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1、2.Quantitative2.1.进阶题Q-l.Thetablebelowshowsthreemutuallyexclusive$2,000,000mortgagechoices.Eachofthethreechoicesiscompoundedmonthly.MortgagetypeQuotedannualinterestrateatinitiation32-yearfixedrate6.5%24-yearfixedrate6.0%32-yearadjustablerate4.5%Theadjustable-ratemortgagewillresetitsinterestrateto6.2
2、%attheendoftheyear4.Afterresettingtheinterestrateattheendofyear4,whichmortgagewillhavethelargestmonthlypayment?A.32-yearfixedratemortgage.B.24-yearfixed-ratemortgage.C.32-yearadjustable-ratemortgage.Solution:B.Afteryear4,the24-yearfixed-ratemortgagehasthelargestpayment.Theloanpaymentsaresummarizedin
3、thetablebelow.MortgagetypeInitialPayment($)Paymentafteradjustment($)32-yearfixed12z3S9.9212,389.9224-yearfixed13,119.5613,119.5632-yearadjustable9,836.9311,785.90Paymentonthe32-yearfixediscalculatedas:N=1232=384zl/Y=6.5/12,PV=-2,000,000,FV=0;CPTPMT=12,389.92Paymentonthe24-yearfixediscalculatedas:N=1
4、224=288,l/Y=6/12,PV=-2,0f000zFV=O;CPTPMT=13,119.56Paymentonthe32-yearadjustableiscalculatedas:InitialpaymentN=1232=384zl/Y=4.5/12,PV=2,000,000;FV=0;CPTPMT=9,836.93Balanceatendofyear4:N=1228=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;
5、FV=0;CPTPMT=llz785.90Q-2.Whenrollingtwosix-sideddiceandsummingtheiroutcomes,whichofthefollowingsumsismostlikelytooccur?A.NineB.SixC.FiveSolution:B.Thisscenarioprovidesanexampleofadiscreterandomvariable.Thepairedoutcomesforthediceareindicatedinthefollowingtable.Theoutcomeofthedicesummingtosixisthemos
6、tlikelytooccurofthethreechoicesbecauseitcanoccurinfivedifferentways,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.Independentsamplesdraw
7、nfromnormallydistributedpopulationsexhibitthefollowingcharacteristics:SampleSizeSampleMeanSampleStandardDeviationA2821050B2119565Assumingthatthevariancesoftheunderlyingpopulationsareequal,thepooledestimateofthecommonvarianceis3,377.13.Thet-teststatisticappropriatetotestthehypothesisthatthetwopopulat
8、ionmeansareequalisclosestto: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.Whichonehas
9、thelargermedian?A.DistributionwithnegativeskewB.DistributionwithpositiveskewC.ThesameSolution:A.Asshowninthefollowingfigure,themedianissmallerthanthemeanforthepositiveskew.Incontrastthemedianislargerthanthemeanforthenegativeskew.ModeMedianMeanMeanMedian1.711,thenullhypothesisthatthemeandifferenceisz
10、eroisrejected.Q-7.Ifthepopulationdistributionisunknown,themethodthatwillleadtotheleastreliableestimationofaparameteristo:A.usepointestimatesinsteadofconfidenceintervalestimates.B.uset-distributioninsteadofstandardnormaldistributiontoestablishconfidenceintervalsC.drawmoresamplesSolution:A.Pointestima
11、tesarelessreliablethanconfidenceintervalestimates.Usingthet-distributionratherthanthenormaldistributionisamoreconservativeapproachtoconstructconfidenceintervals,andthusincreasethereliabilityoftheconfidenceinterval.Increasingthesamplesizecanalsoincreasethereliabilityoftheconfidenceinterval.Q-8.Thetab
12、lebelowreportstheannualreturnsfortwoactiveportfoliosinthesameindustry,namely,theirreturnsaredependentwitheachother.YearPortfolioA(%)PortfolioB(%)2013119201440420151-320168122017212320182-4Ifwewanttotestwhetherthetwoportfolioshavethesamemeanreturnata5%significancelevel,theteststatisticsweshalluseiscl
13、osestto:A.1.96.B.1.66.C.0.45.Z = Lyd 一 L33%HJ iSolution:CFirst,calculatethereturndifferenceeachyear:YearPortfolioA(%)PortfolioB(%)Differences(%)2013119q201440414201513T20168124201721232201824=6Andcalculatethemeandifferenceofreturnsusingafinancialcalculator:Then,calculatethesamplestandarddeviationand
14、thestandarderrorofthemeandifferenceusingafinancialcalculator: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.Headandshoulderspatter
15、n:Pricetarget=neckline-(head-neckline)=23-(33-23)=13.Q-10.Ananalysthasestablishedthefollowingpriorprobabilitiesregardingacompany,snextquartersearningspershare(EPS)exceeding,equaling,orbeingbelowtheconsensusestimate.PriorPrababilitiesEPSexceedconsensus23%EPSequalconsensus56%EPSarelessthanconsensus21%
16、Severaldaysbeforereleasingitsearningsstatement,thecompanyannouncesacutinitsdividend.Giventhisnewinformation,theanalystreviseshisopinionregardingthelikelihoodthatthecompanywillhaveEPSbelowtheconsensusestimate.Heestimatesthelikelihoodthecompanywillcutthedividend,giventhatEPSexceeds/meets/fallsbelowcon
17、sensus,asreportedbelow.ProbabilitiestheCompanyCutsDividends,ConditionalonEPSEXCeeding/Equaling/FallingbelowconsensusP(Cutdiv/EPSexceed)3%P(Cutdiv/EPSequal)11%P(Cutdiv/EPSbelow)86%UsingBayesformula,theupdated(posterior)probabilitythatthecompanysEPSarebelowtheconsensusisclosestto:A.73%.B.84%.C22%Solut
18、ion:A.Bayesformula:P(AB)=P(BA)P（八）P(B)Updatedprobabilityofeventgiventhenewinformation:whereUpdatedprobabilityofeventgiventhenewinformation:P(EPSbelowCutdiv);Probabilityofthenewinformationgivenevent:P(CutdivEPSbelow)=86%;Unconditionalprobabilityofthenewinformation:P(Cutdiv)=P(Cutdiv/EPSexceed)P(EPSec
19、eed)+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.
20、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isapooledestimatoro
21、fthecommonvariance.ThemostappropriateteststatistictodeterminetheequalityofthetwopopulationmeansassumingXiandX2areindependentandnormallydistributedis:UP1寸旦-(一口广.C7H27)r(-RJl2vsSolution:C.Themostappropriateteststatisticforthedifferencebetweentwopopulationmeans(unequalandunknownpopulationvariances)ist=
22、五2.Q-12.MonteCarlosimulationisbestdescribedas:A.arestrictiveformofscenarioanalysis.B.providingadistributionofpossiblesolutionstocomplexfunctions.C.anapproachtobacktestdata.Solution:B.MonteCarlosimulationprovidesadistributionofpossiblesolutionstocomplexfunctions.Thecentraltendencyandthevarianceofthed
23、istributionofsolutionsgiveimportantcluestodecisionmakersregardingexpectedresultsandrisk.Q-13.WhichofthefollowingmostaccuratelydescribeshowtostandardizearandomvariableX?A.SubtractthemeanofXfromX,andthendividethatresultbythestandarddeviationofX.B.SubtractthemeanofXfromX,andthendividethatresultbythesta
24、ndarddeviationofthestandardnormaldistribution.C.DivideXbythedifferencebetweenthestandarddeviationofXandthestandarddeviationofthestandardnormaldistribution.Solution:A.TherearetwostepsinstandardizingarandomvariableX:SubtractthemeanofXfromX,andthendividethatresultbythestandarddeviationofX.Thisisreprese
25、ntedbythefollowingformula: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.
