforeitshouldbetransformedinto

DKKusing the correspondingexchange rates foreachof theobservations in thedataset. In

addition,noneofthechosenmutualfundsusethisindexasabenchmark,whereasthechosen

oneshowsupasmuchmorefrequent.

However, this index is probably the best possible alternative to the chosen index, and

thereforeitisalsoverylikelythattheywillhavepositivecorrelationscloseto1.

Europeanstocks

Again,therearedifferentbenchmarkstochooseamongforthisinvestmentcategory,however

themain concern for this analysis is theMSCIBarra indexes.Here thereare three relevant

onestochoosebetween.

Page33

Name of Benchmark Markets included (countries) MSCI Europe Developed countries (DM) MSCI EM Europe Emerging countries (EM) MSCI AC Europe All countries

ThechoiceofbenchmarkfallsonthestandardMSCIEuropeindex,whichonlyincludesthe16

developedcountriesinEurope.Thereasonforchoosingthisspecificbenchmarkissimple.It

mustbeassumedthatthisbenchmarkmatchestheinvestmentobjectiveofmostofthemutual

funds,becauseitisusedby18outof21mutualfundsintheEuropesample.Secondly,italso

follows the trendwith the literature, e.g. (Christensen, 2003).Therefore, this benchmark is

foundtobeappropriatefortheanalysis.

Globalstocks

AsforthechoiceofbenchmarkfortheEuropeanstock,thebenchmarkfortheglobalstocks

willbechosenamongtheMSCIBarraindexes.Thesametypesofindexesexistfortheglobal

stocksasfortheEuropeanstocks.Theyareshownbelow:

Name of Benchmark Markets included (countries) MSCI World Developed countries (DM) MSCI EM World Emerging countries (EM) MSCI AC World All countries

The choiceofbenchmark for theglobal stocks is abitmoredifficult than for theEuropean

stocks. The reason of that is that 9 out of 23mutual funds in the sample use theMSCIAC

World index,whereas the remaining 14 use theMSCIWorld index as benchmark. For this

analysis it has been decided to follow themajority, and therefore choose the MSCIWorld

indexasbenchmarkfortheanalysis,whichconsistsof24developedcountriesspreadaround

theentireworld.Thisalsocorrespondstothechoicemadeby(Christensen,2003)

Conclusion

It should be noted that the MSCI indexes fulfil the requirements for an appropriate

benchmark.Firstofall,theyarecappedandfreeflowadjustedandsecondlytheycanchosen

as gross indexes, so that dividends are included in the indexes. However, the indexes are

measured in USD, so therefore they are converted into DKK in order to avoid currency

deviationsaffectingtheresults.

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TheMSCI indexesareobtained fromMSCIBarra,andthechosen indexesaregross indexes,

followingthediscussionfromprevious.

Onefinalthingtonoteisthatthesamebenchmarkhasbeenchosenforallthemutualfundsin

eachcategory.However, itcanbeabitmisleading,sincesomemutual fundsholdportfolios

only investing in small cap stocks. Therefore, it could be discussed that it would bemore

appropriatetouseabenchmarkthattakesthisintoaccountforthesespecificfunds.Though,

the purpose of the analysis is to divide the mutual funds into these three investment

categories,andseeifanyofthemareabletooutperformthepassivebenchmarks.However,if

mutual funds show up to have significant Jensen’s index, an evaluation about appropriate

benchmarkwillbedone,inordertoavoidmakingthesamemistakeasIppolito(1989).

Thebelowboxconcludesonthechosenbenchmarks:

Investment category Benchmark - Danish stocks OMX Copenhagen Cap_GI - European stocks MSCI Europe - Global stocks MSCI World

ThethreeregressionmodelsfortheJensenindexthereforeareasfollows:

Danishstocks:

€

rpt − rft =α + βOMXCCap _GI (rOMXCCap _GI − rft ) + et

Europeanstocks:

€

rpt − rft =α + βMSCI Europe (rMSCI Europe − rft ) + et

Globalstocks:

€

rpt − rft =α + βMSCIWorld (rMSCIWorld − rft ) + et

4.2.3Benchmarkforthemulti‐indexmodel

Aspreviouslydiscussed,themutualfundshavethepossibilitytoinvestupto25%inanything

offreechoice.Therefore,thepurposeofthemulti‐indexmodelmustbetocoversomeofthat

free choice. In addition, the benchmarks chosen for the Jensen index might not cover

everything in themain investmentcategoryof themutual funds.Anexampleof thatwasas

previouslydiscussed,themutualfundsinvestinginsmallcapstocks.Themulti‐indexmodel

willthereforealsotrytocoverthat.

Themulti‐indexmodelwillbeacombinationofstockindexesandbondindexes.Theindexes

chosenforeachofthethreeinvestmentcategoriesarenotthesame,andthereforetheywill

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bediscussedindividually.However,thebondindexeswillbethesameforallthreeinvestment

categories.

Thebondindexesareasfollow:

• J.PMorganDenmarkGovernmentBondIndex(JPMGBIDenmark)

• J.P.MorganGlobalBroadex.Denmarkindex(JPMGBIGlobal)

ThedataforthosetwoindexesareobtainedthroughDatastream.

The choice of bond indices follows Christensen (2003). He concludes that the EFFAS bond

indexes could just as well be chosen, however it has been decided to use the J.P: Morgan

indexesinthisthesisaswell.

Danishstocks

ThebenchmarkchosenfortheJensenindexwas“OMXCopenhagenCap_GI”,andaspreviously

discussed,thisindexincludesallstocks,whicharelistedontheCopenhagenstockexchange.

Thiswould therefore also imply that the index includes small cap stocks aswell. Thus, no

need is foundto includeasmallcap index in themulti‐indexmodel forDanishstockssince

“OMXCopenhagenCap_GI”shouldbeabletocoverthat.

AnotherindexthatwillbeincludedistheMSCIWorldindex,sinceitisbelievedthatthiscould

berelevantforthemutualfundsinvestinginDanishstocks.However,oneshouldthoughkeep

inmindthatthisindexonlyincludesmidandlargecapstocksonthedevelopedmarkets.

Finally,thetwobondindexeswillbeincluded,andthereforetheregressionreadasfollows:

€

rpt − rft =α + βOMXCCap _GI (rOMXCCap _GI − rft )+βMSCIWorld (rMSCIWorld − rft ) + βJPMGBIGlobal (rJPM GBIGlobal − rft )+ βJPM GBI Denmark (rJPM GBI Denmark − rft ) + et

Europeanstocks

In the single index model for European stocks, the MSCI Europe was used as benchmark.

However, this index only includes mid and large cap stocks for the developed markets in

Europe.Therefore, itwouldbe interesting to includean indexon theemergingmarkets. In

ordertocoverthat,theMSCIEMEuropeindexhasbeenincludedinthemodel.Inaddition,it

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wouldalsobeinterestingtocoverthedevelopmentinthesmallcapstocks,andthereforeit

hasbeendecidedtoincludetheMSCIEuropeSCaswell.

Finally,themulti‐indexmodelfortheEuropeanstockswill includetheMSCIWorldindexto

coverthedevelopmentontheworldmarketaswellastheOMXCopenhagenCap_GItocover

the development on the Danish market. In order to cover the development on the bond

markets,thetwopreviouslydiscussedbondindexeswillbeincluded.

Therefore,themulti‐indexmodelforEuropeanstocksreadsasfollows:

€

rpt − rft =α + βMSCI Europe (rMSCI Europe − rft )+ βMSCI EM Europe (rMSCI EM Europe − rft ) + βMSCI EuropeSC (rMSCI EuropeSC − rft )+ βMSCIWorld (rMSCIWorld − rft ) + βOMXCCap _GI (rOMXCCap _GI − rft )+ βJPM GBIGlobal (rJPMGBIGlobal − rft )+ βJPM GBI Denmark (rJPM GBI Denmark − rft ) + et

Globalstocks

For themutual funds investing in global funds, the regressionmodel will follow the same

patternastheonefortheEuropeanstocks.

It isfirstofalldesiredtocoverthedevelopmentinboththeemergingmarketsaswellasin

the small cap markets, since those two things are not included in the MSCI World index,

followingthesamediscussionasfortheMSCIEuropeindex.

Therefore,besides theMSCIworld index, theMSCIWorldSC indexaswell as theMSCIEM

index will be included. These two indexes should cover the development in the small cap

stocksaswellasintheemergingmarkets,respectively.

Finally, OMX Copenhagen Cap_GI index will be included along with the two previously

discussed bond indexes. Therefore, the multi‐index model for the global stocks read as

follows:

€

rpt − rft =α + βMSCIWorld (rMSCIWorld − rft )+ βMSCIWorld SC (rMSCIWorld SC − rft ) + βMSCI EM (rMSCI EM − rft )+ βOMXCCap _GI (rOMXCCap _GI − rft ) + βJPMGBIGlobal (rJPM GBIGlobal − rft )+ βJPM GBI Denmark (rJPM GBI Denmark − rft ) + et

Page37

4.3Choiceofriskfreerate

Therisk freerateshouldrepresent the interest ratean investorcouldobtain in themarket

withouttakinganyriskatall.Intheliteraturedifferentproxiesoftheriskfreerateareused,

suchas“1‐monthT‐bill”,“1‐monthCIBOR”,etc.

This analysis will follow the analysis by Christensen (2003),where he uses the 1‐month

CIBORasaproxyfortheriskfreerate.

The CIBOR (Copenhagen Interbank Offered Rate) is the interest rate, which the Danish

NationalBankdeterminesonadailybasis,basedonreportsfromseveralDanishbanks.

In practical terms, the CIBOR rate is the reference interest rate, which is used for lending

moneyontheinterbankmarket.TheCIBORratesdifferdependentontheperiod,whichgoes

fromoneweekupto12months.

Another argument for using the1‐monthCIBOR rate is, that it is a good representative for

whatthemutualfundscouldhaveearnedontheinterbankmarketinthespecificmonthofan

investment. It shouldalsobenoted, that it isveryunlikely thatother reasonablechoicesof

proxyfortheriskfreeratewouldchangetheconclusionsoftheanalysis.Thereasonofthatis,

thattheriskfreerateisdeductedfromboththereturnofthemutualfunds,aswellasfrom

thereturnofthebenchmark.

Thedataforthe1‐monthCIBORhasbeenobtainedfromthehomepageoftheDanishNational

Bank.Theinterestratesaremeasuredonanannualbasis,andthereforethefollowingformula

hasbeenusedinordertoobtaincontinuousinterestrates:

Monthlyinterestrate=

€

(1+ annual int erest rate)1/12 −1

4.4Survivorshipbias

Survivorshipbiasisdefinedasthebiasthatoccursinyourdatasetifnon‐survivingfundsare

systematicallyignoredintheperformanceanalysis.Theresultofthatwilloftenbethatyour

analysiswouldsignificantlyoverstatethereturns,simplybecausethattheworstperforming

funds(andthereforenon‐surviving)areleftoutoftheanalysis(Christensen,2005)

Eltonet al. (1996) argues that thedisappearanceofmutual fundsoccur eitherdue topoor

performanceorduetoasufficientlysmallmarketvalueofthefund.Inaddition,theyconclude

that lowmarketvaluesareoftencausedbypoorperformance. Theyfurtherarguethat it is

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rarethatfundstotallydisappear,butinsteadtheyareoftenmergedintofundsthatoftenfall

intothesamefamilyoffundsandhaveperformedbetter.Theadvantageforthemutualfunds

ofdoingsois,thattheycontinuetoearnfeesontheinvestmentintheoriginalfund,whilethe

recordofthefund’spoorperformancebecomesdifficulttotrack.

Malkiel (1995) furtheradds to thediscussionofsurvivorshipbias thatmutual funds,which

acceptveryhighriskhaveveryhighprobabilitiesoffailureaswell.Therefore,survivingfunds

doingsoarecomparedtotakingalargebetandhavingwon.Thisclearlyshows,thatifonly

survivingfundsareincluded,thentheywillpositivelybiastheaveragereturnofthemutual

funds.

Asimplified,yetextremeexamplecouldbe10mutual fundsthatallhave invested inhighly

risky securities in a specific area over a 10‐year period. One of these funds has earned a

returnof200%overtheperiod,whereastheother9fundshavenotsurvivedtheperioddue

to poor performance. If a performance analysis was carried out at the end of the 10‐year

period, only including the surviving funds, then the conclusion would be that the average

return within this specific area of investment had been 20%. However, this result is very

misleading,sincetheother9fundsearningnegativereturnwherenotincludedduetolackof

datafortheentireperiod.Theactualaveragereturnwithinthisareaofinvestmentislikelyto

behighlynegative.Thisistheessenceofsurvivorshipbias.

Malkiel (1995) also analysed the effect of survivorship bias. The conclusion was, that the

largertheperiodofanalysis,thelargertheeffectofsurvivorshipbiaswouldbe.Theanalysis

showed that over a 10‐year period, the average annual return of only surviving fundswas

1,5%‐points higher than the average annual return for all funds including non‐surviving

funds.Theanalysiswasalsoconductedona15‐yearperiod,andtheresultswereevenmore

surprising here. The results were that the average annual return of surviving funds was

18,7%, while it was only 14,5% including the non‐surviving funds. This amounts to a

differenceof4,2%‐points,whichmustbeconsideredquitealot.

However, most of the analyses regarding survivorship bias has been conducted on the

Americanmarket.IntheanalysisbyChristensen(2005),heconcludesthattheDanishmutual

fundsarealmostfreeofsurvivorshipbias,becausenofundshavedefaultedduringhisperiod

Page39

ofanalysis.Itshouldthoughbenoted,thattheperiodofhisanalysiswas1996‐2003,whereas

theperiodofthisthesisis2001‐2010.

Itshouldfinallybenoted,thatthisthesisonlydealswithfundsthathaveexistedfortheentire

periodofanalysis.Therefore,fundsthathavedefaultedormergedintheperiodarenottaken

into consideration. In other words, survivorship bias is ignored, and therefore one should

keepthatinmind,wheninterpretingtheresultsoftheanalysis.

4.5Assumptionsforthetests

Thedataused for theanalysis isclassifiedas timeseriesdata, since theanalysisdealswith

observationsovertime.Timeseriesdataisoftenmoredifficulttoanalysethanregularcross‐

sectionaldata, as it is rare that timeseriesdata is independentacross time.Therefore,one

shouldstronglyconsideranddiscusstheassumptionsrequiredforthetesttobevalidbefore

onedraws conclusionsupon them. In case the assumptions arenot fulfilled, one shouldbe

carefullywheninterpretingtheresults.Ifassumptionsshowupnottobefulfilled,comments

willbemadeintheanalysis.

The discussion of assumptions for time series regression will follow the Gauss‐Markov

assumptionsfromWooldridge(2008).

1.Linearinparameters

The assumption states that there should be a linear relationship between the dependent

variable and the independent variables in the model. Eviews will therefore estimate the

modelasifitwaslinear,whetherornotthisisactuallythecaseornot.However,aspreviously

discussed,researchuponCAPMhasshownthat itseemsas if therelationshipbetweenpast

return and beta is linear. This analysiswill also estimate beta based on past returns, and

thereforeitisexpectedthattherelationshipwillbelinearinthiscaseaswell.Inaddition,the

performance measurements used in this analysis are built upon CAPM, which requires

linearity.Therefore,itisassumedthatthisassumptionisfulfilled.

2.Noperfectcollinearity

This assumption states that none of the independent variables can be constant or a linear

combination of the others. If two independent variables show up to be a perfect linear

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combinationofeachother,itissaidthattheyareperfectlycorrelatedwitheachother.Thisis

aproblem,becauseonecannotestimateamodel,whichhasperfectlycorrelatedvariables.If

onefacestheproblemofperfectcollinearity,oneshouldeitherreformulatetheentiremodel,

orleaveoneofthevariablesoutofthemodel.

Itisimportanttokeepinmindthattheassumptionallowsforhighcorrelationsbetweenthe

variables,howevertheyarejustnotallowedtobeperfectlycorrelated.

Intermsofthisanalysis, itmustbeexpectedthatindependentvariables(benchmarks)have

ratherhighcorrelations,astheytosomeextentsufferfromthesamesystematicrisk.Stillitis

notexpectedthatanyofthemsufferfromperfectcollinearity.

Thereisnoreasontotestforthisassumption,asEviewscannotestimateregressions,which

sufferfromperfectcollinearity.Inotherwords,Eviewsautomaticallydoesthetestforyou.If

opposedtoallexpectations,somemodelsshowuptobesufferfromperfectcollinearity,the

modelwillbereformulated,andcommentsregardingthatwillbemade.

3.Zeroconditionalmean

Thisassumptionrequiresthattheexpectedvalueoftheerrortermiszerogivenanyperiodof

theexplanatoryvariable.

Instatisticalterms,theassumptionrequiresthatE(u)=0.Iftheerrortermisindependentof

theobservationsoftheexplanatoryvariablesinthemodel,thisassumptionwillhold.

Tobemorespecific,theassumptiondoesnotputanyrestrictiononthecorrelationbetween

theindependentvariablesinthemodelorintheerrortermacrosstime.Theassumptiononly

states, that the average value of the error term should be unrelated to the independent

variablesinalltimeperiods.

The two most common causes of breaches of this assumption are omitted variables and

measurementerrorsinsomeoftheregressors.

Thisassumption ismuchrelated to the lastassumptionaboutnormality, so the test for the

assumptionwillberelatedtothat.

4.Homoskedasticity

The assumption abouthomoskedasticity requires that the varianceof the error term is the

sameforallvaluesintheperiod.Ifhomoskedasticityisnotpresent,itissaidthatthemodel

suffers for heteroskadasticity. A more graphical illustration would show that if

Page41

heteroskadasticity is present in themodel, a scatterplot with the residuals (errors) would

showa funnel shapecompared to theactual regression line.The funnel shapewouldcause

thatthevarianceiseitherincreasedordecreasedovertime,whichisaproblemforthemodel.

Theassumptionabouthomoskedasticityplaysnoroleinshowingthatthemodelisunbiased,

howevertheassumptionisstillimportant.Ifamodelsuffersfromheteroskadasticityitisvery

likelythattheestimatesofthestandarderrorsarewrong.Thisisofgreatimportanceforthe

conclusionof themodel, as the standarderrorsareused in the calculationsofwhether the

variablesinthemodelaresignificantornot.

Thetestonbetacoefficientslooksasfollows:

€

H 0 :β = 0H1 :β ≠ 0

€

ˆ β x − β x 0

se( ˆ β x )Tn−k−1

The illustration above shows that the standard error of the beta coefficient is used in

calculatingtheteststatisticinthetestonthesignificanceofthebetacoefficient.Aspreviously

discussed,ifheteroskedasticityispresent,theestimateofthestandarderrorsareunreliable,

andtherefore thesignificanceof thecoefficientswillbeunreliableaswell.This isofcrucial

importance for the assignment, since it is the significance of the beta coefficients, which

determineswhetherthefundshaveperformedneutrallyornot.

The assumption about homoskedasticity will therefore be dealt with carefully in the

assignment.

Thespecifichypothesiswhicharetestedlooksasfollows:

H0:Var(ux)=σ2(Homoskedasticity)

H1:Var(ux)≠σ2(Heteroskadasticity)

Therearevariousteststhatcanbeusedtotestthehypothesis,howeverthemostcommonly

usedare theBreusch‐Pagan test and theWhite test.The two tests are rather similar, since

theybothusethesquaredresidualsasdependentvariablesinaregressionmodel,wherethey

try to determine whether these can be explained using different forms of the original

independent variables. It does notmattermuch,which one is used, since they should both

Page42

yield the same result. Therefore this analysis will use the Breusch‐Pagan test to test for

variancehomogeneity.

Theevaluationofthehypothesiswillsolelybebasedonthep‐valuefromtheBreusch‐pagan

test.Asignificancelevelof5%ischosen,andifthep‐valueisbelow5%,thenull‐hypothesisis

rejected,meaningthatheteroskedasticityispresent.

Ifheteroskedasticityshowsuptobepresentinthemodel,oneshouldcorrectforthatbyusing

White’sheteroskedasticity‐robuststandarderrors.

5.Noserialcorrelation

Thisassumptionstatesthattheerrorsshouldbeuncorrelatedinthetimeseriesregression.In

statisticaltermsthemodelstatesthat:Corr(ut,usX)=0,forallt≠s.

Inordertotestwhetherthisassumptionholds,thefollowinghypothesisshouldbetested:

H0:Noserialcorrelation

H1:Serialcorrelation(autocorrelation)

If the null hypothesis is rejected, the model is said to suffer from serial correlation or

autocorrelation,whichmeansthattheerrorsarecorrelatedovertime.

There are several tests to choose between in order to test for serial correlation. Themost

commonly used are the Durbin‐Watson test and the Breusch‐Godfrey test. However, the

DurbinWatson test is primarily used to test for first order serial correlation,whereas the

Breusch‐Godfrey test is used to test for higher order serial correlation. In terms of this

analysis, it isdesired to test for12thorder serial correlation.Thereasonof that is that this

thesisisworkingwithmonthlyobservations,andthereforeitwouldbeappropriatetotestfor

12thorderserialcorrelationinordertoavoidanyseasonality.Thus,inordertotestforserial

correlation, the Breusch‐Godfrey test for serial correlation is used, which is a Lagrange

multipliertestforserialcorrelation.

TheBreusch‐Godfreytestworksbymakinganauxiliaryregressionusingtheresiduals from

theoriginalregressionregressedonasetoflaggedresidualsaswellasthevariablesusedin

theoriginalmodel.What thetestdoes, is that it testswhether thecoefficientsof the lagged

regressionintheauxiliaryregressionareallequaltozero.Thetestwillgiveaχ2‐teststatistic.

If the p‐value of that test statistic is below the significance level of 5%, then the null

Page43

hypothesis is rejected, meaning that the model suffers from serial correlation. In case the

model suffers from serial correlation, this should be corrected by using the Newey West

heteroskedasticityandautocorrelationconsistent(HAC)standarderrors.

The consequences of serial correlation are that the estimates of the standard errorwill be

unreliable, which again causes the significance of the coefficient to be unreliable as well,

following the discussion fromprevious. In addition, it is said that in the presence of serial

correlation,OLSisnolongerBLUE(bestlinearunbiasedestimator).

Ithasshownthatpositiveserialcorrelationcausesthestandarderrorstobeunderestimated

thereby causing the t‐statistic to be biased upwards. In other words this means, that if

positiveserialcorrelation ispresent, it ismore likelythatthecoefficientwillshowuptobe

significantduetoahighert‐statistic.Positiveserialcorrelationwillalsocausethevarianceof

theerrortermtobeunderestimated,therebyexaggeratingR2.

Again,theabovediscussionshowswhyitisimportanttotakethisassumptionundercareful

consideration.

6.Normality

ThefinalassumptionisthattheerrorsareindependentofXandthattheyareindependently

andidenticallydistributed.Instatisticaltermsitwouldlooklikethis:X∼normaldistribution

(0,σ2),meaningthatitisrequiredthattheerrorsshouldfollowanormaldistributionwitha

meanof0andavarianceofσ2.

Thislastassumptionissaidtobethestrongestofthesix.Somearguethatbecausetheerroris

thesumofmanydifferentthingsaffectingthedependentvariable, theargumentationofthe

centrallimittheoremcanbeused,whichstatesthatifthesamplesizeissufficientlylarge,it

canbeassumedthatthevariableinquestionisapproximatelynormallydistributed.

Inordertotestfornormalityoftheerrors,theJarque‐Beratestfornormalitywillbeused.

Thehypothesiswhicharebeingtestedare:

H0:Theerrorsarenormallydistributed

H1:Theerrorsarenotnormallydistributed.

Again,ifthep‐valueoftheteststatisticisbelow5%,thenthenullhypothesisisrejected.

Page44

The consequences of non‐normality are that the errors terms will be either left or right

skewed, and thereby it is like that the results are affected by extreme values. This would

thereforealsocausethatoneshouldbecarefulwheninterpretingtheresults if itshowsup,

thattheerrorsarenotnormallydistributed.

Ifitshowsupthatvariableshaveerrorsthatarenotnormallydistributed,onewaytocorrect

foritistouselogarithmicfunctiononsomeoftheindependentvariables.Thiswouldnarrow

inthetailsofthedistribution,sothatitbecomesclosertothenormaldistribution.Howeverin

termsofthisanalysis,nocorrectionwillbemade,howevercommentswillbemadeinstead.

Page45

5.Performanceanalysis

Thissectionwillcoveradiscussionoftheresults,whichhavebeenreachedintheanalysis

usingthedifferentperformancemeasurements.

5.1Jensen’salpha

TheresultsfortheJensenindexareshowninappendix1alongwiththeassumptionsforthe

tests.Generallyspeaking,manyofthemodelssufferedfromproblemswiththeassumptions.

33outofthe60fundsshowedproblemswithheteroskedasticityand13outofthe60funds

showed problems with serial correlation. However, corrections for that was made in the

regression model, using White’s heteroskedasticity‐robust standard errors to correct for

heteroskedasticityandNeweyWestheteroskedasticityandautocorrelationconsistent(HAC)

standard errors to correct for serial correlation. Due to the corrections, these two

assumptions should not cause problems for the interpretation, and therefore no further

commentswillbemaderegardingthat.

However,only20outofthe60fundsshoweduptohavenormallydistributederrors.Thisisa

problem,becausethismeansthat40outofthe60fundsarenotabletofulfiltheassumption

aboutnormality.Nothingcanbedonetocorrectforthat,sothereforecommentswillbemade

regardingthiswheninterpretingtheresults.

When interpreting the results from the Jensen index, a tablewill be shownwith the funds

showingsignificantperformance.

Thefollowingcolourshasbeenusedtoshowsignificanceatthedifferentlevels:

= Significantata5%level = Significantata10%level

Thecomments thataremadewillprimarilybeconcernedwith themutual funds that show

significant performance. However, if something is specifically noticeable in the neutrally

performancefunds,asmallcommentwillbemade.

Page46

5.1.1Danishstocks

16 mutual funds were analysed in the category Danish stocks. The significant results are

showninthetablebelow:

TheresultsfromtheanalysisofthemutualfundsinvestinginDanishstocksshowsthat12out

ofthe16fundshaveperformedneutrallycomparedtothebenchmark.Thereasonofthatis

thattheintercept(=α)showsuptobeinsignificantintheregressionmodel.

However, more interesting is the 4 mutual funds that showed up to have significant

performance.

2mutual funds showed significant negative performance at a 5% significance level, and 1

mutual fund showed significant negative performance at a 10% significance level. On the

morepositiveside,1mutualfundshoweduptohavesignificantpositiveperformanceata5%

significancelevel.Theresultsforthenegativeperformingfundsshowunderperformanceof

‐0,2349% to ‐0,2730% per month compared to the benchmark, whereas the positive

performing fund shows an over performance of 0,3271% per month compared to the

benchmark.Inaddition,itshouldbenotedthatallthe4significantlyperforminginvestment

fundsshowbetavaluesaboveone,meaningthattheyhaveahighersystematicriskthanthe

market.

Generally,theresultsseemreliable.Howeverfor“EgnInvestDanmark”oneshouldbecareful

in the interpretationof theresultsbecause the Jarque‐Bera testshowsastrongrejectionof

normality,which isabreachof theassumptions.For the threeremaining funds, the Jarque‐

Bera tests show up insignificant, meaning that the errors of those funds are normally

distributed. In addition the 4 mutual funds show rather high adjusted R2, ranging from

0,932580to0,981023,meaningthatthechosenbenchmarkisrathergoodatexplainingthe

variationinthereturnsofthese4investmentfunds.

Page47

Onefinalthingtonoteintheinsignificantlyperformancefundsisthat2ofthefundsthatare

included invest in small cap stocks. These two funds show adjusted R2 of 0,390177 and

0,790927, which must be considered rather low. However, small cap stocks should be

includedinthechosenbenchmark,andtherefore it issurprisingthattheiradjustedR2 isso

low.

5.1.2Europeanstocks

21mutual fundswereanalysed in thecategoryEuropeanstocks.Thesignificant resultsare

showninthetablebelow:

WhenlookingattheabovetableforthemutualfundsinvestinginEuropeanstocks,onecan

see that 8 out of the 21mutual funds showed significantly negative performance at a 5%

significance level, and2mutual funds showed significantly negative performance at a 10%

level.Onthemorepositiveside,1mutualfundshowedsignificantpositiveperformanceata

10%level.Asumupwouldthenshowthat10mutualfundsperformedneutrally.

The negative performance range from ‐0,1734% to ‐0,5430% per month compared to the

benchmark,howeveroneshouldbecarefulintheinterpretationoftheseresults,sinceonly3

outofthe10negativelyperformingmutualfundsshowsignsofnormallydistributederrors.

Therefore,thereisabreachoftheassumptionintheremaining7mutualfunds,whichcould

biastheresults.Forthenegativeperformingfunds,adjustedR2looksreasonablewhere9out

of 10 of them lie between 0,914246 to 0,987777. 1 fund falls below these values with an

adjusted R2 of 0,876721, however this is still relatively high considering that only one

benchmarkisusedinthemodel.Thereforethisdoesnotraiseanyconcern.

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“Danske Invest Europa Small Cap” shows significant positive performance at a 10%

significance level. The result shows an over performance of 0,5383% compared to the

benchmark, however one should be careful when interpreting this result. First of all, the

Jarque‐Beratestshowsupsignificantmeaningthatthehypothesisregardingnormalityinthe

errorsisrejected.Secondly,theadjustedR2isratherlowat0,781052,meaningthatthereis

roomforimprovementinthebenchmark’sabilitytoexplainthevariationinthereturnofthe

mutualfunds.Thisisprobablyconnectedtothefact,thatthefundinvestsinsmallcapstocks,

andthereforethechosenbenchmark ispossiblynotthebest inthiscase. Itmustthoughbe

expected that the multi‐index model will correct for that, since a small cap index will be

includedasbenchmarkhere.

Itcouldalsobenotedthatonly3outofthe21investmentfundsinvestinginEuropeanstocks

show up to have normally distributed errors. This is of course a problem for the

interpretationoftheresults.Lastly, itcouldbenotedthat3of themutual funds includedin

thissampleinvest insmallcapstocks.These3mutualfundsalsoshowupwithsomeofthe

highestbetavalues,rangingfrom1,11792to1,254541,howeverthisisnotsurprising,sinceit

must be assumed that the small cap stocks are more volatile than the larger and more

establishedcompanies.Generally,theadjustedR2isratherlowforthese3fundscomparedto

theothers,howeveraspreviouslydiscussed,itmustbeexpectedthatthiswillchangeinthe

multi‐indexmodel.

5.1.3Globalstocks

23 mutual funds were analysed in the category Global stocks. The significant results are

showninthetablebelow:

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When looking at the above table, one can see that 14 out of the 23 mutual funds show

significantperformance.Thisfirstofallmeansthat9ofthefundshaveperformedneutrally

comparedtothebenchmark.

7 fundsshowsignificantnegativeperformanceata5%significance leveland3 fundsshow

significantnegativeperformanceata10%significancelevel.The10underperformingfunds

have under performed the benchmark by ‐0,0921% to 1,0576% permonth. However, one

shouldbecarefulwheninterpretingtheresults,sinceonly5ofthe10fundsshowuptohave

normallydistributederrors.

For the underperforming funds, the adjusted R2 looks reasonable for all of them, besides

“Danske Invest Global Plus”. In this case, the adjusted R2 only shows a value of 0,622307,

whichmustbeconsideredratherlow.Thesamefundhasareallyhighbetavalueof1,735854,

whichbyfaristhelargestintheentiresample.Thisindicatesthatthechosenbenchmarkdoes

notmatchthe investmentstrategyof thisspecific fundwell.Oneshouldagainnote that the

Jarque‐Berateststronglyrejectsnormality intheerrorsforthisspecific fund,andtherefore

thiscouldaffecttheresults.

4fundsshowsignificantpositiveperformanceata5%significancelevel.These4fundshave

outperformed the benchmark by between 0,4405% to 1,0109% per month. However, one

shouldagainbecarefulabouttheinterpretationoftheseresults.Firstofall,noneofthese4

fundshavenormallydistributederrors,andsecondlyallofthemshowratherlowadjustedR2.

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However,“SkagenGlobal”deviatesabitfromtheother3.Firstofall,thisitistheonlyoneof

the4,whichhasabetavalueabove1,anditsadjustedR2is0,797165,whichissignificantly

higherthantheother3.

For the remaining 3 funds, they all show beta values that are way below 1, ranging from

0,623612to0,748632.Thismeansthattheinvestmentsbythesefundsaremuchlessvolatile

than the market. This could indicate that this is not an appropriate benchmark, simply

because the market is much more volatile than the portfolios of the mutual funds, and

therefore it could become easier to show significant performance. Finally, the adjusted R2

showsvaluesfrom0,635523to0,726274forthese3funds,againindicatingthatthechosen

benchmarkcouldbebetteratexplainingthevariationinthereturnsoftheinvestmentfunds.

5.1.4ConclusiononJensen’salpha

ToconcludeonJensenindex,thetablebelowhasbeenmadeinordertoillustratetheresults:

Thegeneralconclusionbasedonthetableaboveisthatthemutualfundshavenotbeenable

tooutperformthebenchmarkaftertheexpensesfromthemutualfundshavebeendeducted.

23 fundsshowsignificantlynegativeperformance,meaning that theyhavenotbeenable to

cover their expenses,whereas31 funds show insignificantperformance,meaning that they

have justbeenable tocover theirexpenses.Therearethough6 funds,whichshowpositive

performance, however large uncertainty is connected to most of these results due low

adjustedR2andproblemswithnormality,whichhasbeendiscussedpreviously.

Thebestperformingfundinthesampleis“SEBinvestDanskeAktier”,whichshowedanover

performanceof0,3271%permonthcomparedtothechosenbenchmark.Thereasonwhythis

fund is chosenas thebest is, that it is theonly fundof the6overperforming funds,which

showsnormalityandhighadjustedR2.

“SEBinvest Danske Aktier” shows a p‐value of 0,180253 in the Jarque‐Bera test, which

strongly maintains the assumption of normality. In addition, it shows an adjusted R2 of

0,953366,whichmeans that the chosenbenchmarkmatches the investment strategyof the

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fundratherwell.Therefore, itmustbeconcluded that theresults for thisspecific fundsare

reliable,andbasedonthatitisclassifiedasthebestperformingfundintheJensenindex.

In otherwords, this simplymeans that thismutual fund has shown superior performance

evenafteritsexpenseshasbeendeducted.

5.2Multi‐indexmodel

Whenestimatingtheregressionmodelforthemulti‐indexmodel,insignificantvariableshave

beenremovedonebyone,startingwiththeonewiththehighestp‐value.Eachtimeavariable

hasbeenremoved,anewregressionmodelhasbeenestimated.

IfalladditionalbenchmarksbesidestheonefromtheJensenindexshowupinsignificant,then

themulti‐indexmodelwillendupwithexactlythesameresultsastheJensenindex.

ThisespeciallyoccurredforthemutualfundsinvestinginDanishstocks.

In appendix 2, tables have beenmade, showingwhich benchmarks showup significant for

eachofthemutualfunds.Appendix2alsoincludestablesfortheassumptionsaswellasfor

theresults.

The results for each of the three investment categorieswill now be discussed individually

followedbyaoverallconclusiononthemulti‐indexmodel.

5.2.1Danishstocks

Aspreviouslydiscussed,themulti‐indexmodeldidnotseemtodomuchgoodforthemutual

funds investing in Danish stocks. For 8 out of the 16 mutual funds, all of the additional

benchmarksshowedupinsignificant.For3fundstheMSCIworldindexshowedupsignificant,

and for5 funds, the “JPMGBIBroadex.Denmark” showedupsignificant.However for2of

these,itwasonlysignificantata10%level,indicatingthatitseffectonthemodelisrelatively

uncertain.

TheJPMGBIDenmarkdidnotshowupsignificantatall,therebyshowingthatthisindexhas

noabilitytoexplainthereturnsoftheinvestmentfundsatall.

Generally, itmustbeconcluded that theOMXCopenhagenCap_GI indexdoesratherwell in

explainingthereturnsoftheinvestmentfundsinvestinginDanishstocks,sinceinhalfofthe

cases,themulti‐indexmodeldoesnotaddanythingadditionalcomparedtotheJensenindex.

In addition, for the mutual funds, which have more than one significant benchmark, the

adjustedR2hasonlyincreasedslightly.TheadjustedR2hasincreasedbylessthan1%forall

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ofthe8funds,therebyindicatingthatthemulti‐indexmodelonlyimprovesthepredictability

ofthereturnsoftheinvestmentfundsslightly.

In terms of the actual results, little have changed. 12 funds still showneutral performance

compared to the benchmarks, and it is still the same 4 funds, which show significant

performance as in the Jensen Index. The only thing that has changed is that the

underperformanceby“SparindexOMXC20aktier”isnowsignificantata5%level,compared

tothe10%levelintheJensenindex.

Intermsofthecoefficientsofα,thechangesaresmallaswell.Themostnoticeablechangeis

for Carnegie WorldWide/Danske aktier, which went from a negative, though insignificant

coefficientofαintheJensenindextoaninsignificantpositiveαinthemulti‐index,wherethe

MSCIWorld index showedup significant.However, for the remaining 15 funds, the sign in

frontoftheαcoefficientremainedthesame.

In terms of the assumptions, nothing of importance changed. One additional fund showed

signs of heteroskadasticity, however this was corrected for by using White’s

heteroskedasticity‐robust standard errors, and therefore this should cause no problems. In

termsofserialcorrelationandnormality,theconclusionwasthesameasintheJensenindex.

Therefore, the conclusion for themutual funds investing inDanish stocks is that themulti‐

indexmodeldoesnotseemtoaddmuchadditionalcomparedtotheJensenindex.

5.2.2Europeanstocks

ForthemutualfundsinvestinginEuropeanstocks,themulti‐indexmodelseemedtohavea

muchlargereffectcomparedtothemulti‐indexmodelfortheDanishstocks.

The table below summarizes the number of times the different benchmarks showed up

significant.

Page53

One thing,which isnoticeable is that theMSCIEurope index,whichwasused in the Jensen

indexshowedupinsignificanttwotimes.However,thisdoesnotcomeasthebiggestsurprise,

sincethese2fundsinvestinsmallcapstocks,andthereforeitmustbeassumedthatthesmall

cap indexwould be the primary source in explaining the variation in the returns of these

mutual funds. This was probably also the reason why these funds showed relatively low

adjustedR2intheJensenindexmodel.

ItshouldprobablyalsobenotedthattheMSCIEMEuropeshowedupinsignificantinallofthe

models,therebyindicatingthattheemergingmarketsinEuropearenotrelevantinexplaining

thereturnsoftheseinvestmentfunds.

Intermsoftheactualresults,relevantchangesoccurredinthemulti‐indexmodelcompared

totheJensenindex.

11 funds showed negative performance at a 5% significance level, and 3 funds showed

negativeperformanceata10%level.ComparedtotheJensenindex,thisisanincreaseof3at

the5%levelandanincreaseof1atthe10%level.

However, none of the funds showed significant positive performance in the multi‐index

model, and therefore it must be concluded, that the multi‐index model strengthens the

negative performance of the mutual funds, since it indicates significantly negative

performancefor4additionalfunds.

Whenlookingatthecoefficientforα,onecanseethatthedirectionofthechangediffersfrom

fundtofund,howeverthemostcommonchangeisadecreaseinα,meaningthemulti‐index

modelgenerallyshowsamorenegativeperformancethantheJensenindex.

The increase intheadjustedR2wasalso larger fortheEuropeanstocksthanfortheDanish

stocks.Thegeneralpictureshowedincreasesof1‐2%‐pointwithsomedeviations.Especially,

the mutual funds investing in small cap stocks showed large increases in the adjusted R2

showing increases between 10‐12%‐point. However, again this does not come as a big

surprise,sincethesmallcapindexisamoreappropriatebenchmarkforthesefunds.

Thetestfortheassumptionlookedrathersimilarforthemulti‐indexmodelasfortheJensen

index.IntheJensenIndex,13fundsshowedsignsofheteroskedasticity,5fundsshowedsign

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of serial correlation, and 3 funds showed normally distributed errors. For themulti‐index

model,thesenumberschangedto11,8and4respectively.

5.2.3Globalstocks

For themutual funds investing in global stocks, all of the includedbenchmarks showedup

significant.Again,atablehasbeenmadeinordertosummarizethenumberoftimeseachof

thebenchmarksshowedupsignificant.

Intotal23mutualfundswereanalysedinthiscategory,andfromthetableonecanseethat

theMSCIworld indexonlyshowedupsignificant21 times.However, in thiscase it ismore

surprisingthanfortheEuropeanstocks,sincenoneofthemutualfundsinthiscategoryhave

investments in small cap stocks as their primary investment objective, thereby explaining

whytheMSCIWorldindexwouldnotbetheprimarybenchmark.Themutualfundswherethe

MSCIWorld index showed up insignificant even have the MSCIWorld index as their own

benchmark.

In addition, it is seen from appendix 2, that all of the additional benchmarks showed up

insignificant for 4 of themutual funds. This alsomeans that themulti‐indexmodel added

additionalinformationintheregressionmodelfortheremaining19mutualfunds.Thiswas

also seen by a general increase in adjusted R2 of 2‐3%‐point in many cases. Of course

deviation from that occurred,where the largest one being “Sparinvest Value aktier”which

rosefrom0,635523intheJensenindexto0,854505inthemulti‐indexmodel.

Theresultsforthemulti‐indexmodelalsochangedsignificantlyfortheGlobalstocks.Inthe

Jensenindex,7fundsshowedsignificantnegativeperformanceata5%level,3fundsshowed

significant negative performance at a 10% level, and 4 funds showed significant positive

performanceata5%level.Inthemulti‐indexmodel,thesenumberschangedto14,1,and1

respectively. Again, this indicates that the multi‐index model strengthens the weak

performanceofthemutualfunds.

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In the Jensen index, 4 funds showed significant positive performance at a 5% level, and as

previouslymentioned, thisnumberdecreased to1 in themulti‐indexmodel.Theremaining

three funds showed neutral performance in the multi‐index model. The reason of that

probablyis,thatthese3fundsshowedratherlowadjustedR2intheJensenindex,andasthe

returnpredictabilityofthesemutualfundsroseinthemulti‐indexmodel,thefundsshowed

insignificant performance. An example of this increase in adjusted R2 is as previously

mentioned “Sparinvestvalueaktier” showingsignificantpositiveperformance in the Jensen

index,with a low adjusted R2, however in themulti‐indexmodel adjusted R2 increased by

approximately22%‐point, and theadditional returnpredictability in themulti‐indexmodel

resultedininsignificantperformanceinthemulti‐indexmodel.

It should thoughbenoted, that theα coefficient for these3 funds remainspositive, though

highlyinsignificantwithp‐valuesrangingfrom0,1492to0,9240.

“ValueInvestDanmark,BlueChip”showedsignificantpositiveperformanceinthemulti‐index

aswell,howevertheexcessreturndecreasedfrom0,4405%permonthintheJensenindexto

0,3376% in the multi‐index model. It should though be noted that this fund showed a

relativelylowadjustedR2inthemulti‐indexmodelaswell,withavalueof0,771921.

The overall picture of the multi‐index model for the Global stocks is that the sign on the

coefficients remain similar to the Jensen index. However, for the funds with negative α

coefficientsthemainconclusionisthatmorefundsshowsignificantnegativeperformancein

themulti‐indexmodel.

The tests for the assumptions looked better for themulti‐indexmodel than for the Jensen

index.3fewerfundsshowedsignsofheteroskedasticityand1lessfundshowedsignsofserial

correlation, however correctionswheremade for that. Butmore importantly, therewas an

increase in the number of investment funds showing normally distributed errors. This

number increased from9 in the Jensen index to14 in themulti‐indexmodel.This is still a

positive development, since the results become more valid when the assumption about

normally distributed errors is fulfilled. Therefore, the results from the multi‐index model

mustbeconsideredmorereliablethantheresultfortheJensenindex.

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5.2.4Conclusiononthemulti‐indexmodel

Again,atablehasbeenmadeinordertosummarizetheresultsforthemulti‐indexmodel:

As previouslymentioned,multi‐indexmodel has strengthened thenegative performance of

themutualfunds.Thelargestchangeisseenintheglobalstocks,wherethenumberoffunds

showing significant negative performance has doubled from the Jensen index to themulti‐

indexmodel.

Thenumberof significantlynegativeperforming fundshas increased from23 in the Jensen

index to 31 in themulti‐indexmodel. Thereby again indicating that themulti‐indexmodel

weakenstheperformanceofthemutualfunds.

Thenumberofsignificantlypositiveperformingfundsdecreasedfrom5intheJensenindexto

2inthemulti‐indexmodel.

ThepositiveperformingfundinthecategoryDanishstocksremainsbeing“SEBinvestDanske

Aktier”,howeveralloftheadditionalbenchmarksshowedupinsignificantforthisfund,and

thereforetheresultsandconclusionsremainthesameasintheJensenIndex.

Thesecondpositiveperformancefundwas“ValueinvestDanmark,BlueChip”.However,this

mutual fundwashighlyaffectedbythemulti‐indexmodel,sinceallbenchmarksshowedup

significantbesidestheMSCIEMindex.Thefundshowedanoverperformanceof0,3376%per

monthcomparedtothebenchmarks,however itstillshowedarelatively lowadjustedR2of

0,771921.This is inspiteof the fact that the investment fundusestheMSCIworld indexas

benchmark,andthereforeitwouldbeexpectedthemutualfundshowedahigheradjustedR2.

Theerrorsalsoshoweduptobenormallydistributedforthisfund,meaningthattheresults

mustbeconsideredvalid.

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Bothofthepositiveperformingfundsshownormallydistributederrors,sotheevaluationof

thebestfundwillsolelybebasedonthehighestα.

“SEBinvestDanskeAktier”showsanoverperformanceof0,3271%permonth,whereas

“ValueinvestDanmark,BlueChipshowsanoverperformanceof0,3376%permonth.

The conclusion is that “ValueinvestDanmark,BlueChip” is thebestperforming fund in the

sample.

Ontheotherhand,theworstperformingfundinthesamplewas“DanskeInvestGlobalPlus”

showing an under performance of ‐1,0576% per month compared to the benchmarks. It

shouldbenoted,thatthemulti‐indexmodelissimilartotheJensenindexincaseofthisfund,

anditsadjustedR2isrelativelylowat0,622307.Inaddition,itshowsastrongrejectionofthe

hypothesisregardingnormallydistributederrors.

5.3Markettiming

Theresultsfromthemarkettiminganalysis,usingthequadraticTreynor&Mazuymodel, is

showninappendix3.

5.3.1Danishstocks

Theresults fromtheTreynorandMazuyregressiongavequitesurprisingresults,since8of

the16fundsshowedmarkettimingabilities.Theresultsforthose8fundsareshownbelow:

However,firstofall,theestimatesofαchangedsignificantlyforthemutualfundsinvestingin

Danish stocks by including the quadratic term. In the Jensen index, 2 funds showed

significantlynegativeαata5%level,1showedsignificantlynegativeαata10%level,and1

fund showed significantlypositiveα at a 5% level. In theTreynor andMazuymodel, these

numberschangedto8,1,and1,respectively.However,oneshouldthoughnotethatfor3of

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these funds, the quadratic term shows up insignificant, and therefore the estimates are

affectedbysomething,whichhasnosignificanteffectonthemodel.

Whengenerallylookingattheestimatedα‐values,onecansee,thattheyaremuchlowerthan

for the Jensen index. This indicates weak selection ability by the mutual funds using the

Treynor&Mazuymodel,sinceonly1fundshowssignificantselectionabilities.

Aspreviouslymentioned,theresultsfortheinvestmentcategory“Danishstocks”werequite

surprising, since 14 out of the 16 analysed mutual funds showed up to have a positive

coefficient of the quadratic term. Of these 14, 8 of them showed up significant at a 5%

significancelevel.Inotherwords,halfofthemutualfundsinthiscategoryshowedsignificant

timingabilities.

TheconclusiontotheTreynorandMazuymodelis,thattheestimateofαismuchlowerthan

in the Jensen index. Thismust be caused by the fact that 8mutual funds show significant

timingabilities.

The Jensen index only focuses on selection abilities, and therefore better performance

attributabletotimingbecomessomewhatincludedintheselectionabilities.However,inthe

TreynorandMazuymodelseparationismadebetweenselectionandtimingabilities.

Generally, the selection abilities of the funds are verypure,which ismuch in linewith the

theoryofefficientmarkets.However,8fundspossesstimingabilities,whichpositivelyaffects

the results. This means that these 8 funds to some degree have managed to adjust their

systematicrisktothedevelopmentinthemarket.

5.3.2EuropeanandGlobalstocks

The results for the European andGlobal stocks are quite similar, so therefore theywill be

discussedjointly.

In total, 44 fundswere analysed in these two investment categories, and themain results

werethatnoneofthesefundsshowedanysignofmarkettiming.Only4outofthe44funds

showed a positive coefficient on the quadratic term, though all of them being highly

insignificant.For theremaining40 funds,almosthalfof themshowedasignificantnegative

coefficientonthequadraticterm,meaningthattheyhaveadjustedthesystematicriskonthe

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portfoliooppositelyofthedirectionofthemarket.Thishasaffectedtheresultsofthefunds

negatively.

Since close tohalf of thequadratic terms showedup significant, it alsomeans that it has a

significanteffectonthereturnsofthemutualfunds.

Whenlookingattheαvalues,onecanseethattheyhave increasedcomparedtotheJensen

index. In the Treynor and Mazuy model, there are 8 funds showing significant selection

abilities at a 5% level, and 2 funds showing significant selection abilities at a 10% level.

Compared to the Jensen index, this means that there are funds, which show significant

positive selectionabilities,howeverdue toweak timingabilities, theyarenot able to show

significantpositiveperformanceintheJensenindex.

5.3.2ConclusionontheTreynorandMazuymodel

Theconclusionontheanalysisofmarkettimingisthat8outofthe60analysedfundsshow

significantmarkettimingabilities.Itshouldbenotedthatallofthe8fundsarefoundinthe

investmentcategory,Danishstocks.

The results, which has been reached does not match the general results reached in the

literature.E.g.TreynorandMazuy(1966)reachedtheconclusionthatonly1outof57mutual

fundshadbeenable to time themarket.A similaranalysisof theDanishmutual fundswas

conducted by Christensen (2004). He reached the conclusion that only 2 out of 47 Danish

mutualfundsshowedsignificanttimingabilities.

Therefore, the results reachedaboutDanishmutual funds investing inDanish stocks raises

some concern, since these results are opposite of the results reached in the literature. In

termsof theresults,only2of the8 fundsshownormallydistributederrors,and it is likely

thatthatcouldhaveaneffectontheresults,howeveritisveryunlikelythatthiswouldexplain

the entire effect. The results have thoughbeen checked, andnomistakeshavebeen found.

Thereforetheresultsmustberegardedasreliable,meaningthatthegeneraltendencyisthat

itispossibletotimethemarketforthemutualfundsinvestinginDanishstocks.

Forthemutualfundsshowingsignificanttimingabilities,theresultsconfirmstheresearchby

Grant(1977)sayingthatthetruevalueofαislikelytobeunderestimatedinthepresenceof

timingabilities.ThereasonofthatisthatintheTreynorandMazuymodel,thevaluesofαare

smallerinpresenceoftimingabilitiesthantherespectiveαvaluesfromtheJensenregression.

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Finally,theresultsforthemutualfundsinvestinginEuropeanandGlobalstocksaresimilarto

theresultsreachedintheliterature.Theresultisthattimingabilitiesarenotpresentforany

ofthemutualfundsinthesetwocategories.

5.4ConclusionontheperformanceanalysisThegeneralconclusionontheperformanceanalysisisthatafewoftheanalysedfundshave

beenabletooutperformthemarketinboththesingle‐andmulti‐indexmodel.However,the

majority of the results show either that the investment funds are just able to cover their

expensesorthattheycannotcovertheirownexpenses.

Intermsofmarkettiming,8fundsinvestinginDanishstocksshowedmarkettimingabilities,

whereasallremainingfundsshowednosignsofmarkettiming.

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6.Discussionandreflections

Thediscussionandreflectionswillbeusedtocomparetheresultsofthisthesistosimilar

analyses.Inaddition,itwillalsobediscussedwhichthingscouldhavebeeninterestingto

includeinthisthesis.

InApril 2011, “DanskAktionærforening” published an analysis of theDanishmutual funds

(Danskaktionærforening, 2011).Thepurposeof this analysiswasnot to seewhichmutual

fundsthatwereabletooutperformcertainbenchmarks,butinsteadtorankthemutualfunds

basedontheirperformanceandinvestmentcategory.Itwouldthereforebeinterestingtosee

how they ranked themutual funds that performedwell in this thesis. It should though be

notedthatthemethodofanalysisusedby“DanskAktionærforening”differssignificantlyfrom

theoneusedinthisanalyses,howeveracomparisonisstillfoundinteresting.

First of all, “Dansk Aktionærforening” divides the investment fund into three different risk

categories (low,medium, and high),where Global stocks belong to the low risk group and

DanishandEuropeanstocksbelongtothemediumriskgroup.

The best performing fund in the thesiswas “ValueInvest Danmark, Blue Chip” investing in

globalstocks.Intheanalysisby“DanskAktionærforening”,thisfundisrankednumber7out

of70fundsinthislowriskgroup.5outofthe6betterperformingfundsarenotincludedin

the sampleof this thesis,howeveronDAF’s4thplace comes “ValueInvestDanmark,Global”

whichwasalsoincludedinthesampleofthisthesis.Inthisthesis,thefundshowedsignificant

positive performance in the Jensen index, though with a low R2. The low R2 was though

improved in the mutual index model, and this caused that the fund performed neutrally

comparedtothebenchmark.Thedifference intheresults fromthisanalysisandtheoneby

“DanskAktionærforening”isprobablythattheirperiodofanalysisismuchshorter,sincethey

usereturnsona1‐,3‐,and5‐yearbasis,weightingthenewestreturnshighest.

Forthemutualfundsshowingsignificantnegativeperformanceinthisthesis,theresultsare

prettyconsistentwiththosereachedbyDAF,sincetheyplacethemajorityofthosefundsin

thelowerhalfoftheirranking.

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TheDanishandEuropeanstockswereput in the sameriskgroupbyDAF,namelymedium

risk.Intermsofthisthesis,thebestperformingfundinthosetwoinvestmentcategorieswas

“SEBinvest Danske aktier”. This fund was ranked as the 16th best mutual fund in the risk

groupoutof91fundsintotal.ThemajorityofthefundsrankedbetterintheanalysisbyDAF

werenot included in the sampleof this thesis.However therewas1, and thiswas “Danske

investEuropaSmallCap”,rankedasnumber5intheanalysisbyDAF.Intheanalysisofthis

thesis, “Danske Invest Europa Small Cap” showed significant positive performance in the

Jensen index, though with a low adjusted R2. In the multi‐index model, the adjusted R2

increased,therebycausingthefundtoshowneutralperformance.

However, one really interesting aspect is that “Danske Invest Europa Small Cap” was

appointedas thebestperforming fund in2010witha returnof46,03percent in thatyear,

whichmustbeconsideredimpressive.Though,italsoshowsthatgreatperformancein1year

doesnotguaranteetopperformanceoverlongerperiods,sincethefundwas“only”rankedas

number5inthetotalanalysisbyDAF.Thisalsoexplainswhytheresultsfromthisthesiscan

differ fromtheresults fromtheanalysisbyDAF.Besidesperiodofanalysis,differencescan

alsobecausedbydifferentmethodsofanalysis.

Again, the general picture for this risk group is that themutual funds that has performed

worstintheanalysisofthisthesis,arealsorankedrelativelylowintheanalysisbyDAF.The

three worst performing funds in multi‐index model for Danish and European stocks was,

startingwiththeworst:

1. AlfredBergInvest,Europæiskeaktier (α=‐0,005940)

2. SEBinvestEuropa,Stockpicking (α=‐0,003940)

3. Alm.BrandInvest,Europæiskeaktier (α=‐0,003370)

These three funds where ranked 87, 86, and 65 out of the 91 funds in their sample. The

general tendency is again, that the funds showing significant negative performance in this

thesisisrankedrelativelylowintheanalysisbyDAF.

Therefore, the conclusion must be that there is consistency between the funds that have

performedwellinthisthesis,andthefundsthathaveperformedwellintheanalysisbyDAF.

Whenthinkingabouttheconsistencybetweentheresultsfromthisanalysis,andtheresults

form the analysisbyDAF, one cannot stopwondering about the fact themutual funds that

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haveperformedwellinthisthesiswithaperiodofanalysisfrom2001to2010alsoseemto

haveperformedwellinthe5yearperiodendingin2010byDAF.Someofthisconsistencyis

naturallycausedbythefactthattheentireperiodofanalysisbyDAFisincludedinthelonger

periodofanalysisinthisthesis,andthiswillofcoursecausesomeconsistency.Howeverone

cannotavoidrelatingthistoperformancepersistency,whichconcernswhetherperformance

ofamutualfundinafutureperiodcanbepredictedbaseduponpreviousperiods.Research

hasbeendonewithinthis field,oneof thosefortheDanishmutual fundsbeingChristensen

(2004). His conclusion is that performance persistency is not a general tendency for the

Danishmutual funds, and that it is only seen in a few and one‐off situations. His advice is

therefore that one should not select future mutual funds based on their previous

performance.However, theanalysisbyChristensen(2004) isrelativelyoldwithaperiodof

analysisrangingfrom1996to2003,anditwouldthereforehavebeenreallyinterestingtosee

whetherthesamepictureshowedupinthisperiodofanalysis.However,duetothelimitation

ofthethesis,thishasbeenleftout.

TheFederationofDanish InvestmentAssociates (IFR)publishes statisticson the returnsof

theinvestmentfundsontheirhomepage.Theresultsareshownasanaverageannualreturn

on different time periods of which the 10 year period is themost relevant for this thesis.

Along with the average annual return of the mutual fund, the average annual return of

relevantbenchmarksisgivenaswell.However,anactualperformanceanalysisisnotcarried

out on their homepage, and therefore one has to draw relative conclusions based on the

returnstatistics.

For the mutual funds investing in Danish stocks, the benchmark given by IFR is “OMX

KøbenhavnTotalindeksCap_GI”.Fromthereturnstatistics,itisseenthattheaveragereturn

forthisindexhasbeen8,72%peryearforthe10‐yearperiod.Theyalsohave16investment

funds included in the10yearperiod, and judging from theblinkof aneye, it seemsas if3

fundsshowsignificantlyhigherperformancethanthebenchmark.Thebestperformingfund

in their statistics is SEBinvestDanske aktier, yielding an average annual return of 13,74%.

The performance analysis in this thesis showed that SEBinvestDanske aktierwas the only

significantly positive performing fund for the investment category “Danish stocks”, so

thereforeconsistencyisfoundhere.However,forthetworemainfundsshowinghighannual

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returns in theanalysisby IFR, the resultsdeviated in thisanalysis, since theyboth showed

neutralperformanceinthisthesis.However,oneshouldkeepinmindthatinthisanalysisan

actualperformanceanalysisiscarriedout,andthereforethemorevolatileaportfoliois,the

moredifficultitistoshowsignificantdifferencebetweentheportfolioandthebenchmark.

FortheEuropeanstocks,IFRusesMSCIEuropeGIindexasbenchmark.Thisindexshowsan

averageannualreturnof0,75%peryear.Mostofthereturnsforthemutualfundseemtofall

relatively close or below the return of the benchmark, which is fairly consistent with the

resultsreachedinthisanalysis.Theonlymutualfundwhichpositivelydeviateslargelyfrom

the benchmark is “Danske Invest Europa Small Cap” showing an average annual return of

7,97%whichismuchhigherthanthebenchmark.Thisfundshowedpositiveperformancein

theJensenindex,howeverneutralperformanceinthemulti‐index.Thereasonofthatisthat

theinthemulti‐indexmodel,asmallcapindexhasbeenincluded,whichismoreappropriate

asbenchmarkforthisfund.ItislikelythattheEuropeansmallcapstockhasyieldedamuch

higherreturn in theperiod than theMSCIEurope index,and therefore this fundstandsout

whenitiscomparedtotheMSCIEuropeindex.

However,thegeneralconclusionfortheEuropeanstocksisthattheresultsofthisanalysisare

relativelysimilartotheresultsbyIFR.

Finally, for the Global stocks, IFR shows MSCI World GI as benchmark. This indexes has

yieldedareturnof‐0,65%peryearforthe10‐yearperiod.Themutualfundsperformingwell

in this thesis also showed high returns compared to the benchmark. However, the most

noticeableaspectisprobablythetwo“Skagen”funds,whichshowedthehighestreturnsinthe

statisticsbyIFR.Inthisthesis,oneofthemwasincluded,anditshoweduptohavepositive

performanceintheJensenindex,thoughneutralperformanceinthemulti‐indexmodel.

However,asageneralconclusionitmustbeconcludedthattheresultsreachedinthisanalysis

arerelativelyconsistentwiththestatisticsbyIFR.Itdoesnotseemasifthisanalysisdraws

any significant conclusions, which cannot be supported by the statistics by IFR. However,

therearea few funds that showhighperformance in the statisticsby IFRand thenneutral

performanceinthisanalysis.Though,aspreviouslydiscussed,thiscanbeduetofactorsasthe

volatilityoftheportfolio,choiceofbenchmark,etc.

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Duetothelimitationsofthethesis,therearecertainaspects,whichhavenotbeenanalysed.

Aspreviouslydiscussed, itwouldhavebeenreally interestingtoseewhetherthereturnsof

theinvestmentfundsarepersistentornot.

Inaddition,itcouldalsohavebeeninterestingtoseewhetherthereisapositiverelationship

between the costsof investing ina fundand the returns thismutual fund is able toobtain.

Thiscouldhavebeenobtainedbyregressingthecostsofthefundsagainstthereturnsofthe

funds. If apositive linear relationshipshowsup, itwould indicate thatmutual fundswitha

highcostareabletoearnhigherprofits.

In the thesis, market timing was analysed using the Treynor & Mazuy model, and as

previously discussed the results for the mutual funds showed quite surprising results. It

wouldthereforehavebeeninterestingtouseanothermethodtoanalyseformarkettiming.

OneinterestingmethodcouldhavebeentousetheHenriksson&Merton(1981)model,which

isanon‐parametricmodelandsee if thismodelwouldshowthesamesurprisingresultsas

the method used. However, Christensen (2005) analysed the timing abilities using both

methods,andhisconclusionwasthattheresultswerequitesimilarbetweenthetwomodels.

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7.Conclusion

The theoretical framework of the thesis was based on the theory of efficientmarkets and

CAPM, and therefore the performance of the mutual funds were analysed using theories

withinthatfield.TheperformancemeasurementschosenwastheJensenindexfollowedbya

multi‐indexmodel.Finally,theperformanceofthemutualfundswasseparatedintoselection

andtimingabilitiesusingtheTreynorandMazuymodel.

The period of analysis was 2001‐2010, and the analysis was focused on 60 Danish equity

mutualfundsinvestinginDanishstocks,Europeanstocks,andGlobalstocks.Theresultsfor

eachoftheinvestmentcategorieswillbesummarizedindividually.Itshouldbenoted,thatin

theconclusionsignificantperformancecoversbothsignificanceata5%and10%significance

level.

The results for the mutual funds investing in Danish stocks were relatively similar in the

Jensenindexandthemulti‐indexmodel.Inbothmodels,12ofthe16analysedfundsshowed

neutralperformance,3 funds showed significantnegativeperformance, and1 fund showed

significant positive performance. The second best performing fund was found in this

investmentcategory,namely“SEBinvestDanskeaktier”showingamonthlyoverperformance

of0,3271%comparedtothebenchmark.

Secondly, the results for themutual funds investing inEuropeanstockswereanalysed,and

here therewasasmalldeviationbetweentheresults fromthe Jensen indexandtheresults

fromthemulti‐index.IntheJensenindex,10outofthe21analysedfundsshowedsignificant

negative performance, 10 showed neutral performance, and 1 showed significant positive

performance. However, in the multi‐index model, the number of significantly negative

performingfundsincreasedto13,andnofundsshowedsignificantpositiveperformance.The

increaseinthenumberofnegativeperformingfundswasprimarilycausedbyanincreasein

theadjustedR2.

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Finally,themutualfundsinvestinginGlobalstockswereanalysed,andherelargerdeviation

betweentheJensenindexandthemulti‐indexmodeloccurred.Outofthe23analysedfunds,

10showedsignificantnegativeperformance,9 showedneutralperformance, and4 showed

significantpositiveperformance. Inthemulti‐indexmodel, thesenumberschangedto15,7,

and 1, again caused by a better ability of the benchmarks to explain the variation in the

returnsoftheinvestmentfunds.

Thebestperformingfundintheentiresamplewasfoundinthisinvestmentcategory,namely

“Valueinvest Danmark, Blue Chip” showing a monthly overperformance of 0,3376%

comparedtothebenchmarks.

The general conclusion on the twomodels is that the performance of themutual funds is

weakened in themulti‐indexmodel compared to the Jensen index.Aspreviouslydiscussed,

this is caused by the fact that more benchmarks are able to capture a larger part of the

variationinthereturnsoftheinvestmentfunds.

Totally,inthemulti‐indexmodel,31fundsshowedsignificantnegativeperformance,27funds

showed neutral performance, and only 2 showed significant positive performance. The

generalconclusionistherefore,thattheDanishmutualfundsinvestinginthose3investment

categorieshavenotbeenabletooutperformthemarket,whichismuchinlinewiththetheory

ofefficientmarkets.

Inthelastpartoftheanalysis,theTheynorandMazuymodelwasusedtoanalysewhetherthe

mutualfundspossessedtheabilitytotimethemarket.Theresultwasthat8mutualfundsall

investinginDanishstocksshowedsignificanttimingabilities,thoughthiscausedtheselection

abilityoftheseinvestmentfundstolookworse.Fortheremaining52fundsnosignsoftiming

abilitieswerefound.

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Appendix1:Jensen’salpha

IfJensen’salphashowuptobesignificant,thefollowingcolourshasbeenusedtomarkit:

= Significantata5%level = Significantata10%level

Danishstocks

Results

Assumptions

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Europeanstocks

Results

Assumptions

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Globalstocks

Results

Assumptions

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Appendix2:Multi‐indexmodel

Ifalphashowsuptobesignificant in themulti‐indexmodel, the followingcolourshasbeen

usedtomarkit:

= Significantata5%level = Significantata10%level

Danishstocks

Significantbenchmarks

Assumptions

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Results

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Europeanstocks

Significantbenchmarks

Assumptions

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Results

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Globalstocks

Significantbenchmarks

Assumptions

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Results

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Appendix3:Markettiming

Danishstocks

Results

Assumptions

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Europeanstocks

Results

Assumptions

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Globalstocks

Results

Assumptions

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Appendix4:BenchmarksoftheinvestmentfundsThetablebelowshowthebenchmarksthattherespectivefundsuse: