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ysisandtheUseofFRMPartITheIntroductionofTimeStatisticalProbabilityProbabilityPointEstimateandConfidenceHypothesisLinearDate2-BAII3-示调用FORMAT功能，出现DEC=2.00。若要改为四位小数，输入6，再按【ENTER】，出现DEC=6.00004-括号的使负号功1/X功ex功yx功nCr功

1/2：【2】【1/Xe2：【2】【2nd】【LN23：【2】【yx】【3】【=21/3：【2】【yx】【3】【1/X】【= 5-TheIntroductionofTimeStatisticalProbabilityPointEstimateandHypothesisLinearDate6-TimeTimeRequiredinterestrateonaNPVand7-TimeValueofRequiredrateofreturnaffectedbythesupplyanddemandoffundsinthethereturnthatinvestorsandsaversrequiretogetthemtolendtheirusuallyforparticularDiscountratetheinterestrateweusetodiscountpaymentstobemadeintheusuallyusedinterchangeablywiththeinterestOpportunitycostalsounderstoodasaformofinterestrate.Itisthevaluethatforgobychoosingaparticularcourseof8-TimeValueofposerequiredrateofNominalrisk-freerate=realrisk-freerate+expectedinflationRequiredinterestrateonasecurity=nominalrisk-freerate+riskpremium+liquidityriskpremium+maturityrisk9-TimeValueofEAR 𝑝−

1+ 1+ =那么如果是semim=2如果是quarterly=ThegreaterthecompoundingthegreatertheEARwillbeincomparisontothestatedthegreaterthedifferencebetweenEARandthestated10-TimeValueofFuturevalue(FV):Amounttowhichinvestmentgrowsafteroneormorecompoundingperiods.Presentvalue(PV):CurrentvalueofsomefuturecashAnnuities:isafinitesetoflevelsequentialcashequalequalamountofcashsameN=numberofI/Y=interestrateperPV=presentPMT=amountofeachperiodicFV=future11-投资100元 计算结果为：FV=259.3742 计算结果为：PV=-55.839512- =计算结果为：N143.80因为计算结果以“月”为单位，需换算成“年”：143.8÷1211.98，13-Acompanyplanstoborrow$50,000forfiveyears.Thecompany’sbankwilllendthemoneyatarateof9%andrequiresthattheloanbepaidoffinfiveequalend-of-yearpayments.CalculatetheamountofthepaymentthatthecompanymustmakeinordertofullyamortizethisloaninfiveN=5,I/Y=9,PV=50,000,FV=0;CPT:PMT=-Usingtheloandescribedintheprecedingexample,determinethepaymentamountifthebankrequiresthecompanytomakequarterlypayments.N=5×4=20,I/Y=9/4=2.25,PV=50,000,FV=0;CPT:PMT=-14-N=20*12=240,I/Y=6.2/12,PV=700,000,CPTPMT=-15-Ordinaryannuities（后付年金）:Thefirstcashflowoccurs firstExample：mortgageloans,investments,Annuitydue（先付年金）:Thefirstcashflowoccurs Example：rentalfees,tuitionfees,livingexpenses,Usecalculator,putthecalculatorintheBGNmodeand16- 的话，假设回报率为BGN模式：([2NDBGN2ND]SET2NDN=20,I/Y=4,PMT=100,000CPTPV=-17-BGN模式：([2NDBGN2ND]SET2NDN=4,I/Y=4,PMT=50,000,CPTPV=-18-TimeValueofAboutDefinition:Aperpetuityisasetoflevelnever-endingsequentialcashflows,withthefirstcashflowoccurringoneperiodfromnow.AAAAAAA0AAAAAAAPV1=A/(1+r)PV2=A/(1+r)2PV3=A/(1+r)3PV4=

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1+𝐴InternalRateofReturnWhenNPV=0,thediscount21-B公司计划以 一台新机器，希望的投资回报率为10%未来五年内公司预计现金流如下表所示，试求净现值 收益。01234522-

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24-TheIntroductionofTimeStatisticalProbabilityProbabilityPointEstimateandConfidenceHypothesisLinearDate25-StatisticalConceptsandMarketStatisticalBasicTypesofmeasurementFrequencyMeasuresofcentralMeasuresof26-StatisticalConceptsandMarketDescriptiveDescriptivestatisticsisthestudyofhow anbeeffectivelytodescribetheimportantaspectsoflargedataByconsolidatingamassofnumericaldetails,descriptivestatisticsdataintoInferentialMakesestimationsaboutalargesetofdata(apopulationwithsmallergroupofdata.27-StatisticalConceptsandMarketDefinitionofPopulation:ApopulationisdefinedasallmembersofspecifiedAnydescriptivemeasureofapopulationcharacteristiciscalledDefinitionofSample:AsampleisasubsetofaAsamplestatistic(orstatistic)isa tycomputedfromorusedtodescribeasample.28-StatisticalConceptsandMarketRelativeTherelativefrequencyofobservationsinanintervalisthenumberofobservations(theabsolutefrequency)intheintervaldividedbythetotalnumberofobservations.FrequencyAfrequencydistributionisatabulardisplayofdatasummarizedintorelativelysmallnumberofintervals.FrequencydistributionsysttoevaluatehowdataareCumulativefrequency/cumulativerelativeThecumulativerelativefrequencycumulates(addsup)therelativefrequenciesaswemovefromthefirstintervaltothelast..29-StatisticalConceptsandMarketConstructingafrequencyReal(Inflation-Adjusted)EquityReturns:NineteenMajorEquityArithmeticMeanArithmeticMeanNewSouthUnitedUnited30-StatisticalConceptsandMarketConstructingafrequencyFrequencyDistributionofAverageRealEquityFrequency(%)5.0to336.0to477.0to58.0to49.0to331-StatisticalConceptsandMarketMeasuresofcentraltendency:mode,median,Thearithmetic = Theweighted𝑤= 𝑖𝑖=1+⋯Thegeometric𝐺=𝑁= Theharmonic

32-

ExpectedExpectedValue:Ameasureofcentraltendency–thefirst𝑬𝑿= 𝑷𝑖𝑷11𝑷2𝑷3𝑷𝑎PropertiesofExpectedIfbisaconstant,E(b)=Ifaisaconstant,E(aX)=Ifaandbareconstants,thenE(aX+b)=aE(X)+E(b)=aE(X)+E(X2)≠E(X+Y)=E(X)+Ingeneral,E(XY)≠E(X)E(Y);IfXandYareindependentrandomvariables,thenE(XY)=E(X)E(Y).33-StatisticalConceptsandMarketAbsolutedispersion:istheamountofvariabilitypresentcomparisontoanyreferencepointor Range umvalue–minimum 𝑖− ForFor

𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 𝑖𝑖=1 −34-PropertiesofThevarianceofaconstantiszero.Bydefinition,aconstanthasIfaisconstant,then:σ2(aX)=Ifbisaconstant,then:σ2(X+b)=Ifaandbareconstant,then:σ2(aX+b)=IfXandYaretwoindependentrandomvariables,σ2(X+Y)=σ2(X)+ σ2(X–Y)=σ2(X)+IfXandYareindependentrandomvariablesandaandbarethenσ2(aX+bY)=a2σ2(X)+Forcomputationalconvenience,wecanget:σ2(X)=E(X2)–[E(X)]2,𝐸 35-36- 37-Covariancemeasureshowonerandomvariablemoveswithanotherrandomvariable.CovariancerangesfromnegativeinfinitytopositivePropertiesofIfXandYareindependentrandomvariables,theircovarianceisCov(X,X)=E[(X-E(X))(X-Cov(X+Y,Z)=Cov(X,Z)+Cov(Y,Cov(a+bX,c+dY)=bdCov(X,IfXandYareNOTindependent,σ2(X±Y)=σ2(X)+σ2(Y)±38-Correlation

𝐶PropertiesofCorrelationCorrelationhasnounits,rangesfrom-1toCorrelationmeasuresthelinearrelationshipbetweentwoIftwovariablesareindependent,theircovarianceiszero,therefore,thecorrelationcoefficientwillbezero.Theconverse,however,isnottrue.Forexample,Y=X2.Variancesofcorrelatedσ2(X±Y)=σ2(X)+σ2(Y)±39-Correlationr=perfectpositive0<r<positivelinearr=nolinear-1<r<negativelinearr=-perfectnegativeperfectpositivecorrelationrperfectpositivecorrelationr=perfectpositivecorrelationr=0.8perfectpositivecorrelationr=perfectnegativecorrelationr=-0.7perfectnegativecorrelationr=-StatisticalConceptsandMarketPositive(right)Negative(left)AdistributionthatisnotsymmetricaliscalledPositiveskewed：Mode<median<mean,havingarightfatNegativeskewed：Mode>media>mean,havingaleftfat41-StatisticalConceptsandMarketKurtosisisthestatisticalmeasure lsuswhenadistributionisorlesspeakedthananormalLeptokurticvs.AdistributionthatismorepeakedthannormaliscalledAdistributionthatislesspeakedthannormaliscalledSample42-StatisticalConceptsandMarket43-TheIntroductionTimeStatisticalProbabilityPointEstimateandConfidenceHypothesisLinearDate44-ProbabilityTwodefiningpropertiesofMultiplicationruleandAdditionDependentandindependentExpected45-46-47-48-ProbabilityBasicRandomvariablea tywhosevalueisuncertain.Thereturnonariskyassetisanexampleofarandomvariable.esarethepossiblevaluesofarandomAneventisaspecifiedset Mutuallyexclusiveevents—cannotbothhappenatthesameExhaustiveevents—includeall Probabilityofan𝐧𝑷𝑨

TwodefiningpropertiesofTheprobabilityofanyeventEisanumberbetween0and1:0≤P(E)≤Thesumoftheprobabilitiesofanysetofmutuallyexclusiveexhaustiveeventsequals1:P(E1)+P(E2)+……+49-ProbabilityUnconditionalprobability(marginalprobability):Supposethequestionis“Whatistheprobabilitythatthestockearnsareturnabovetherisk-freerate(eventA)?”TheanswerisanunconditionalprobabilitythatcanbeviewedastheratiooftwoConditionalprobability:Supposewewanttoknowtheprobabilitythatthestockearnsareturnabovetherisk-freerate(eventA),giventhatthestockearnsapositivereturn(eventB).Withthewords“giventhat,”wearerestrictingreturnstothoselargerthan0percent—anewelementincontrasttothequestionthatbroughtforthanunconditionalprobability.Theconditionalprobabilityiscalculatedastheratiooftwo P(A|B)P(AB);P(B)0 P(B|A)P(AB);P(A)0 P(A)50-ProbabilityJointprobability:MultiplicationThejointprobabilityofAandBcanbeIfAandBaremutuallyexclusivethen:ProbabilitythatatleastoneoftwoeventswillAdditionrule:GiveneventsAandB,theprobabilitythatAorBoccurs,orbothoccur,isequaltotheprobabilitythatAoccurs,plustheprobabilitythatBoccurs,minustheprobabilitythatbothAandBoccur.P(AorB)=P(A)+P(B)-IfAandBaremutuallyexclusiveevents,then:P(AorB)=P(A)+P(B)51-ProbabilityIndependentDefinitionofIndependentEvents:TwoeventsAandBareifandonlyifP(A|B)=P(A)or,equivalently,P(B|A)=MultiplicationRuleforIndependentEvents.Whentwoeventsareindependent,thejointprobabilityofAandBequalstheproductoftheindividualprobabilitiesofAandB:P(AB)=P(A)×P(B)IndependenceandMutuallyExclusivearequiteIfexclusive,mustnotCauseexclusivemeansifAoccur,Bcannotoccur,Ainfluents52-TheIntroductionofTimeStatisticalProbabilityPointEstimateandConfidenceHypothesisLinearDate53-CommonProbabilityCommonProbabilityPropertiesofdiscretedistributionandcontinuousDiscreteuniformContinuousuniformNormal54-CommonProbabilityDescribetheprobabilitiesofallthe esforaDiscreteandcontinuousrandomDiscreterandomvariablestakeonatmostacountablenumber esbutdonotnecessarilytobeContinuousrandomvariables:cannotdescribethe esacontinuousrandomvariableZwithalistz1,z2,...becausee(z1+z2)/2,notinthelist,wouldalwaysbeP(x)=0eventhoughxcanP55- Fordiscreterandom0≤p(x)≤Probabilitydensityfunction(p.d.f):ForcontinuousrandomvariableCumulativeprobabilityfunction(c.p.f):56-CumulativeDistributionFunction0

P(a≤X≤b)=F(b)– P(a≤X≤b)=Areaunderbetweenaand=F(b)–

57-BinomialBernoulliP(X=1)= P(X=0)=1–BinomialTheprobabilityofksuccessesinn

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1−ExpectationsandBernoullirandompp(1–Binomialrandomnp(1–58-BinomialBinomialk012……n59-BinomialTheBinomialDistribution–Binomial emoresymmetricasn60-DiscreteDiscreteuniformdistributionwouldbeaknown,finitenumberofesequallylikelytohappen.Everyoneofn eshasequalprobabilityForexample,rollingadicewillhave6 esInthatcase,theprobabilityforeach eis0.167[i.e.61-ContinuousUniformAllintervalsofthesamelengthontheContinuousUniformDistribution'ssupportareequallyprobable.Thesupportisdefinedbythetwoparameters,aandb,whichareminimum umPropertiesofContinuousuniformForalla≤x1<x2𝑃𝑋≤ P(X<aorX>b)=62-63-CommonProbabilityTheshapeofthedensityxX~N(µ,Symmetricaldistribution:skewness=0;Alinearcombinationofrandomvariablestheseareinnormallydistributionisalsonormallydistributed.Asthevaluesofxgetsfartherfromthemean,theprobabilitydensitygetsmallerandsmallerbutarealwayspositive.64-CommonProbabilityTheconfidence,𝜇+,𝜇+,𝜇+,𝜇+ 65-CommonProbabilityStandardnormalN(0,1)orStandardization:ifX~N(µ,σ²),Z-P(Z>z)=1

ZX

~66-67-CommonProbabilityDefinition:IflnXisnormal,thenXislognormal,whichisusedtothepriceofRightBoundedfrombelowbyzero,soitisusefulformodelingasset 1068-EstimationandHypothesisWhatisStatisticalConcernedwithdrawingconclusionsaboutthenatureorsomepopulation(e.g.,thenormal)onthebasisofarandomsamplethatsupposedlybeendrawnfromthatpopulation.Looselyspeaking,isthestudyoftherelationshipbetweenaandasampledrawnfromthatSamplingandsamplepopulation69-TheIntroductionofTimeStatisticalProbabilityProbabilityPointEstimateandConfidenceHypothesisLinearDate70-SamplingandPointthestatistic,computedfromsampleinformation,whichisusedtoestimatethepopulationparameterConfidenceintervalarangeofvaluesconstructedfromsampledatasotheparameterwithinthatrangeataspecified71-EstimationandHypothesisEstimationofPointUsingasinglenumericalvaluetoestimatetheparameterofthe𝑋→ 72-EstimationandHypothesisConfidenceintervalUsinganintervaltoestimatethescopeofthe73-CharacteristicsofProbabilityFromthePopulationtotheSampleThesamplemeanofarandomvariableXisgenerallydenotedbythesymbol(readasXbar)andisdefinedas=

ThesamplemeanisknownasanestimatorofE(X),whichwecannowcallthepopulationmean.Anestimateofthepopulationissimplythenumericalvaluetakenbyanestimator.74-SampleSample Thesamplevariance,denotedby𝑆𝑆2whichisanestimatorof𝜎𝜎2,whichwecannowcallthepopulationvariance.Thesamplevarianceis 2

=

𝑖−2−Theexpression(n–1)isknownasthedegreesofIfthesamplesizeisreasonablylarge,wecandividebyninsteadof(nSX(thepositivesquarerootofSamplevariance),iscalledthestandard75-TheCentralLimitTheCentralLimitTheoremIfX1,X2,…,Xnarandomsamplefromanypopulationprobabilitydistribution)withmeanand𝜎𝜎2,thesample2 𝑋tendstobenormallydistributedwithmean𝜇𝜇 asthesamplesizeincreases y(technically, y)Ofcourse,iftheXihappentobefromthenormalpopulation,thesamplemeanfollowsthenormaldistributionregardlessofthesamplesize.76-StatisticalInference:EstimationandHypothesisEstimationandHypothesisTesting:TwinBranchesOfStatisticalPricetoearning(P/E)ratiosof28companiesontheNewYorkstockexchange(NYSE)TCMean= Samplestandarddeviation=77-BestLinearUnbiasedEstimatorPropertiesofpointThemeanoftheestimatorscoincideswiththetrueparameter

=Anunbiasedestimatorisalsoefficientifthevarianceofitssamplingdistributionissmallerth ltheotherunbiasedestimatorsoftheparameteryouaretryingtoestimate.e.g.𝑁𝜇,

Theaccuracyoftheparameterestimateincreasesasthesamplesizeincreases(seethestandarderror).78-BestLinearUnbiasedEstimatorAnotherpropertyofapointestimateislinearity.Apointestimateshouldbealinearestimator(i.e.,itcanbeusedasalinearfunctionofthesampledata).Iftheestimatoristhebestavailable(i.e.,hastheminimumvariance),exhibitslinearity,andisunbiased,itissaidtobethebestlinearunbiasedesti