Upload
lammien
View
263
Download
9
Embed Size (px)
Citation preview
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)