CFA一级百题进阶题：数量.docx

2.Quantitative2.1.进阶题Q-l.Thetablebelowshowsthreemutuallyexclusive$2,000,000mortgagechoices.Eachofthethreechoicesiscompoundedmonthly.MortgagetypeQuotedannualinterestrateatinitiation32-yearfixedrate6.5%24-yearfixedrate6.0%32-yearadjustablerate4.5%Theadjustable-ratemortgagewillresetitsinterestrateto6.2%attheendoftheyear4.Afterresettingtheinterestrateattheendofyear4,whichmortgagewillhavethelargestmonthlypayment?A.32-yearfixedratemortgage.B.24-yearfixed-ratemortgage.C.32-yearadjustable-ratemortgage.Solution:B.Afteryear4,the24-yearfixed-ratemortgagehasthelargestpayment.Theloanpaymentsaresummarizedinthetablebelow.MortgagetypeInitialPayment($)Paymentafteradjustment($)32-yearfixed12z3S9.9212,389.9224-yearfixed13,119.5613,119.5632-yearadjustable9,836.9311,785.90Paymentonthe32-yearfixediscalculatedas:N=12×32=384zl/Y=6.5/12,PV=-2,000,000,FV=0;CPTPMT=12,389.92Paymentonthe24-yearfixediscalculatedas:N=12×24=288,l/Y=6/12,PV=-2,0f000zFV=O;CPTPMT=13,119.56Paymentonthe32-yearadjustableiscalculatedas:InitialpaymentN=12×32=384zl/Y=4.5/12,PV=2,000,000;FV=0;CPTPMT=9,836.93Balanceatendofyear4:N=12×28=336,l/Y=4.5/12,FV=0,PMT=9,83693;CPTPV=1,877,349.82Paymentaftertheendofyear4:N=336,l/Y=6.2/12,PV=-1,877,349.82;FV=0;CPTPMT=llz785.90Q-2.Whenrollingtwosix-sideddiceandsummingtheiroutcomes,whichofthefollowingsumsismostlikelytooccur?A.NineB.SixC.FiveSolution:B.Thisscenarioprovidesanexampleofadiscreterandomvariable.Thepairedoutcomesforthediceareindicatedinthefollowingtable.Theoutcomeofthedicesummingtosixisthemostlikelytooccurofthethreechoicesbecauseitcanoccurinfivedifferentways,whereasthesummationtofiveandninecanoccurinonlyfourdifferentways.SummedOutcomePairedOutcomes(Die1,Die2)PossibleCombinations5(1,4)/2,3),E2),and1)46(1,5),(2,4),(3,3),(4,2),and(5,1)59(3,6),(4,5),(5,4),and(6,3)4Q-3.Independentsamplesdrawnfromnormallydistributedpopulationsexhibitthefollowingcharacteristics:SampleSizeSampleMeanSampleStandardDeviationA2821050B2119565Assumingthatthevariancesoftheunderlyingpopulationsareequal,thepooledestimateofthecommonvarianceis3,377.13.Thet-teststatisticappropriatetotestthehypothesisthatthetwopopulationmeansareequalisclosestto:A.1.80.B.0.31.C0.89.Solution:C.Thet-statisticforthegiveninformation(normallydistributedpopulations,populationvariances0.89assumedequal)iscalculatedas:(210-195)-0t=3377.133377.13)0.5二Q-4.Twodistributionshavethesamemean.Oneisnegativelyskew,theotherispositiveskew.Whichonehasthelargermedian?A.DistributionwithnegativeskewB.DistributionwithpositiveskewC.ThesameSolution:A.Asshowninthefollowingfigure,themedianissmallerthanthemeanforthepositiveskew.Incontrastthemedianislargerthanthemeanforthenegativeskew.Mode<Median<MeanMean<Median<ModePositive(right)skewNegative(left)skewTherefore,ifthetwomeansequal,themedianofthenegativeskewislargerthanthatofpositiveskew.Population12Samplesizen二6n三6SamplevarianceSi2=5&=30Q-5.Thesamplesaredrawnindependently,andbothpopulationsareassumedtobenormallydistributedUsingtheabovedata,ananalystistryingtotestthenullhypothesisthatthepopulationvariancesareequal(Ho：妾=底)againstthealternativehypothesisthatthevariancesarenotequal(Ha:娈娈)atthe5%levelofsignificance.ThetableoftheF-Distributionisprovidedbelow.TableoftheF-DistributionPanelA:Criticalvaluesforright-handtailareasequalto0.05dfl(readacross)df2(readdown)123451161200216225230218.519.019.219.219.3310.19.559.289.129.0147.716.949.596,396,2656.615795,415195+05PanelB:Criticalvaluesforright-handtailareasequalto0.025dfl(readacross)df2(readdown)123451648799864900922238.5139.0039.1739.2539.30317.4416,0415.4415.1014.88412.2210,659.989.60936510.018.437.767.397.15Whichofthefollowingstatementsismostappropriate?Thecriticalvalueis:A.B.C.9.36andrejectthenull.9.60anddonotrejectthenull.7.15anddonotrejectthenull.Solution:C.Identifytheappropriateteststatisticandinterprettheresultsforahypothesistestconcerning1)thevarianceofanormallydistributedpopulation,and2)theequalityofthevariancesoftwonormallydistributedpopulationsbasedontwoindependentrandomsamples.numerator.Here,theteststatisticis30÷5=6.Thedegreesoffreedomare5by5.Becauseitisatwo-tailedtest,thecorrectcriticalvalueat=5%is7.15.Andbecausetheteststatisticislessthanthecriticalvalue,wecannotrejectthenullhypothesis.Q-6.Usingthefollowingsampleresultsdrawnas25pairedobservationsfromtheirunderlyingdistributions,testwhetherthemeanreturnsofthetwoportfoliosdifferfromeachotheratthe1%levelofstatisticalsignificance.Assumetheunderlyingdistributionsofreturnsforeachportfolioarenormalandthattheirpopulationvariancesarenotknown.Portfolio1Portfolio2DifferenceMeanreturn15.0020.255.25Standarddeviation15.5015.75625t-statisticfor24degreesoffreedomandatthe1%levelofstatisticalsignificance=IJllNullhypothesis(H0):Meandifferenceofreturns=0Basedonthepairedcomparisonstestofthetwoportfolios,themostappropriateconclusionisthatHOshouldbe:A.acceptedbecausethecomputedteststatisticexceeds1.711.B.rejectedbecausethecomputedteststatisticexceeds1.711.C.acceptedbecausethecomputedteststatisticislessthan1.711.Solution:B.Theteststatisticis:,.wheredisthemeandifference,7卜吞thehypothesizeddifferenceinthemeans,Sdisthesamplestandarddeviationofdifferences,andnisthesamplesize.Inthiscase,theteststatisticequals:(525-(6.252S)=4.20.Because4.20>1.711,thenullhypothesisthatthemeandifferenceiszeroisrejected.Q-7.Ifthepopulationdistributionisunknown,themethodthatwillleadtotheleastreliableestimationofaparameteristo:A.usepointestimatesinsteadofconfidenceintervalestimates.B.uset-distributioninsteadofstandardnormaldistributiontoestablishconfidenceintervalsC.drawmoresamplesSolution:A.Pointestimatesarelessreliablethanconfidenceintervalestimates.Usingthet-distributionratherthanthenormaldistributionisamoreconservativeapproachtoconstructconfidenceintervals,andthusincreasethereliabilityoftheconfidenceinterval.Increasingthesamplesizecanalsoincreasethereliabilityoftheconfidenceinterval.Q-8.Thetablebelowreportstheannualreturnsfortwoactiveportfoliosinthesameindustry,namely,theirreturnsaredependentwitheachother.YearPortfolioA(%)PortfolioB(%)2013119201440420151-320168122017212320182-4Ifwewanttotestwhetherthetwoportfolioshavethesamemeanreturnata5%significancelevel,theteststatisticsweshalluseisclosestto:A.1.96.B.1.66.C.0.45.Z = Lyd 一 L33%HJ iSolution:CFirst,calculatethereturndifferenceeachyear:YearPortfolioA(%)PortfolioB(%)Differences(%)2013119q201440414201513T20168124201721232201824=6Andcalculatethemeandifferenceofreturnsusingafinancialcalculator:Then,calculatethesamplestandarddeviationandthestandarderrorofthemeandifferenceusingafinancialcalculator:Vn-L 7.23%丁2.95% = 0.45Finally,calculatethet-statistic:t-Q-9.Inaheadandshoulderspatternzifthenecklineisat$23,theshouldersat$28,andtheheadat$33.Thepricetargetisclosesttowhichofthefollowing:A.$13.B.$19.C.$40.Solution:A.Headandshoulderspattern:Pricetarget=neckline-(head-neckline)=23-(33-23)=13.Q-10.Ananalysthasestablishedthefollowingpriorprobabilitiesregardingacompany,snextquarter'searningspershare(EPS)exceeding,equaling,orbeingbelowtheconsensusestimate.PriorPrababilitiesEPSexceedconsensus23%EPSequalconsensus56%EPSarelessthanconsensus21%Severaldaysbeforereleasingitsearningsstatement,thecompanyannouncesacutinitsdividend.Giventhisnewinformation,theanalystreviseshisopinionregardingthelikelihoodthatthecompanywillhaveEPSbelowtheconsensusestimate.Heestimatesthelikelihoodthecompanywillcutthedividend,giventhatEPSexceeds/meets/fallsbelowconsensus,asreportedbelow.ProbabilitiestheCompanyCutsDividends,ConditionalonEPSEXCeeding/Equaling/FallingbelowconsensusP(Cutdiv/EPSexceed)3%P(Cutdiv/EPSequal)11%P(Cutdiv/EPSbelow)86%UsingBayes'formula,theupdated(posterior)probabilitythatthecompany'sEPSarebelowtheconsensusisclosestto:A.73%.B.84%.C22%Solution:A.Bayes'formula:P(AB)=P(BA)P（八）P(B)Updatedprobabilityofeventgiventhenewinformation:whereUpdatedprobabilityofeventgiventhenewinformation:P(EPSbelowCutdiv);Probabilityofthenewinformationgivenevent:P(CutdivEPSbelow)=86%;Unconditionalprobabilityofthenewinformation:P(Cutdiv)=P(Cutdiv/EPSexceed)P(EPSe×ceed)+P(Cutdiv/EPSequal)P(EPSequal)+P(Cutdiv/EPSbelow)P(EPSbelow)=23%*3%+56%*ll%+21%*86%=0.69%+6.16%+18.06%=24.91%;Priorprobabilityofevent:P(EPSbelow)=21%.Therefore,theprobabilityofEPSfallingbelowtheconsensusisupdatedas:P(EPSbelowCutdiv)=P(CutdivEPSbelow)P(Cutdiv)×P(EPSbelow)=(0.86/0.2491)×0.2173%Q-ll.Samplesofsize(n,n)aredrawnrespectivelyfromtwopopulations(X,X)with12.12associatedsamplemeansandstandarddeviationsof(舛卜，苻F)and俄，S)andassociatedpopulationmeansandstandarddeviations0f(1,2)and(b2)where(12).Inaddition,7thesamplemeanof寸,汁卜withastandarderrorof甜升7andapopulationmeanof喜舌and制isapooledestimatorofthecommonvariance.ThemostappropriateteststatistictodeterminetheequalityofthetwopopulationmeansassumingXiandX2areindependentandnormallydistributedis:UP1寸旦¾-("一口广.C7H27)r(-RJl2vsSolution:C.Themostappropriateteststatisticforthedifferencebetweentwopopulationmeans(unequalandunknownpopulationvariances)ist=五2.Q-12.MonteCarlosimulationisbestdescribedas:A.arestrictiveformofscenarioanalysis.B.providingadistributionofpossiblesolutionstocomplexfunctions.C.anapproachtobacktestdata.Solution:B.MonteCarlosimulationprovidesadistributionofpossiblesolutionstocomplexfunctions.Thecentraltendencyandthevarianceofthedistributionofsolutionsgiveimportantcluestodecisionmakersregardingexpectedresultsandrisk.Q-13.WhichofthefollowingmostaccuratelydescribeshowtostandardizearandomvariableX?A.SubtractthemeanofXfromX,andthendividethatresultbythestandarddeviationofX.B.SubtractthemeanofXfromX,andthendividethatresultbythestandarddeviationofthestandardnormaldistribution.C.DivideXbythedifferencebetweenthestandarddeviationofXandthestandarddeviationofthestandardnormaldistribution.Solution:A.TherearetwostepsinstandardizingarandomvariableX:SubtractthemeanofXfromX,andthendividethatresultbythestandarddeviationofX.Thisisrepresentedbythefollowingformula:Z=(X-).Q-14.Adescriptivemeasureofapopulationcharacteristicisbestdescribedasa:A.parameter.B.frequencydistribution.C.samplestatistic.Solution:A.Anydescriptivemeasureofapopulationcharacteristiciscalledaparameter.Q-15.Thediscrepancybetweenastatisticallysignificantresultandaneconomicallymeaningfulresultisleastlikelytheresultof:A.transactioncosts.B.samplingerrors.C.risktolerance.Solution:B.Samplingerrorswillresultinstatisticalerror.Astatisticallysignificantresultmightnotbeeconomicallymeaningfulafterananalystaccountsfortherisk,transactioncosts,andapplicabletaxes.