2025CFA《数量方法》历年真题解析

2025CFA《数量方法》历年真题解析考试时间：______分钟总分：______分姓名：______一、选择题1.Aprobabilitydistributionshowsthepossibleoutcomesofanexperimentandtherelativelikelihoodofeachoutcome.Whichofthefollowingbestdescribesaprobabilitydistribution?I.Italwayssumsto1.II.Theoutcomesmustbemutuallyexclusive.III.Theprobabilitiesmustbenon-negative.Selectthecorrectanswerusingthefollowingformat:(e.g.,A,B,C,D)2.IfXisarandomvariablefollowingaPoissondistributionwithameanof3,whatistheprobabilitythatXequals0?Provideyouranswerroundedtofourdecimalplaces.3.Theexpectedreturnofanassetis12%withastandarddeviationof18%.Ifthereturnsarenormallydistributed,approximatelywhatrangeofreturnswouldyouexpecttosee68%ofthetime?Statethelowerandupperboundsoftherangeastwoseparatepercentagesroundedtoonedecimalplace.4.Afinancialanalystistestingthenullhypothesisthattheaveragereturnofastockiszero.Sheusesasignificancelevel(α)of0.05.Ifthecalculatedp-valueis0.03,whatistheanalyst'sconclusion?Indicatewhetherthenullhypothesisisrejectedornotrejected.5.Inasimplelinearregressionanalysis,thecoefficientofdetermination(R-squared)is0.65.Whatdoesthisvalueimply?Provideabriefexplanationinnomorethan30words.6.TheaverageannualreturnonStockAis10%withastandarddeviationof15%.TheaverageannualreturnonStockBis8%withastandarddeviationof10%.Whichstockhasahigherrelativedispersionofreturns?Statethenameofthestock.7.Whichofthefollowingstatementsaboutthenormaldistributioniscorrect?I.Itissymmetricarounditsmean.II.Themean,median,andmodeareallequal.III.Thetotalareaunderthecurveisnotequalto1.Selectthecorrectanswerusingthefollowingformat:(e.g.,A,B,C,D)8.Aresearchercollectsasampleof30observationstoestimatethepopulationmean.Thesamplestandarddeviationis5.Iftheresearcherwantstoconstructa95%confidenceintervalforthepopulationmean,whatisthedegreesoffreedomforthet-distributionused?Provideyouranswerasaninteger.9.Inamultipleregressionmodelwith3independentvariablesand1dependentvariable,theF-statisticiscalculatedtobe12.5.ThecriticalF-valueatthe0.05significancelevelwith3numeratordegreesoffreedomand96denominatordegreesoffreedomis2.61.Whatistheconclusionregardingtheoverallsignificanceofthemodel?Indicatewhetherthenullhypothesisisrejectedornotrejected.10.Thecovariancebetweenthereturnsoftwoassetsis50.IfthestandarddeviationofAsset1is20%andthestandarddeviationofAsset2is30%,whatisthecorrelationcoefficientbetweenthetwoassets?Provideyouranswerroundedtothreedecimalplaces.二、计算题1.Aninvestmentportfolioconsistsofthreeassets:StockX,StockY,andBondZ.Thefollowinginformationisavailable:*ExpectedreturnofStockX(Rx)=14%*ExpectedreturnofStockY(Ry)=9%*ExpectedreturnofBondZ(Rz)=6%*WeightofStockX(wx)=0.4*WeightofStockY(wy)=0.3*WeightofBondZ(wz)=0.3Calculatetheexpectedreturnoftheportfolio.2.Acompanyisconsideringtwomutuallyexclusiveprojects.Theinitialinvestmentandexpectedend-of-yearcashflowsforeachprojectareasfollows:|Project|InitialInvestment|Year1CashFlow|Year2CashFlow|Year3CashFlow||--------|-------------------|-----------------|-----------------|-----------------||A|$100,000|$40,000|$50,000|$60,000||B|$150,000|$60,000|$70,000|$0|CalculatetheNetPresentValue(NPV)ofProjectAassumingadiscountrateof10%.3.ArandomvariableXfollowsanormaldistributionwithamean(μ)of50andastandarddeviation(σ)of5.CalculatetheprobabilitythatXisgreaterthan60.Provideyouranswerroundedtofourdecimalplaces.4.Youaregiventhefollowingdatafortwovariables,XandY:*n(numberofobservations)=25*ΣXi=125*ΣYi=150*ΣXi^2=650*ΣYi^2=875*ΣXiYi=737.5a.Calculatetheslope(b1)andintercept(b0)ofthesimplelinearregressionlineY=b0+b1X.b.Calculatethestandarderroroftheestimate(Sε).三、解释题1.ExplainthedifferencebetweenaTypeIerrorandaTypeIIerrorinthecontextofhypothesistesting.Provideanexampleforeachtypeoferror.2.Describetheconceptofskewnessinaprobabilitydistribution.Whatdoesanegativeskewnessvalueindicateaboutthedistributionofdata?3.Afinancialanalystusesmultipleregressiontomodelthereturnofanasset(Y)basedonthreeindependentvariables:marketreturn(X1),interestrate(X2),andinflation(X3).Theregressionoutputprovidesthefollowingcoefficients:*b0(intercept)=1.5%*b1(marketreturncoefficient)=0.8*b2(interestratecoefficient)=-0.3*b3(inflationcoefficient)=2.0Interpretthemeaningofthecoefficientsb1,b2,andb3inthecontextofthemodel.试卷答案一、选择题1.A2.0.04983.9.0%to15.0%4.Rejected5.Itindicatesthat65%ofthevariationinthedependentvariableisexplainedbytheindependentvariable(s)intheregressionmodel.6.StockA7.A8.299.Rejected10.0.577二、计算题1.ExpectedReturn=(0.4*14%)+(0.3*9%)+(0.3*6%)=5.6%+2.7%+1.8%=10.1%2.NPV_A=(-$100,000)+$40,000/(1+0.10)^1+$50,000/(1+0.10)^2+$60,000/(1+0.10)^3NPV_A=(-$100,000)+$36,363.64+$41,322.31+$45,078.63=$22,764.583.Z=(60-50)/5=2.0P(X>60)=P(Z>2.0)=1-P(Z<=2.0)=1-0.9772=0.02284.a.*Cov(X,Y)=ΣXiYi-(ΣXi)(ΣYi)/n=737.5-(125)(150)/25=737.5-750=-12.5*Var(X)=ΣXi^2-(ΣXi)^2/n=650-(125)^2/25=650-625=25*b1=Cov(X,Y)/Var(X)=-12.5/25=-0.5*b0=mean(Y)-b1*mean(X)=150/25-(-0.5)*125/25=6+2.5=8.5TheregressionlineisY=8.5-0.5X.b.*Sε=sqrt[(ΣYi^2-b0ΣYi-b1ΣXiYi)/(n-2)]*Sε=sqrt[(875-8.5*150-(-0.5)*737.5)/(25-2)]*Sε=sqrt[(875-1275+368.75)/23]*Sε=sqrt[(-131.25+368.75)/23]=sqrt[237.5/23]=sqrt[10.32608696]≈3.215三、解释题1.ATypeIerroroccurswhenthenullhypothesis(H0)istrue,butitisincorrectlyrejected.Example:Concludingthatanewinvestmentstrategysignificantlyoutperformsthemarketwhenitactuallydoesnot.ATypeIIerroroccurswhenthenullhypothesis(H0)isfalse,butitisincorrectlynotrejected.Example:Concludingthatanewinvestmentstrategydoesnotsignificantlyoutperformthemarketwhenitactuallydoes.2.Skewnessmeasurestheasymmetryofaprobabilitydistribution.Anegativeskewnessva