CFA Level 1 Notes 2017

CFA Level 1 Notes 2017 Page 2 to 191 CFA Level 1 Ethics Summary Notes Page 192 to 197

Ethicalconductimprovesoutcomesforstakeholders,bybalancingselfinterestwithimpactonothersCodeofethics rulesandstandardsthatrequireminimumlevelofethicalbehaviourProfessionalcodeofethics wayforprofessiontocommunicatetopublicthatitsmemberswilluseknowledge/skilltoserveclientsinhonestandethicalmanneràincreasepublicconfidenceandtrustthatmemberswillactethically

Challengestoethicalbehaviour:• Overestimatingownethicalcharacteràoverconfidencebias• Consideringonlyneartermconsequences• Lettingsituational(external)influencessuchaspeerpressure,loyaltyundulyaffect

decisionsandbehaviouro Overateethicalqualityofownbehaviourandoveremphasizeimportanceof

ownpersonaltraitsàsituationalinfluencesaremoreimportantthanpersonal/internal

Unethicalbehaviour(suchasprovidingmisleadinginfo)couldaffectallocationofcapitalraisedànegativeconsequencesonallstakeholdersineconomyEthicalprinciplesoftensethigherstandardofbehaviourthanlawsandregulations.Newlawsresultfrominstancesofunethicalbehaviour(e.g.SecuritiesAct1933,Glass-StegalAct,SecuritiesAct1934,Sarbanes-Oxleylaws(EnronandWorldcom),Dodd-FrankAct(GFC).Ethicaldecisionsrequiremorejudgmentandconsiderationofimpactofbehaviouronstakeholdersthanlegal.

Ethicaldecisionmakingframework:• Identify:facts,stakeholders,duties,ethicalprinciples,conflictsofinterest• Consider:situationalinfluences,additionalguidance(e.g.friends,compliance

department),alternativeactions(shortandlongtermeffects)• Decideandact• Reflect:evaluateoutcomesofactions-wasoutcomeasanticipated?

CFAInstituteProfessionalConductProgram(PCP)• PCPinconjunctionwithDRCareresponsibleforenforcementoftheCodeand

Standards• CoveredbyCFAInstituteBylawsandRulesofProcedureforProceedingRelatedto

ProfessionalConduct• Basedonprinciplesoffairnessofprocesstomembersandmaintaining

confidentialityofproceedings• AllCFAmembersandcandidatesenrolledinCFAarerequiredtocomplywithcode

andstandards• DisciplinaryReviewCommittee(DRC)hasresponsibilityforprogramand

enforcementofCodeandStandards volunteercommittee• INQUIRY:CFAInstituteProfessionalConductstaffconductsinquiresrelatedto

professionalconductàcanbepromptedby4things:o Self-disclosurebymembersonProfessionalConductStatementsof

involvementincivillitigationorcriminalinvestigationo WrittencomplainstoPCPaboutmember’sconducto Evidenceofmisconductmymemberreceivedthroughpublicsourceso ReportbyCFAexamproctorofpossibleviolationo AnalysisofexammaterialandmonitoringsocialmediabyCFAInstitute

• INVESTIGATION:Onceinquirybegins,staffmayrequest(inwriting)explanationfrommemberandmayinterviewthemember,interviewcomplainant/3 dpartyandcollectdocuments/recordsrelevanttoinvestigation

o Nodisciplinarysanctionsappropriateo Issuecautionarylettero Disciplinememberàmembermustacceptorrejectsanction.Ifrejected,

matterreferredtoDRCofMembers.§ Sanctionsincludepubliccensure,membershipsuspension,

revocationofcharter

6Codeofethics1. Actwithintegrity,competence,diligenceandrespect2. Placeintegrityofprofessionandclientsabovepersonalinterests3. Reasonablecareandexerciseindependentprofessionaljudgmentwhenmaking

investmentrecommendations4. Practiceandencourageotherstopracticeinethicalmanner5. Promoteintegrityandviabilityofglobalcapitalmarkets6. Maintainandimproveprofessionalcompetence

7standardsofprofessionalconduct1. PROFESSIONALISM

A. Knowledgeofthelaw(includingcodeofethicsandstandardsofprofessionalconduct) intheeventofaconflict,thestricterlaw,ruleorregulationapplies.

B. Independenceandobjectivity notofferoracceptgiftorcompensationthatwouldcompromiseindependence/objectivity

C. Misrepresentation analysis,recommendationsoractions§ Creditingsourcenotrequiredwhenusingstatistics,tablesand

projectionsfromrecognisedfinancialandstatisticalreportingservices

D. Misconduct notengageinconductinvolvingdishonesty,fraud,deceit2. INTEGRITYOFCAPITALMARKETS

A. Materialnonpublicinfo thatcouldaffectvalueofinvestment§ Publiconceitisannouncedtothemarketplace§ Mosaictheory=reachinginvestmentconclusionthroughanalysisof

publicinfo+non-materialnonpublicinfo§ Membersshouldmakeefforttoachievepublicdisseminationbythe

firmofinformationtheypossess.Firmsshouldreviewemployeetradesandmaintainwatchlists.

B. Marketmanipulation notdistortpricesorartificiallyinflatetradingvolumeàonlyifthereisINTENTtomislead.

3. DUTIESTOCLIENTSA. Loyalty,PrudenceandCare actinbenefitofclient,placeclientsinterest

beforeemployer’s/owninterest§ Submitatleastquarterlystatementsshowingsecuritiesincustody

andalldebits,creditandtransactions.Notvoteonallproxies.B. FairDealing dealingwithclientswhenmakinganalysis,recommendations,

engagement§ E.g.donottakesharesofanoversubscribeIPO

C. Suitability riskandreturnobjectives,suitableinvestments,consistentwithobjectivesandconstraintsofportfolio

§ Membersgatherinfoatbeginningofrelationshipintheformofaninvestmentpolicystatement(IPS)

§ Iftradehasmaterialimpact,youcanupdateIPSsoclientacceptschangedriskprofilethatwouldpermitthetrade.Ifclientwont

acceptthechangedIPS,managermayfollowfirmpolicywhichmayallowclient-directedaccount

D. Performancepresentation fair,accurateandcomplete§ Membermustnotstateorimplyabilitytoachieverateofreturn

similartothatachievedinthepast§ Membershouldpresentperformanceofweightedcompositeof

similarportfoliosratherthansingleaccount§ Includeterminatedaccountsandstatewhenterminated

E. Preservationofconfidentiality keepinfoaboutclients(currentandpast)confidentialunless3exceptions:illegalactivities,disclosurerequiredbylaw,clientpermitsdisclosure

4. DUTIESTOEMPLOYERSA. Loyalty actforbenefitofemployerandnotdivulgeconfidentialinfo

§ Norequirementtoputemployerinterestsaheadoffamilyandpersonalobligations

§ Whenleavinganemployer,membermustcontinuetoactuntilresignationiseffective

§ Violationsincludemisappropriationoftradesecretsandclientlists,misuseofconfidentialinfo,solicitingemployer’sclients,self-dealing.

B. AdditionalCompensationArrangements notacceptgifts,benefitsthatmightcreateconflictofinterestunlessobtainwrittenconsentfromallpartiesinvolved

§ Ifclientoffersbonusdependingonfutureperformance,thisisancompensationarrangementàrequireswrittenconsentinadvance

§ Ifclientoffersbonusdependingonpastperformance,thisisagiftàrequiresdisclosuretoemployertocomplywithStandardI(B)IndependenceandObjectivity

C. ResponsibilitiesofSupervisors makesurepeoplecomplywithlaws,regulationandCodeandStandards

5. INVESTMENTANALYSIS,RECOMMENDATIONSANDACTIONSA. DiligenceandReasonableBasis reasonablebasissupportedbyresearch

andinvestigationforanalysis,recommendation§ Applicationdependsoninvestmentphilosophyadheredto,

members’rolesininvestmentdecisionmakingprocess,andresourcesandsupportprovidedbyemployer

§ Considerationsincludeeconomicconditions,firmsfinancialresults/operatinghistory,feesandhistoricalresults,limitationsofquantmodels,peergroupcomparisonsforvaluationareappropriate

§ Membersshouldencouragefirmtoadoptpolicyforperiodicinternalreviewofqualityof3 dpartyresearch

B. CommunicationwithClients disclosebasicprinciplesofinvestmentprocessandconstructportfoliosandanychangesthatmightmateriallyaffectprocesses,significantlimitationsandrisks,identifyingimportantfactorsandcommunicatethem,distinguishbetweenfactandopinion.

§ Expectationsbasedonmodeling/analysisarenotfacts§ Communicategains/lossesintermsoftotalreturns§ Explainlimitationsofmodel/assumptionsusedandofthe

investmentitself e.g.liquidityandcapacityC. RecordRetention developandmaintainrecordstosupportanalysisand

recommendationwithclients(e.g.documentingdetailsofconvo)

§ Memberwhochangesfirmsmustre-createanalysisdocumentationsupportingrecommendationandmustnotrelyonmaterialcreatedatpreviousfirm

§ Ifnoregulatorystandards/firmpoliciesinplace,recommends7-yearminimumholdingperiod

6. CONFLICTOFINTERESTA. DisclosureofConflicts mattersthatcouldimpairindependenceand

objectivityorinterferewithdutytoclientsandemployer§ E.g.ownershipofstockincompanythatrecommending§ E.g.compensation/bonus/commissions

B. PriorityofTransactions clients/employerspriorityoverown§ LimitationsonemployeeparticipationinequityIPO,private

placement§ Blackoutperiod nopersonalpurchase/saleofsecurityinadvance

ofclient/employerC. ReferralFees compensationreceivedorpaidtoothersfor

recommendationofproducts/services7. RESPONSIBLEASACFAINSTITUTEMEMBER/CANDIDATE

1. ConductasParticipantsinCFAInstitutePrograms notcompromisereputationorintegrityofCFA

§ e.g.examcheating,improperlyusingdesignation,notrevealconfidentialinforegardingCFA,misrepresentinginfoonProfessionalConductStatement(PCS)

2. ReferencetoCFAInstitute,DesignationandProgram notmisrepresentorexaggeratemeaning/implications

§ MembersmustsignthePCSannually,andpayCFAmembershipduesannuallyàiffailtodothis,personwillnolongerbeanactivemember

READSTANDARDSOFPRACTICEHANDBOOK11 EDITION 15%ofquestionsbasedfromthisbookand2shortreadingsonGIPSINTRODUCTIONTOTHEGLOBALINVESTMENTPERFORMANCESTANDARDS(GIPS)

GIPSareasetofethicalprinciplesbasedonstandardised,industry-wideapproach• InvestmentfirmscanvoluntarilyfollowGIPSinpresentationofhistoricalresultsto

clientsàStandardsseektoavoidmisrepresentationofperformance• Onlyinvmgmtfirmsthatactuallymanageassetscanclaimcompliance• Createdtomakeiteasiertocomparedifferentinvestmentmanagementfirmsà

GIPSallowclientstomoreeasilycompareperformanceAcompositeisanaggregationofoneormoreindividualdiscretionaryportfoliosrepresentingasimilarinvestmentstrategyorobjective e.g.largecapgrowthstocks.

• Reportingoncompositesgivesclientsinfoaboutfirm’ssuccessinmanagingvarioustypesofsecuritiesorresultsforvariousinvestmentstyles

• Mustincludeallactualfeepayingdiscretionaryportfolios(pastandpresent)o Actual=nodummyportfolio.Feepaying=nocharity.Discretionary=

controloverstockselection

• Firmshouldidentifycompositebeforeperformanceisknownàpreventsfirmfromchoosingtoincludecompositeinordertocreatecompositewithsuperiorreturns

o Mustbedeterminedonex-antebasis(i.e.beforeperiodwhichcomposite’sperformancewillbecalculated)àpre-establishedcriteria

OncefirmclaimsGIPScompliance,theymustvoluntarilyhirea3rdpartytoperformverification

• VerificationofGIPScomplianceisoptional,butifissuedby3 dpartythenreportmustbeissuedwithrespecttowholefirm(firm-widebasis)

• Mustattest2things:o Whetherfirmhascompliedwithallcompositeconstructionrequirementso Firmsprocessesandproceduresareestablishedtopresentperformance

accordingwithcalculationmethodology,datarequirementsandformatrequiredbyGIPS

GIPSObjectivesàGoalsoftheGIPSExecutiveCommittee:

• Obtainglobalacceptanceofcalculationandpresentationstandardsinfair,comparableformatwithfulldisclosure

• Ensureconsistent,accurateperformancedatainareasofreporting,records,marketingandpresentations

• Promotefaircompetitionamonginvmgmtfirmsinallmarketswithoutunnecessaryentrybarriersfornewfirms

• Promoteglobal“selfregulation”KeyCharacteristicsofGIPS

• Toclaimcompliance,firmsmustdefineits“firm”àreflectthedistinctbusinessentitythatisheldouttoclientsandprospectsasthefirm

• Ethicalstandardsforperformancepresentationwhichensurefairrepresentation• Includeallactualfee-paying,discretionaryportfoliosincompositesformin5years

orsincefirm/compositeinception.o After5yearsofcompliantdata,firmmustaddannualperformanceeach

yeargoingforwarduptomin10years.• Firmsrequiredtousecertaincalculationandpresentationstandardsandmake

specificdisclosures• GIPScontainbothrequiredandrecommendedprovisionsàfirmsencouragedto

adoptrecommendedprovisions• Encouragedtopresentallpertinentadditionalandsupplementalinfo• FollowlocallawswheninconflictwithGIPS,butdisclosetheconflict• SupplementalprivateequityandrealestateprovisionscontainedinGIPSaretobe

appliedtothoseassetclasses Requirement RecommendationDefinitionofthefirm • ApplyGIPSonfirm-widebasis

• Firmdefinedasdistinctbusinessunit• Totalfirmassetsincludetotalmktvalueof

discretionaryandnon-discretionaryassets,includingfee-payingandnon-fee-payingaccounts

• Includeassetperformanceofsub-advisors,aslongasfirmhasdiscretionoversub-advisorselection

• Includebroadestdefinitionofthefirm,includingallgeographicalofficesmarketedundersamebrandname

• Iffirmchangesitsorganisation,historicalcompositeresultscannotbechanged

Documentpoliciesandprocedures

• Document,inwriting,policiesandproceduresfirmusestocomplywithGIPS

Claimofcompliance • Compliancestatement“ABChasprepared…incompliancewithGIPS”

• Nosuchthingaspartialcompliance• Nostatementsreferringtocalculation

usedincompositepresentationorperformanceofindividualclientasbeing“inaccordancewithGIPS”(unlesscomplaintfirmisreportingresultsdirectlytoclient)

Firmfundamentalresponsibilities

• Compliantpresentationtoallprospects(withinprevious12months)

• Compositelistanddescriptiontoallprospectsthatmakearequest.Listdiscontinuedcompositesformin5years

• Ifjointlymarketingwithanotherfirm,clearlydefinefirmseparationifnoncompliant

Firmmustpresentmin5yearscompliantperformanceunlessfirmorcompositehasbeeninexistencelessthan5yearsà“sinceinception”mustbepresented.

• Afterinitialcompliantpresentation,1yearofcompliantperformancemustbeaddedeachyeartoarequiredminimumhistoryof10years.

• Firmsmaypresentperiodsofnoncompliantperformanceaslongasnononcompliantperformanceifpresentedforanyperiodafter1s January2000

FirmsthatpreviouslypresentedperformanceincompliancewithparticularCountryVersionofGIPS(CVG)mayclaimGIPScompliancepriorto1Jan2006.

• Mustcontinuetoincludedatauntilmin10yearscompliantperformancepresented• WherecountryspecificregulationsconflictwithGIPS,firmsmustfollowcountry-

specificregulationsbutmustalsodisclosurenatureofconflictwithGIPS9majorsectionsoftheGIPSstandards

0. FundamentalofComplianceo Definitionoffirm

§ DistinctbusinessentityheldtoclientsclaimingGIPScompliance§ Totalfirmassets=fairvalueofallassetsincludingdiscretionary,

nondiscretionary,fee-payingandnon-fee-payingaccountso Documentationoffirmpolicies&procedureswithrespecttoGIPS

compliance§ E.g.makecompletelistofcompositestoanyprospectiveclientà

listmustincludecurrentandanyterminatedcompositeswithinlast5years

o ComplyingwithGIPSupdateso Claimingcomplianceinappropriatemanner

§ Notclaimpartialcompliance.§ NotstatespecificcalculationisGIPScompliant

o Appropriateverificationstatementwhen3 dpartyverifierisemployed1. Inputdata

o Consistentinordertoestablishfullandcomparableperformance2. Calculationmethodology

o Uniformityrequiredsoresultsarecomparableo Mustnotuseestimatedtradingexpenses

3. Compositeconstructiono Meaningful,asset-weightedcompositewithperformancebasedon

performanceof1ormoreportfoliowithsameinvestmentstrategy4. Disclosure

o E.g.claimofcompliance,currency,feeschedule,benchmarkdescription,compositecreationdate

5. Presentationandreportingo InvestmentperformanceaccordingtoGIPS.Otherfirm-specificinfonot

specificallyrequiredbyGIPSshouldbeincludedwhenappropriate6. Realestate

o ProvisionsapplytoALLrealestateinvestment(land,buildings)regardlessoflevelofcontrolfirmhasoverit,orwhetherassetisproducingrevenueorleverageinvolved

o REITS,Commercialmortgagebackedsecurities(CMBS)andprivatedebtinstrumentsarenotconsideredrealestateandmustfollowsections0-5

7. Privateequityo PEInvestmentsmustbevaluedaccordingtoPrivateEquityValuations

Principlesunlessinvestmentisopen-endedorevergreenfundàfollowGIPSo Notpubliclytradedregardlessofstageofbusinessdevelopment(e.g.

venturecapital,ownershipofpreviouslypubliccompany,mezzaninefinancing,limitedpartnershipshares,andfund-of-fundsinvestments

8. Wrapfee/SMAportfolioo Section0to5aresupplementedorreplacedbyrequirementsspecificinthis

section

TimeValueofMoney• Thevalueoftheinvestment’scashflowsmustbemeasuredatsomecommonpoint

intimeàatendofinvestmenthorizon(FV)orbeginningofinvestmenthorizon(PV)InterestratesareameasureoftheTVM.

• Equilibriuminterestratesaretherequiredrateofreturnforaparticularinvestment.• Interestratesaretheopportunitycostofcurrentconsumption• DiscountrateforcalculatingPVoffuturecashflows• Realrisk-freerateisatheoreticalrateonasingle-periodloanwhenthereisno

expectationofinflationàinvestorsincreaseinpurchasingpower(afteradjustingforinflation)

o T-billratesarenominalrisk-freeratesbecausetheycontainaninflationpremium

o Nominalrisk-freerate=realrisk-freerate+expectedinflationrate

Typesofsecurityrisks:

• Defaultrisk riskthatborrowerwillnotmakepromisedpaymentsintimelymanner• Liquidityrisk riskofreceivinglessthanfairvalueforinvestmentifitmustbesold

forcashquickly• Maturityrisk long-termbondsaremorevolatilethanshort-termbondsandhence

havematurityriskrequiringamaturityriskpremiumEachriskfactorisassociatedwithriskpremiumthatisaddedtonominalriskfreeratetoadjustforgreaterdefaultrisk,lessliquidityandlongermaturityrelativetoveryliquid,short-term,defaultriskfreeratesuchasthatonT-bills

• Requiredinterestrate=nominalriskfreerate+defaultriskpremium+liquiditypremium+maturityriskpremium

o Note:nom na r skfreerate=rea r skfreerate+ nf at onprem umStatedannualintrate–quotedintratethatdoesnotaccountforcompoundingEffectiveannualrate–annualrateofreturnearnedafteradjustmentsmadefordifferentcompoundingperiods.

• Therateofinterestthatinvestorrealizesasresultofcompounding.(i.e.8%savingsratecompoundedquarterlyassupposedto2%perquarter)

• EAR=(1+periodicrate)m–1o M=numberofcompoundingperiodsperyearo Periodicrate=statedannualrate/m

• EARincreasesatadecreasingrateascompoundingfrequencyincreases• Itisnecessarywhencomparinginvestmentthathavedifferentcompoundingperiodsà

apples-to-applesratecomparison• Continuouscompounding–ifnumberofcompoundingperiodsbecomeinfinite

o FVN=PV(ern)

o er-1=EARFuturevalue amountwhichcurrentdepositwillgrowovertimewhenplacedinaccountpayingcompoundinterestàcompoundvalue

• FV=PV(1+I/Y)No PV=moneyinvestedtodayo I/Y=rateofreturnpercompoundingperiodo N=totalnumberofcompoundingperiodso (1+I/Y)Nreferredtoasfuturevaluefactor

Presentvalue–today’svalueofacashflowthatistobereceivedatsomepointinthefutureàamountinvestedtodayatagivenratetoendupwithspecificFV

• discountingfuturecashflowsbacktothepresent• interestrate=discountrate,oppcost,rateofreturn,costofcapital• PV=FV/(1+I/Y)N

o KnownaspresentvaluefactorAnnuity streamofequalcashflowsthatoccuratequalintervalsoveragivenperiod

• Ordinaryannuity cashflowsoccurringatendofeachcompoundingperiod• Annuitydate cashflowsoccurringatbeginningofeachcompoundingperiod

𝐹𝑉𝐴𝑛𝑛𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇1 + 𝑟 1 − 1

𝑟

𝑃𝑉𝐴𝑛𝑛𝑢𝑖𝑡𝑦 = 𝑃𝑀𝑇1 − 1 + 𝑟 31

𝑟

• Annuitydueà1lessdiscountingperiodsince1s CFisatt=0andhencealreadyitsPV

o PVAnnuityDue=PVOrdinaryAnnuity*(1+r)§ PVofannuitydue>PVordinaryannuity

Perpetuity financialinstrumentthatpaysfixedamountofmoneyatsetintervalsoveraninfiniteperiodoftime.

• i.e.perpetualannuity(i.e.setofnever-endingsequentialcashflows)• e.g.BritishConsolbondsormostpreferredstock

𝑃𝑉𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 =𝑃𝑀𝑇𝑟

IncreaseincompoundingfrequencyàincreaseeffectiveintrateàincreaseFVàdecreasePVLoanamortisation processofpayingoffaloanwithaseriesofperiodloanpayments,wherebyaportionofoutstandingloanispaidoff,oramoritsed,witheachpayment.e.g.Amortisationschedulefor10kloanat10%for5yearsPeriod BegBal

PMT Interest

(begbal*intrate)Principal(PMT interest)

EndBal(begbal principal)

1 10,000 $2637.97 1000 1637.97 8362.032 8362.03 $2637.97 836.203 1801.767 6560.2633 6560.263 $2637.97 656.03 1981.94 4578.324 4578.32 $2637.97 457.83 2180.14 2398.185 2398.18 $2637.97 239.82 2398.18 0

𝐶𝑜𝑚𝑝𝑜𝑢𝑛𝑑𝐴𝑛𝑛𝑢𝑎𝑙𝐺𝑟𝑜𝑤𝑡ℎ𝑅𝑎𝑡𝑒 𝐶𝐴𝐺𝑅 =𝐶𝑢𝑟𝑟𝑒𝑛𝑡𝑃𝑟𝑒𝑣𝑖𝑜𝑢𝑠

B/D− 1

Cashflowadditivityprinciple PVofanystreamofcashflowsequalssumofPVofthecashflows

DiscountedCashFlowApplicationsNetPresentValue(NPV)ofaninvestmentprojectisthePVofexpectedcashinflowsassociatedwithprojectlessPVofprojectsexpectedoutflows,discountedattheappropriatecostofcapitalInternalRateofReturn(IRR)–discountrateforwhichNPVofinvestmentis0

• i.e.rateofreturnthatequatesthePVofaninvestment’sexpectedbenefits(inflows)withthePVofitscosts(outflows)

• Givesyousenseofreturnonproject-‘internal'becausedependsoninvestmentCF• Discountrate<IRR=+veNPV• Discountrate>IRR=-veNPV• IfDiscountrate(i.e.hurdlerate)=IRRà0NPV

NPVDecisionRule ifprojecthaspositiveNPV,thisamountgoestofirm’sshareholders

• AcceptprojectswithpositiveNPVàincreaseshareholderwealth

• RejectprojectswithnegativeNPVàdecreaseshareholderwealth• When2projectsaremutuallyexclusive,projectwithhigherpositiveNPVshouldbe

acceptedIRRdecisionrule providesanalystwithanalysinganinvestmentresultintermsofrateofreturn.IRRusescostofcapitalasthehurdlerateàisthereturnoftheprojecthigherthanthecostofcapital?

• AcceptprojectswithIRR>firm’srequiredrateofreturn• RejectprojectswithIRR<firm’srequiredrateofreturn

IRRandNPVrulesgivesameaccept/rejectdecisionwhenprojectsareindependent

• Ifmutuallyexclusive(i.e.conflict)àusewhichonegiveshigherNPVHoldingperiodreturn(HPR)-%changeofvalueofaninvestmentovertheperioditisheld

𝐻𝑃𝑅 =𝑒𝑛𝑑𝑖𝑛𝑔𝑣𝑎𝑙𝑢𝑒 + 𝐶𝐹𝑟𝑒𝑐𝑒𝑖𝑣𝑒𝑑

𝑏𝑒𝑔𝑖𝑛𝑛𝑖𝑛𝑔𝑣𝑎𝑙𝑢𝑒− 1

2typesofportfolioreturnmeasurementtools:

1. MoneyWeightedReturn(=IRR)-internalrateofreturnonaportfoliotakingintoaccountallcashinflowsandoutflowsàPV f ows=PVou f ows

o Inflows=beginningvalueofaccountandalldepositsintoaccounto Outflows=endingvalueandallwithdrawalso ReturnsdependsontimingandamountofCF.o AppropriateifPMhascontroloftimingandamountofinvestment

2. TimeWeightedReturnàmeasurescompoundrateofgrowthof$1initiallyinvestedintheportfolioaspecifiedperformancehorizon

o Processofaveragingasetofvaluesovertimeo Preferredmethodofperformancebecauseitisnotaffectedbythetimingof

cashinflowsandoutflowso Iffundscontributedtoportfolioatfavourabletimeàmoney

weightedreturn>time-weightedrateofreturn§ Timeweightedremovesthisdistortion

o Returnsdonotdependontimingandamountofcashflow.o AppropriateifPMdoesnotcontroltimingandamountofinvestmento [(1+rt1)×(1+rt2)×…(1+rtn)]-1

AnnualizedTimeweightedreturn=[(1+HPR )(1+HPR2)] / -1

• N=numberofper odsàonlydo1/nifannualisedMoneymarketisthemarketforshort-termdebtinstruments.

• T-bills(TreasuryBills)arepurediscountinstrumentsandquotedonabankdiscountbasis

5differentyieldmeasures BankDiscountYield,HoldingPeriodYield,MoneyMarketYield,EffectiveAnnualYield,BondEquivalentYield.BankDiscountYield(BDY) expressesthedollaramountfromthefacevalueasafractionoffacevalue,notthemarketprice.

• Annualizesusingsimpleinterestandignoreseffectsofcompoundinterest• Basedon360-dayyearratherthan365

𝐵𝐷𝑌 =𝐹𝑎𝑐𝑒𝑉𝑎𝑙𝑢𝑒 − 𝑝𝑢𝑟𝑐ℎ𝑎𝑠𝑒𝑝𝑟𝑖𝑐𝑒

𝐹𝑎𝑐𝑒𝑉𝑎𝑙𝑢𝑒×

360𝑑𝑎𝑦𝑠𝑡𝑖𝑙𝑙𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦

D scou = V u c ase ce

HoldingPeriodYield(HPY) totalreturnaninvestorearnsbetweenpurchasedateandthesaleormaturitydate(i.e.returnforholdingsecurityofaperiod)

• Akaholdingperiodreturnàactualreturninvestorreceivesifheldtomaturity

𝐻𝑃𝑌 =𝑝𝑟𝑖𝑐𝑒𝑎𝑡𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦 − 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 + 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡𝑝𝑎𝑦𝑚𝑒𝑛𝑡

𝑖𝑛𝑖𝑡𝑖𝑎𝑙𝑣𝑎𝑙𝑢𝑒

EffectiveAnnualYield=(1+HPY)365/ 1

• AnnualizedHPYonbasisof365daysincorporatingeffectsofcompoundingMoneymarketyield(akaCDequivalentyield)istheannualizedHPY,assuminga360dayyearàmakesyieldonT-billcomparabletoyieldquotesforinterest-bearingmoneymarketinstrumentsthatpayinterestona360daybasis

o Annualizedyieldbasedonpriceof360dayyearthatdoesnotaccountforcompoundingàassumessimpleinterest

o MMY>BDY

𝑀𝑀𝑌 = 𝐻𝑃𝑌×360

#𝑑𝑎𝑦𝑠𝑡𝑜𝑚𝑎𝑡𝑢𝑟𝑖𝑡𝑦

𝑟QQ =360×𝑟RS

360 − (𝑡×𝑟RS)

Bondequivalentyieldrefersto2xthesemiannualyield

• =2*[(1+yieldonannualbond)0 5)-1]• UsedbecauseBondspaycouponssemiannually• Note:EAY>BEY

StatisticalConceptsandMarketReturnsCentraltendency-Provideanindicationofinvestment’sexpectedreturn

• Arithmeticmean,geometricmean,weightedmean,medianandmode.Dispersion–indicatetheriskinessofaninvestment

• Range,absolutedeviation,varianceLackofsymmetry(skewness)andwhichdistributionispeaked(kurtosis)Descriptivestatistics summariseimportantcharacteristicsoflargedataInferentialstatistics procedurestomakeforecasts,estimatesandjudgmentsaboutalargesetofdataonbasisofstatisticalcharacteristicsofasmallerset(sample)Population setofallpossiblemembersofastatedgroup.Sample subsetofthepopulationofinterest4typesofmeasurementscales

• Nominalscales dataputintocategoriesthathavenoparticularorderàonlynamesused(e.g.bond1,bond2etc…)

• Ordinalscales dataputintocategoriesthatcanbeorderedwithrespecttosomecharacteristics(e.g.ranking1000smallcapgrowthstocksbyperformance)

• Intervalscales providesrelativerankingplustheassurancethatdifferencesbetweenscalevaluesareequal(e.g.temperature degrees)

• Ratioscales providesrankingandequaldifferencesbetweenscalevaluesandalsoatruezeropointastheorigin(e.g.measureofmoneyin$)

o AbsolutezeroParameter measureusedtodescribecharacteristicofapopulation(e.g.meanreturnandSDofreturnSamplestatistic measureacharacteristicofasampleFrequencydistribution summarisedatabyassigningspecificgroupsorintervalsandmaybemeasuredusinganytypeofmeasurementscale

• Absolutefrequency=actualnumberofobservationsthatfallwithinaninterval• Modalinterval=intervalwiththegreatestfrequency

𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝐹𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑓𝑟𝑒𝑞𝑢𝑒𝑛𝑐𝑦𝑜𝑓𝑒𝑎𝑐ℎ𝑟𝑒𝑡𝑢𝑟𝑛𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙

𝑁𝑢𝑚𝑏𝑒𝑟𝑜𝑓𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛𝑠

Histogram presentationofabsolutefrequencydistribution

• AllowsustoquicklyseewhereobservationsareconcentratedArithmeticmeanistheonlymeasureofcentraltendencyforwhichsumofdeviationsfromthemeanis0àarithmeticmean=sumofobservations/#observations

• Usedtoestimateavgreturnover1-periodtimehorizonWeightedmeanrecognisesdifferentobservationsmayhavedisproportionateinfluenceofthemean.

• ReturnofportfolioisweightedaverageofreturnsofindividualassetsintheportfolioMedianmaybeabettermeasureofcentraltendencybecauseitisnotaffectedbyextremevaluesUnimodal–whendistributionhas1valuethatappearsmorefrequentlyGeometricmean–usedwhencalculatingavginvestmentreturnsovermultipleperiodsorwhenmeasuringcompoundgrowthrates

𝑅Y [ 1 + 𝑅B 1 + 𝑅[ 1 + 𝑅D B/D 1• Geometricmeanalwayslessorequaltoarithmeticmeanàdifferenceincreasesas

dispersionofobservationincreases• Sinceannualreturnsarecompoundedeachperiod,GMisappropriatemeasureofpast

performanceàgivesaverageannualcompoundgrowtho GMbesttoestimatemulti-yearreturns

Harmonicmean–canbeusedtofindaveragepurchaseprice

𝐻𝑎𝑟𝑚𝑜𝑛𝑖𝑐𝑚𝑒𝑎𝑛𝑁

1𝑥^+ 1𝑥[+ 1𝑥D

IfreturnsarevariableàArithmeticmean>GeometricMean>HarmonicMean

• Basisforclaimedbenefitofdollarcostaveraging• Ifconstant,AM=GM=HM• ThegreaterthevariabilityàthemoreAMwillexceedGM

Quantile valueatorbelowwhichastatedproportionofdatainadistributionline

• Quartile(quarters),Quintile(5ths),Decile(10ths),Percentile(hundredths)• Quantilesandmeasureofcentraltendency=measureoflocation• 𝐿` (𝑛 + 1) `

Baa

n observationy percentile

Dispersion–variabilityaroundthecentraltendencyMeanabsolutedeviation(MAD)–avgofabsolutevaluesofdeviationsofindividualobservationsfromarithmeticmean

𝑀𝐴𝐷 = |𝑋^ − 𝑚𝑒𝑎𝑛|D

^dB𝑛

Standarddeviation>MAD

Standarddeviation= =(return–mean)2

Chebyshev’sinequality–%ofobservationsthatliewithinkstandarddeviationsofthemeanisatleast1–1/k2forallk>1

• Basisfornormaldistribution(i.e.75%ofobservationsliewithin+/ 2SD’sfrommean)

• UsedtomeasuremaxamountofdispersionregardlessofdistributionshapeCoefficientofvariation(CV) amountofdispersioninadistributionrelativetoit’smeanàrelativedispersion

𝐶𝑜𝑒𝑓𝑓𝑖𝑐𝑖𝑒𝑛𝑡𝑜𝑓𝑉𝑎𝑟𝑖𝑎𝑡𝑖𝑜𝑛 = 𝑆𝐷𝑥𝐴𝑣𝑔𝑥

• Enablesustomakedirectcomparisonofdispersionacrosssetsofdataàusedtomeasurerisk(variability)perunitofexpectedreturn(mean)

• HigherCV=higherriskSharperatio measuresexcessreturnperunitofrisk

𝑆ℎ𝑎𝑟𝑝𝑒𝑟𝑎𝑡𝑖𝑜 = 𝑟f − 𝑟g𝜎f

rp=portfo oreturn=meanreturn

rf=r sk-freereturnσp=SDofportfo o

• Returnonportfolio risk-freereturn=excessreturnàmeasuresextrareward

investorreceivesforexposingthemselvestorisk• Sharperatioof0.14=investmentearned0.14%ofexcessreturnperunitofrisk• 2limitationsofsharperatio

o If2portfolioshave-veSharperatio,itisnotnecessarilytruethathigherSharperatioimpliessuperiorrisk-adjustedreturns

o InvestmentswithoptioncharacteristicshaveasymmetricreturndistributionsreflectinglargeprobabilityofsmallgainswithsmallprobabilityoflargelossàSDmayunderestimateriskandproducehigherSharperatio

Skewness extenttowhichdistributionisnotsymmetrical• Positivelyskewed outliersinupperregion(righttail)à

skewedtorighto Mean>Median>Mode

• Negativelyskewed outliersinlowerregion(lefttail)àskewedtoleft

o Mode>Median>Meano Indicatesreturnsbelowmeanaremoreextreme

àmoreriskyi.e.meanis“pulled”indirectionoftheskewKurtosis measureofdegreetowhichdistributionormoreorless“peaked”thannormaldistributionàprobabilityofextremeoutcomes(thicknessoftail)

• Leptokurtic distributionthatismorepeakedthannormaldistributiono K>3o Morereturnsclusteredaroundmeanandmorereturnswithlargedeviations

(i.e.morepeakedwithfattertails moreextremeoutliers)§ e.g.equitydistributionsareleptokurtic

• Platykurtic distributionthatislesspeaked(i.e.flatter)thannormaldistributionàbetterforriskadverseclients

o K<3• Mesokurtic distributionthatissamekurtosisasanormaldistribution

o K=3Excesskurtosis=ifdistributionhasmore/lesskurtosisthannormaldistribution

• Greaterpositivekurtosisandnegativeskewinreturns=morerisk• +ve=leptokurtic(morepeaked,flattail)àprobabilityofextremeoutcomesgreater• -ve=platykurtic(lesspeaked,thintails)• excesskurtosis>1isconsideredsignificant

Sampleskewness=(observation mean)3/SD3

• Rightskeweddistributionà+vesampleskewness• Left-skeweddistributionà-vesampleskeweness

Samplekurtosis=(observation mean)4/SD4

• Excesskurtosis=kurtosis 3Arithmeticmeanappropriateforforecastingsingleperiodreturnsinfuture,whilegeometricmeanappropriateforforecastingfuturecompoundreturnsovermultipleperiods.

• Geometricmeanappropriateforpastperformance• Arithmeticmeanappropriateforforward-lookingcontext

ProbabilityConceptsMutuallyexclusiveevent cannotbothhappenatthesametimeExhaustiveevent includeallpossibleoutcomes

3typesofprobability:• Empiricalprobability establishedbyanalysingpastdata• Prioriprobability determinedusingformalreasoning,logicalanalysis&inspection

process• Subjectiveprobability personaljudgement

Unconditionalprobability(marginalprobability) probablyofeventoccurringregardlessofpastorfutureoccurrenceofothereventsàP(A)Conditionalprobability occurrenceof1eventaffectsprobabilityofoccurrenceofanothereventà“given”àP(A|B)Jointprobability AandBwillbothoccuràP(AB)Probabilitystateasodds:GivenP(E),oddsofE=P(E)/[1-P(E)]

• E.g.Probabilityis1/3àodds=1/3/2/3=1/2• OddsforEof“atob”àprobabilityisa/(a+b)

P(A|B)=i(jR)i(R)

Multiplicationrule:P(AB)=P(A|B)xP(B)

• Determinesjointprobabilityàbothwilloccur• Usedforindependentevents

Additionrule:P(AorB)=P(A)+P(B) P(AB)• Determinesprobabilitythatatleast1of2eventswilloccuràeitherAorB• IfmutuallyexclusiveeventsàP(AB)=0

Totalprobabilityrule:P(A)=P(A|B )P(B )+P(A|B2)P(B2)+….+P(A|B )P(B )• Determinesunconditionalprobabilityofevent,givenconditionalprobabilities• Usedformutuallyexclusiveandexhaustiveevents

And=multiplicationOr=additionIndependentevent occurrenceof1hasnoinfluenceonoccurrenceofothers

• P(A|B)=P(A)Expectedvalue averagevalueforrandomvariablethatresultsfrommultipleexperiments

• Weightedaverageofallpossibleoutcomesofarandomvariable• E(X)=P(X )X +P(X2)X2+….+P(X )X • E(X)=E(X|S)P(S)+E(X|1-S)P(1-S)

Varianceofrandomvariableàexpectedvalueofsquareddeviationsfromrandomvariableexpectedvalue

• s2=[X-E(X)]2xP(X)Covariance measureofhow2assetsmovetogether

• Cov(RA,RB)=P(AandB)x{[RA E(RA)][RB E(RB)]}• Positiveresultàmoveupwardstogetheranddownwardstogether• Difficulttointerpretàsoweusecorrelation

Correlationcoefficient:𝐶𝑜𝑟𝑟 𝐴, 𝐵 = lmnop^oDqr(j,R)

sSt×sSu

• Measuresstrengthoflinearrelationshipbetween2randomvariables(-1to1)• 1=positivecorrelationàsameproportionalmovement

• -1=negativecorrelationàoppositeproportionalmovementVarianceof2assetportfolio=WA

2SDA2+WB

2SDB2+2WAWBCovarianceAB

Varianceof2assetportfolio=WA2SDA

2+WB2SDB

2+2WAWBSDASDBCorrelationABBayes’formula updategivensetofpriorprobabilitiesforgiveneventinresponsetoarrivalofnewinfo

• 𝑈𝑝𝑑𝑎𝑡𝑒𝑑𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦 = ipmwDrx^DgmyDqmDz^{^mDo|fpmwDrx^Dgm

×𝑝𝑟𝑖𝑜𝑟𝑝𝑟𝑜𝑏𝑎𝑏𝑖𝑙𝑖𝑡𝑦𝑜𝑓𝑒𝑣𝑒𝑛𝑡

• 𝑃(𝐸𝑣𝑒𝑛𝑡|𝐼𝑛𝑓𝑜) i(�Dgm �nrD{)i �Dgm

×𝑃(𝐸𝑣𝑒𝑛𝑡)

• allowsustoadjustourviewpointwhenwereceivenewinformationLabelingformulaànumberofwayinwhichlabelscanbeassigned= D

D D� …D�

combinationformula(binomial):nCr D

D p p

• n!=nx(n-1)x(n-2)…..• nCrisnumberofcombinationsofselectionritemsfromsetofnitemsàordernotimportant

Permutationformula–orderofgroupofobjectsàwhenorderisimportant

nPr DD p

CommonProbabilityDistributionsProbabilitydistribution–describesprobabilitiesofallpossibleoutcomesforarandomvariable(e.g.rollingdie)

• SUM=1• 0<P(x)<1

Discreterandomvariable–finitenumberofpossibleoutcomescountedcanbemeasuredandhavepositiveprobability(e.g.#daysitcanraininamonth)

• P(x)=0whenxcannotoccur,orP(x)>0ifitcan• Wecanlistallthepossibleoutcomes(e.g.stockprice)• “howmany”

Continuousrandomvariable–numberofpossibleoutcomesisinfinite(e.g.amountofdailyrainfallbetweenzeroand100)

• P(x)=0eventhoughxcanoccurài.e.probabilityofgeneratingrandomnumberequaltoanyfixedpointunderacontinuousuniformdistributionis0

• E.g.returnonaninvestment• “howmuch”• Probabilitydensityfunction(pdf)–functionwithnon-negativevaluessuchthatprobability

describedbyareasundercurvegraphicthefunctionProbabilityfunction–specifiesprobabilitythatrandomvariableisequaltoaspecificvalueCumulativedistributionfunction(CDF)–givesprobabilitythatrandomvariablewillbelessthanorequaltospecificvalues:F(x)=P(X<x)Discreteuniformrandomvariable–probabilitiesforallpossibleoutcomesforadiscreterandomvariableareequal(e.g.rollingadice)

• F(xn)=np(x)

Binomialrandomvariable–numberof“successes”inagivennumberoftrials• Bernoullirandomvariable=Whennumberoftrialsis1with2possible

outcomes(E.g.Cointoss,stockpriceatendofperiod)• Binomialprobabilityfunctiondefinesprobabilityofxsuccessesinntrials

o P(x)=NCx(PX)(1-p)n-x

§ NCx=1

1 � �

o Mean=E(X)=npo Variance(X)=np(1-p)o Note:trialsareindependentandrandomvariableXisdiscreteo Fixed#trials,independent,2outcomes

TrackingError–differencebetweentotalreturnonportfolioandtotalreturnonbenchmarkContinuousuniformdistribution–definedoverarangethatspansbetweenlowerlimit(a)andupperlimit(b)whichserveasparametersofthedistribution

• ProbabilityofXoutsideboundaries=0• P(x1<X<x2)=(x2 x1)/(b a)

Normaldistribution

• Skewness=0àdistributionissymmetricaboutitsmeanàmean=median=mode• Kurtosis=3àflatdistribution• Probabilitiesofoutcomesfurtheraboveandbelowmeangetsmallerbutdonotgoto0à

tailgetsthinbutextendinfinitelyUnivariatedistribution–distributionofsinglerandomvariableMultivariatedistribution–specifiesprobabilitiesassociatedwithgroupofrandomvariables(canbediscreteorcontinuous)whenbehaviourisdependent

• #correlationsinmultivariatenormaldistribution=0.5n(n 1)• whenbuildingportfolio,itisdesirabletocombineassetshavinglowreturn

correlationbecausethiswillresultinportfoliowithlowervarianceConfidenceinterval–rangeofvaluesaroundtheexpectedoutcomewithinwhichweexpecttheactualoutcometobesomespecifiedpercentageofthetime

• 68%ofoutcomesarewithin1SDoftheexpectedvalue(mean)• 90%CI:µ ± 1.65𝜎• 95%ofoutcomesarewithin2SDoftheexpectedvalue(mean)

o 95%CI:µ ± 1.96𝜎• 99%ofoutcomesarewithin3SDoftheexpectedvalue(mean)

o 99%CI:µ ± 2.58𝜎Standardnormaldistribution–standardisessothatmeaniszeroandSDis1

• tostandardiseobservationfromnormaldist,Z-valueofobservationmustbecalculatedàrepresentsnumberofSD’sanobservationisfromthepopulationmean

• standardisation–processofconvertingobservedrandomvariabletoitszvalue

𝑍 =𝑥 − µ𝜎 =

𝑜𝑏𝑠𝑒𝑟𝑣𝑎𝑡𝑖𝑜𝑛 − 𝑝𝑜𝑝𝑚𝑒𝑎𝑛𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑑𝑒𝑣𝑖𝑎𝑡𝑖𝑜𝑛

Ztable–usedtocalculatecumulativeprobabilitiesforstandardnormaldistribution

• note:z scoreusedforcalculationP(X<x)àwhendoingP(X>x)use1–zscore

Shortfallrisk–probabilitythatportfoliovalueorreturnwillfallbelowparticulartargetvalueorreturnovergivenperiodoftimeRoy’ssafety-firstcriterion–optimalportfoliominimisesprobabilitythatreturnofportfoliofallsbelowsomeminimumacceptablelevel

• minimumacceptablelevel=thresholdlevel(RL)

• 𝑆𝐹𝑅𝑎𝑡𝑖𝑜 = � �� ����

= ��frq{rzpr{�pD3{�pr��m|zpr{�pD�{oDzopzzrn^o{^mD

• PortfoliowithlargerSFRwillbeonewithlowerprobabilityofnegativereturns

o SFR=numberofSD’sbelowthemeanLognormaldistribution–generatedbythefunctioneXwherexisnormallydistributed

• Boundedfrombelowbyzeroàusefulformodelingassetpriceswhichnevertakenegativevalues(minvalueofzero)

o Wilnotresultinassetreturnoflessthan100%• Skewedtotheright• lnex=x

Discretelycompoundreturns:(1+r/n)n

Continuouscompoundingrate(CCR):EAR=eRcc–1• IfHPR–12.5%àCCR=ln1.125• Decreasinglengthofdiscretecompoundperiods(e.g.fromquarterlyto

monthly)àEARincreasesContinuouslycompoundedreturn=Pert=Ln(S1/S0)=ln(1+HPR)HPRT=eRccxT–1MonteCarlosimulation usesrandomlygeneratedvaluesforriskfactorstoproducedistributionofpossiblesecurityvalues

• Toolforconsideringallpossiblecombinationsà‘WHATIFANALYSIS’• USES

o Valuecomplexsecuritieso Simulateprofits/lossesfromtradingstrategyo Calculateestimatesofvalueatrisk(VaR)todetermineriskinessofportfolio

ofassetsandliabilitieso Simulatepensionfundassetsandliabilitiesovertimetoexaminevariability

ofdifferencebetweenthe2o Valueprotfoliosofassetsthathavenon-normalreturnsdistributions

• LIMITATIONSo Fairlycomplexandwillprovideanswernobetterthanassumptionsused

o Outputonlyasaccurateasassumptionsusedofinputso Simulationisnotananalyticmethod,butastatisticaloneàcannotprovide

insightsHistoricalsimulation usesrandomlyselectedpastchangesinriskfactorstogeneratedistributionofpossiblesecurityvalues

• Advantage:usingactualdistributionofriskfactorssothatitdoesnothavetobeestimated(likeMonteCarlo)

• Disadvantage:pastchangesinriskfactorsnotgoodindicationoffuturechangeso Unabletoaddress“whatif”questionthatMonteCarlosimulationcan

SamplingandEstimationSimplerandomsampling–selectingsampleinsuchawaythateachiteminthepopulationhasthesamelikelihoodofbeingincluded

• E.g.systematicsampling–selectingeverynthmemberfromthepopulation

Samplingerror–differenceb/wsamplestatisticanditscorrespondingpopulationparameter• E.g.samplingerrorofmean=samplemean–populationmean

Stratifiedrandomsampling–usesclassificationsystemtoseparatepopulationintosmallergroupsbasedon1ormoredistinguishingcharacteristicsàthensampleispooledfromeachgroup.

• Usedinbondindexing–duetodifficultyandcostofcompletelyreplicatingentirepopulations(categorizedaccordingtobondriskfactorssuchasduration,maturity,couponrateetc..àthensamplesdrawnfromeachcategoryandcombinedtoformfinalsample)

Timeseriesdata–observationstakenoverperiodoftimeatspecificandequallyspacedintervals(e.g.monthlyreturnsonstockfrom1994to2004)Crosssectionaldata–observationstakenatasinglepointintime(e.g.reportingEPSofstockmarketasof31December2004)Longitudinaldata–observationsovertimeofmultiplecharacteristicsofthesameentity(e.g.unemployment,inflation,GDPforcountryover10years)Paneldata–observationsovertimeofsamecharacteristicformultipleentities(e.g.debttoequityratiofor20companiesoverpast2years)Centrallimittheorem(CLT)–samplingdistributionofthesamplemeanapproachesnormalprobabilityasthesamplesizebecomeslarger

• i.e.varianceofsampledecreasesassamplesizeincreases• usedforsamplesdescribedbyanyprobabilityfunction• Allowsustousesamplingstatisticstoconstructconfidenceintervalsforpointestimatesof

populationmeans• Normaldistributioniseasytoapplytohypothesistestingandconstructionofconfidence

intervals• Inferencesaboutpopulationmeancanbemadefromsamplemean,regardlessof

population’sdistribution,aslongassamplesizeis“sufficientlylarge”àn>30• Meanofpopulation=meanofsampledistribution• Varianceofsample=populationvariance/samplesize

𝑆𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟𝑜𝑓𝑠𝑎𝑚𝑝𝑙𝑒𝑚𝑒𝑎𝑛 = �{oDzopzzrn^o{^mDD��wrpmgmw�rpno{^mD�

• Assamplesizeincreasesàstandarderrordecreases• StandarderrorofsamplemeanistheSDofthedistributionofthesamplemean

Desirablepropertiesofanestimatorare:• UnbiasedàExpectedvalueofestimatorisequaltoparameter(samplemean=

populationmean)• Efficientàvarianceofsamplingdistributionsmallerthanunbiasedestimatorsof

parameter• Consistencyàaccuracyofparameterestimateincreasesassamplesizeincreases

(standarderrorofsamplemeanfalls=distributionbunchesmorearoundmean)Pointestimates single(sample)valuesusedtoestimatepopulationparametersEstimator=formulausedtocomputeapointestimateConfidenceinterval rangeofvaluesinwhichpopulationparameterisexpectedtolie

• 𝐶𝐼 = 𝑝𝑜𝑖𝑛𝑡𝑒𝑠𝑡𝑖𝑚𝑎𝑡𝑒 ± (𝑟𝑒𝑙𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑦𝑓𝑎𝑐𝑡𝑜𝑟×𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑𝑒𝑟𝑟𝑜𝑟)t-distribution aballshapedprobabilitydistributionthatissymmetricalaboutitsmean

• Appropriatedistributiontousewhenconstructingconfidenceintervalsbasedonsmallsamples(n<30)withunknownvarianceandnormaldistribution

• Alsoappropriateifvarianceisunknownandsamplesizeislargeenoughthatcentrallimittheoremwillassuresamplingdistributionisapproximatelynormal

• Propertiesincludeo Symmetricalo Definedbyasingleparameter,degreesoffreedom(df)=n-1o Lesspeakedandfatterthannormaldistribution

§ Moreprobabilityinthetails(fatter)àmoreareao Asdfincreasesàshapeofdistributionmorecloselyapproachesstandard

normaldistribution• Hypothesistestingusingt-distributionmakesitmoredifficulttorejectthenull

relativetohypothesistestingusingz-distributionàsincethickertailsmeanmoreobservationsawayfrommean(moreoutliers)

Degreeofconfidence 1 alpha(notealpha=levelofsignificance)loweralpha=widerconfidenceintervalhigherDF=widerconfidenceintervalIfpopulationhasnormaldistributionwithKNOWNvariance,confidenceintervalforpopulationmean=𝑚𝑒𝑎𝑛 ± 𝑍�/[

�D

• Za/2=1.65for90%CIàsignificancelevelis10%(5%ineachtail)• Za/2=1.96for95%CIàsignificancelevelis5%(2.5%ineachtail)• Za/2=2.58for99%CIàsignificancelevelis1%(0.5%ineachtail)

IfpopulationhasnormaldistributionwithUNKNOWNvariance,confidenceintervalforpopulationmean=𝑚𝑒𝑎𝑛 ± 𝑡�/[

�D

• Remembertocomputedf(n-1)beforelookingatttable• 1-tailed(usealpha)or2-tailed(alpha/2)