FRM前导课课件-不公开入门必备前导课程、课件数量_1 5D

QuantitativeAnalysis定量分析,1,Topic1,Probability,2,BasicConcepts,RandomVariableAquantitywhosefutureoutcomesareuncertain.OutcomesPossiblevaluesofarandomvariable.EventAspecifiedsetofoutcomes.,3,BasicConcepts,ProbabilityThemeasureofthelikelinessthataneventwilloccur.Anumberbetween0and1describingthechancethatastatedeventwilloccur.Thehighertheprobabilityofanevent,themorecertainwearethattheeventwilloccur.,4,BasicConcepts,ExhaustiveEventsTheeventscoverallpossibleoutcomes.MutuallyExclusiveEventsEventAwillnotoccurifeventBoccurs,botheventswillnotoccurinthesametime.,5,PropertiesofProbability,TheprobabilityofanyeventEisanumberbetween0and1.Thesumoftheprobabilitiesofanysetofmutuallyexclusiveandexhaustiveeventsequals1.,6,TypesofProbability,EmpiricalProbabilityTheprobabilityofaneventestimatedasarelativefrequencyofoccurrenceusingcollectedempiricalevidence.theratioofthenumberofoutcomesinwhichaspecifiedeventoccurstothetotalnumberoftrials,notinatheoreticalsamplespacebutinanactualexperiment.Inamoregeneralsense,empiricalprobabilityestimatesprobabilitiesfromexperienceandobservation.,7,8,EmpiricalProbability,TypesofProbability,SubjectiveProbabilityAprobabilitydrawingonpersonalorsubjectivejudgment.Subjectiveprobabilitiesdifferfrompersontoperson.Becausetheprobabilityissubjective,itcontainsahighdegreeofpersonalbias.,9,10,TypesofProbability,PrioriProbabilityTheprobabilitydeducedbyreasoningandbasedonlogicalanalysisratherthanonhistoricaldataorpersonaljudgment.Theprioriprobabilitycanmosteasilybedescribedasmakingaconclusionbasedupondeductivereasoningratherthanresearchorcalculation.Uselogictodeterminewhatoutcomesofaneventarepossibleinordertodeterminethenumberofwaystheseoutcomescanoccur.Forexample,considerhowthepriceofasharecanchange.Itspricecanincrease,decreaseorremainthesame.Therefore,accordingtoaprioriprobability,wecanassumethatthereisa1-in-3,or33%.,ExampleQuestion(1),Anempiricalprobabilityisonethatis:A)supportedbyformalreasoning.B)determinedbymathematicalprinciples.C)derivedfromanalyzingpastdata.,11,Answer:CAnempiricalprobabilityisonethatisderivedfromanalyzingpastdata.,BasicConcepts,UnconditionalProbabilityTheprobabilityofaneventwilloccurandnotconditionedonanotherevent,alsoreferredtoasmarginalprobability,denotedP(A).ConditionalProbabilityTheprobabilityofaneventgiven(conditionedon)anotherevent,denotedP(A|B).JointProbabilityTheprobabilityofbotheventAandBhappening,denotedP(AB).,12,13,BasicConcepts,BasicConcepts,IndependentEventsTheoccurrenceofeventAdoesnotaffecttheprobabilityofeventBoccurring,denotedP(A|B)=P(A),equivalently,P(B|A)=P(B).,14,DependentEventsTheprobabilityofeventAoccurringdependson(isrelatedto)theoccurrenceofeventB.,15,BasicConcepts,MultiplicationRuleandAdditionRuleforProbabilities,MultiplicationRuleforProbabilitiesP(AB)=P(A|B)P(B),16,MultiplicationRuleandAdditionRuleforProbabilities,AdditionRuleforProbabilitiesP(AorB)=P(A)+P(B)-P(AB),17,MultiplicationRuleforIndependentEvents,MultiplicationRuleforIndependentEventsP(AB)=P(A|B)P(B)P(AB)=P(A)P(B),18,ExampleQuestion(1),Afirmholdstwo$50millionbondswithcalldatesthisweek.TheprobabilitythatBondAwillbecalledis0.80.TheprobabilitythatBondBwillbecalledis0.30.Theprobabilitythatatleastoneofthebondswillbecalledisclosestto:A)0.24.B)0.86.C)0.50.,19,Answer:BWecalculatetheprobabilitythatatleastoneofthebondswillbecalledusingtheadditionruleforprobabilities:P(AorB)=P(A)+P(B)-P(AB),whereP(AB)=P(A)P(B)P(AorB)=0.80+0.30-(0.80.3)=0.86,Topic2,Statistics,20,21,StatisticsAbranchofmathematicsusedtosummarize,analyze,andinterpretagroupofnumbersorobservations.Ithastwomeans,onereferringtodataandotheronereferringtomethod.,BasicConcepts,DescriptiveStatisticsThestudyofhowdatacanbesummarizedeffectivelytodescribetheimportantaspectsoflargedatasets.,22,StatisticalInferenceInvolvingmakingforecasts,estimates,orjudgmentsaboutalargergroupfromthesmallergroupactuallyobserved.,BasicConcepts,PopulationAllmembersofaspecifiedgroup.SampleAsubsetofapopulation.ParameterAnydescriptivemeasureofapopulationcharacteristic.e.g.,populationmean.SampleStatisticAquantitycomputedfromthesampleandisusedtodescribethesample.e.g.,samplemean.,24,MeasuresofCentralTendency,ArithmeticMean(算术平均值)Thesumoftheobservationsdividedbythenumberofobservations.PopulationMean(总体平均值)Thearithmeticmeanvalueofapopulation.SampleMean(样本平均值)Thearithmeticmeanvalueofasample.,25,MeasuresofCentralTendency,PopulationMeanWhereN=thenumberofobservationsinpopulationSampleMeanWhereN=thenumberofobservationsinsample,samplesize,26,CharacteristicsofarithmeticmeanThearithmeticmeanisthemostfrequentlyusedmeasureofcentraltendencyItdoeshaveshortcomingsthatinsomecasestendtomakeitmisleadingwhendescribingapopulationorsample.Inparticular,thearithmeticmeanissensitivetoextremevalues.Example-9000,1.4,1.6,2.4and3.7.Thearithmeticmeanis-1798.2(-9000+1.4+1.6+2.4+3.7)/5,yet-1798.2haslittlemeaningindescribingourdataset.,27,MeasuresofCentralTendency,MeasuresofDispersion,DispersionThevariabilityaroundthecentraltendency,itaddressestherisk.Sample,28,Measuresofabsolutedispersion,VarianceTheaverageofthesquareddeviationsaroundthemean.Avariancevalueofzeroindicatesthatallvalueswithinasetofnumbersareidentical;allvariancesthatarenon-zerowillbepositivenumbers.Alargevarianceindicatesthatnumbersinthesetarefarfromthemeanandeachother,whileasmallvarianceindicatestheopposite.Varianceisusedinstatisticsforprobabilitydistribution.Sincevariancemeasuresthevariability(volatility)fromanaverageormean,andvolatilityisameasureofrisk,thevariancestatisticcanhelpdeterminetheriskaninvestormighttakeonwhenpurchasingaspecificsecurity.,29,Measuresofabsolutedispersion,PopulationVarianceWhereisthepopulationmeanandNisthesizeofthepopulation.SampleVarianceWherenisthesizeofobservationsinthesample.,30,31,Measuresofabsolutedispersion,StandardDeviationPositivesquarerootofthevariance.Ameasureofthedispersionofasetofdatafromitsmean.Themorespreadapartthedata,thehigherthedeviation.Infinance,standarddeviationisappliedtotheannualrateofreturnofaninvestmenttomeasuretheinvestmentsvolatility.Standarddeviationisalsoknownashistoricalvolatilityandisusedbyinvestorsasagaugefortheamountofexpectedvolatility.,Measuresofabsolutedispersion,PopulationStandardDeviationSampleStandardDeviation,32,MeasureofSkewnessandKurtosis,SkewnessAquantitativemeasureofskew(lackofsymmetry);asynonymofskew.PositivelySkewed=Skewedtotheright=LongtailonrightNegativelySkewed=Skewedtotheleft=LongtailonleftMostsetsofdata,includingstockpricesandassetreturns,haveeitherpositiveornegativeskewratherthanfollowingthebalancednormaldistribution(whichhasaskewnessofzero).Byknowingwhichwaydataisskewed,onecanbetterestimatewhetheragiven(orfuture)datapointwillbemoreorlessthanthemean.,33,34,MeasureofSkewnessandKurtosis,KurtosisThestatisticalmeasurethattelluswhenadistributionismoreorlesspeakedthananormaldistribution.measuresboththepeakednessofthedistributionandtheheavinessofitstail,35,MeasureofSkewnessandKurtosis,36,MeasureofSkewnessandKurtosis,BasicConcepts,CovarianceAmeasureoftheco-movementbetweentworandomvariables,italsomeasuretheextenttowhichtowrandomvariablestendtobeaboveorbelowtheirrespectivemeansforeachjointrealization.Apositivecovariancemeansthattwovariablesmovetogether.Anegativecovariancemeansthattwovariablesmoveinversely.,37,38,BasicConcepts,BasicConcepts,CorrelationCoefficientAstandardizedmeasureoflinearassociationbetweentworandomvariables;itrangesinvaluefrom-1to+1.Perfectpositivecorrelation(acorrelationco-efficientof+1)impliesthatasonesecuritymoves,eitherupordown,theothersecuritywillmoveinlockstep,inthesamedirection.Alternatively,perfectnegativecorrelationmeansthatifonesecuritymovesineitherdirectionthesecuritythatisperfectlynegativelycorrelatedwillmoveintheoppositedirection.Ifthecorrelationis0,themovementsofthesecuritiesaresaidtohavenocorrelation;theyarecompletelyrandom.,39,BasicConcepts,40,Topic3,Distributions,41,BasicConcepts,ProbabilityDistributionAdistributionthatspecifiestheprobabilitiesofarandomvariablespossibleoutcomes.Astatisticalfunctionthatdescribesallthepossiblevaluesandlikelihoodsthatarandomvariablecantakewithinagivenrange.Thisrangewillbebetweentheminimumandmaximumstatisticallypossiblevalues,butwherethepossiblevalueislikelytobeplottedontheprobabilitydistributiondependsonanumberoffactors,includingthedistributionsmean,standarddeviation,skewnessandkurtosis.,42,BasicConcepts,DiscreteRandomVariableArandomvariablethatcantakeonatmostacountablenumberofpossiblevalues.,43,BasicConcepts,ContinuousRandomVariableArandomvariableforwhichtherangeofpossibleoutcomesistherealline(allrealnumbersbetween-and+orsomesubsetoftherealline),44,BasicConcepts,ProbabilityFunctionAfunctionthatspecifiestheprobabilitythattherandomvariabletakesonaspecificvalue,denotedP(X=x).ProbabilityDensityFunctionAfunctionwithnon-negativevaluessuchthatprobabilitycanbedescribedbyareasunderthecurvegraphingthefunction,denotedf(x).,45,BasicConcepts,CumulativeDistributionFunctionAfunctiongivingtheprobabilitythatarandomvariableislessthanorequaltoaspecifiedvalue,denotedF(X)=P(Xx).,46,BasicConcepts,DiscreteUniformDistributionThedistributionhasafinitenumberofspecifiedoutcomes,andeachoutcomeisequallylikely.,47,BasicConcepts,BinomialDistributionItdescribesthedistributionofthenumberofsuccessesinnBernoullitrialsforwhichtheprobabilityofsuccessisconstantforalltrialsandthetrialsareindependent.ItisdenotedXB(n,p).nreferstothenumberofBernoullitrialsandpreferstotheprobabilityofsuccessforeachtrial.,48,BasicConcepts,BernoulliTrialAnexperimentthatcanproduceoneoftwooutcomes.BernoulliRandomVariableArandomvariablehavingtheoutcomes0and1.,49,ExampleQuestion(1),WhichofthefollowingisNOTanassumptionofthebinomialdistribution?A)Theexpectedvalueisawholenumber.B)Thetrialsareindependent.C)RandomvariableXisdiscrete.,50,Answer:AAsimpleexampleshowsusthattheexpectedvaluedoesnothavetobeawholenumber:n=5,p=0.5,expectedvalue=np=2.5.Theotherconditionsarenecessaryforthebinomialdistribution.,BasicConcepts,ContinuousUniformDistributionThedistributiondescribesequallylikelyoutcomeswithinaspecifiedinterval.,51,BasicConcepts,NormalDistributionAcontinuous,symmetricprobabilitydistributionthatiscompletelydescribedbyitsmeananditsvariance.Aprobabilitydistributionthatplotsallofitsvaluesinasymmetricalfashionandmostoftheresultsaresituatedaroundtheprobabilitysmean.Valuesareequallylikelytoploteitheraboveorbelowthemean.Groupingtakesplaceatvaluesthatareclosetothemeanandthentailsoffsymmetricallyawayfromthemean.AlsoknownasaGaussiandistributionorbellcurve.mostcommontypeofdistribution,andisoftenfoundinstockmarketanalysis.Itistheassumptionthatassetreturnsfollowanormallydistributedpattern.,52,BasicConcepts,KeypropertiesofNormalDistributionCompletelydescribedbyitsmeanandvariance.Itissymmetric(skewnessis0),excesskurtosisis0.Alinearcombinationoftwoormorenormalrandomvariablesisalsonormallydistributed.Probabilitiesdecreasefurtherfromthemean,butthetailsgoonforever.Mean=Median=Mode,53,NormalDistribution,54,NormalDistribution,Approximately68%ofallobservationsfallintheinterval(+/-)Approximately95%ofallobservationsfallintheinterval(+/-2)Approximately99%ofallobservationsfallintheinterval(+/-3),55,56,NormalDistribution,BasicConcepts,StandardNormalDistributionAnormaldistributiondescribedbymeanof0andvarianceof1.CharacteristicsofStandardNormalDistributionCompletelydescribedbyitsmeanof0andvarianceof1.Itissymmetric(skewnessis0),excesskurtosisis0.Alinearcombinationoftwoormorenormalrandomvariablesisalsonormallydistributed.Probabilitiesdecreasefurtherfromthemean,butthetailsgoonforever.Mean=Median=Mode,57,BasicConcepts,StandardizationTheprocessofconvertinganobservedvalueforanormallydistributedvariabletoitsZ-value.,58,ExampleQuestion(1),Whichofthefollowingstatementsaboutanormaldistributionisleastaccurate?A)Thedistributioniscompletelydescribedbyitsmeanandvariance.B)Kurtosisisequalto3.C)Approximately34%oftheobservationsfallwithinplusorminusonestandarddeviationofthemean.,59,Answer:CApproximately68%oftheobservationsfallwithinonestandarddeviationofthemean.Approximately34%oftheobservationsfallwithinthemeanplusonestandarddeviation(orthemeanminusonestandarddeviation).,ExampleQuestion(2),Agroupofinvestorswantstobesuretoalwaysearnatleasta5%rateofreturnontheirinvestments.Theyarelookingataninvestmentthathasanormallydistributedprobabilitydistributionwithanexpectedrateofreturnof10%andastandarddeviationof5%.Theprobabilityofmeetingorexceedingtheinvestorsdesiredreturninanygivenyearisclosestto:A)34%.B)98%.C)84%.,60,ExampleQuestion(2),61,Answer:CThemeanis10%andthestandarddeviationis5%.Youwanttoknowtheprobabilityofareturn5%orbetter.10%-5%=5%,so5%isonestandarddeviationlessthanthemean.Thirty-fourpercentoftheobservationsarebetweenthemeanandonestandarddeviationonthedownside.Fiftypercentoftheobservationsaregreaterthanthemean.Sotheprobabilityofareturn5%orhigheris34%+50%=84%.,BasicConcepts,Studentst-DistributionAsymmetricalprobabilitydistributiondefinedbyasingleparameterknownasdegreesoffreedom(df).Atypeofprobabilitydistributionthatistheoreticalandresemblesanormaldistribution.At-distributiondiffersfromthenormaldistributionbyitsdegreesoffreedom.Thehigherthedegreesoffreedom,thecloserthatdistributionwillresembleastandardnormaldistributionwithameanof0,andastandarddeviationof1.DegreesofFreedomThenumberofindependentobservationsused,itequalston-1,nisthesamplesize.,62,t-DistributionvsStandardNormalDistribution,t-distributionhasfattertailsthanthatofstandardnormaldistributiononbothsides.t-distributionislesspeakedthanstandardnormaldistribution.,63,ExampleQuestion(1),WhichstatementbestdescribesthepropertiesofStudentst-distribution?Thet-distributionis:A)symmetrical,anddefinedbyasingleparameter.B)skewed,anddefinedbyasingleparameter.C)symmetrical,anddefinedbytwoparameters.,64,Answer:AThet-distributionissymmetricallikethenormaldistributionbutunlikethenormaldistributionisdefinedbyasingleparameterknownasthedegreesoffreedom.,BasicConcepts,CentralLimitTheoremGivenapopulationdescribedbyanyprobabilitydistributionhavingmeanandfinitevariance2,thesamplingdistributionofthesamplemeancomputedfromsamplesofsizenfromthispopulationwillbeapproximatelynormalwithmean(thepopulationmean)andvariance2/nwhenthesamplesizenislarge.,65,CentralLimitTheorem,ConditionsUsingsimplerandomsampling.Samplesizeisatleast30.PredictionsSamplemeanisapproximatelynormallydistributed.E()=2/n,66,CentralLimitTheorem,67,BasicConcepts,StandardErrorofSampleMeanthestandarddeviationofthesamplemeansoverallpossiblesamples(ofagivensize)drawnfromthepopulationwhenweknowthepopulationstandarddeviationwhenwedonotknowthepopulationstandarddeviationandneedtousesamplestandarddeviation,68,BasicConcepts,69,BasicConcepts,IndependentandIdenticalDistribution(I.I.D)eachrandomvariablehasthesameprobabilitydistributionastheothersandallaremutuallyindependent.Theassumptionisimportantintheclassicalformofthecentrallimittheorem,whichstatesthattheprobabilitydistributionofthesum(oraverage)ofI.I.D.variableswithfinitevarianceapproachesanormaldistribution.,Topic4,Hypothesis,71,BasicConcepts,ConfidenceIntervalArangeforwhichonecanassertwithagivenprobability(1-).Consistofarangeofvalues(interval)thatactasgoodestimatesoftheunknownpopulationparameter.Calculatedfromtheobservations,inprincipledifferentfromsampletosample,thatfrequentlyincludestheparameterofinterestiftheexperimentisrepeated.,72,BasicConcepts,PointEstimatetheuseofsampledatatocalculateasinglevalue(knownasastatistic)whichistoserveasabestguessorbestestimateofanunknown(fixedorrandom)populationparameter.Forexample,samplemeanisthepointestimateofpopulationmean.,73,BasicConcepts,DegreeofConfidenceTheprobabilitythataconfidenceintervalincludestheunknownpopulationparameter,denoted(1-),alsoknownaslevelofconfidence.Thelevelofconfidenceoftheconfidenceintervalwouldindicatetheprobabilitythattheconfidencerangecapturesthistruepopulationparametergivenadistributionofsamples.Thisvalueisrepresentedbyapercentage.Whenwesay,weare99%confidentthatthetruevalueoftheparameterisinourconfidenceinterval,weexpressthat99%oftheobservedconfidenceintervalswillholdthetruevalueoftheparameter.,74,ConfidenceIntervalthismodelrepresentsthekeycharacteristicsorbehaviors/functionsoftheselectedphysicalorabstractsystemorprocess.Themodelrepresentsthesystemitself,whereasthesimulationrepresentstheoperationofthesystemovertime.,MonteCarloSimulationMonteCarlomethodswerefirstintroducedtofinancein1964byDavidB.HertzthroughhisHarvardBusinessReviewarticle,discussingtheirapplicationinCorporateFinance.In1977,PhelimBoylepioneeredtheuseofsimulationinderivativevaluationinhisseminalJournalofFinancialEconomicspaper.,BasicConcepts,MonteCarloSimulationThegenerationofalargenumberofrandomsamplesfromaspecifiedprobabilitydistributionordistributionstorepresenttheroleofriskinthesystem.ApplicationandLimitationsUsedinplanning,infinancialriskmanagement.Usedinvaluingcomplexsecurities.Acomplementtoanalyticalmethodsbutprovidesonlystatisticalestimates,notexactresults,BasicConcepts,BasicConcepts,BasicConcepts,HistoricalSimulationInvolvingrepeatedsamplingfromahistoricaldataseries.Groundedinactualdataandreflectonlytherisksrepresentedinthesamplehistoricaldata.doesnotlenditselfto“whatif”analyses.,Topic9,ModelingCycle,137,BasicConcepts,MovingAverageAwidelyusedindicatorintechnicalanalysisthathelpssmoothoutpriceactionbyfilteringoutthe“noise”fromrandompricefluctuations.Amovingaverage(MA)isatrend-followingorlaggingindicatorbecauseitisbasedonpastprices.ThetwobasicandcommonlyusedMAsarethesimplemovingaverage(SMA),whichisthesimpleaverageofasecurityoveradefinednumberoftimeperiods,andtheexponentialmovingaverage(EMA),whichgivesbiggerweighttomorerecentprices.,MovingAverage,ThemostcommonapplicationsofMAsaretoidentifythetrenddirectionandtodeterminesupportandresistancelevels.MAsareusefulenoughontheirown,theyalsoformthebasisforotherindicatorssuchastheMovingAverageConvergenceDivergence(MACD).,MovingAverage,Autoregressive,AutoregressiveAstochasticprocessusedinstatisticalcalculationsinwhichfuturevaluesareestimatedbasedonaweightedsumofpastvalues.Anautoregressiveprocessoperatesunderthepremisethatpastvalueshaveaneffectoncurrentvalues.AprocessconsideredAR(1)isthefirstorderprocess,meaningthatthecurrentv