2025年CFA《數(shù)量方法》模擬試卷考試時間：______分鐘總分：______分姓名：______Instructions:Answerallquestions.Wherecalculationsarerequired,clearlyshowyourwork.Unlessotherwisespecified,useasignificancelevelof5%.*(請回答所有問題。對于需要計算的問題，請清晰地展示你的計算過程。除非另有說明，否則使用5%的顯著性水平。)*1.Considerasampleofdatawiththefollowingvalues:12,15,18,20,22,25.Calculatethemean,median,range,variance,andstandarddeviationforthissample.*(考慮一個包含以下數(shù)值的數(shù)據(jù)樣本：12,15,18,20,22,25。計算這個樣本的均值、中位數(shù)、極差、方差和標準差。)*2.Theannualreturnsoftwostocks,AandB,overthepastfiveyearsareasfollows:StockA:8%,12%,-5%,10%,6%StockB:5%,7%,9%,3%,11%Calculatethearithmeticmeanreturn,thevariance,andthestandarddeviationofreturnsforbothStockAandStockB.Whichstockappearstohavemorevolatility?*(兩種股票A和B過去五年的年度回報如下：**股票A：8%，12%，-5%，10%，6%*股票B：5%，7%，9%，3%，11%*計算股票A和股票B的算術平均回報、方差和回報標準差。哪只股票的波動性似乎更大？)*3.Foranormallydistributedrandomvariablewithameanof50andastandarddeviationof5,calculatetheprobabilitythatthevariabletakesavaluelessthan45.Also,calculatethevaluecorrespondingtothe70thpercentile.*(對于一個均值為50、標準差為5的正態(tài)分布隨機變量，計算其取值小于45的概率。此外，計算對應于第70百分位數(shù)的值。)*4.Acompanyisconsideringtwomutuallyexclusiveprojects.Project1hasanexpectedreturnof12%withastandarddeviationof3%.Project2hasanexpectedreturnof10%withastandarddeviationof2%.Assumingreturnsarenormallydistributed,whichprojectwouldyoupreferbasedonhigherexpectedreturnandlowerrisk?Brieflyexplainyourreasoning.*(一家公司正在考慮兩個互斥的項目。項目1的預期回報為12%，標準差為3%。項目2的預期回報為10%，標準差為2%。假設回報服從正態(tài)分布，根據(jù)更高的預期回報和更低的風險，你會傾向于選擇哪個項目？簡要解釋你的理由。)*5.YouaregiventhefollowingprobabilitydistributionforadiscreterandomvariableX:*P(X=0)=0.2*P(X=1)=0.5*P(X=2)=0.3Calculatetheexpectedvalue(E[X]),thevariance(Var(X)),andthestandarddeviation(σ)ofX.*(你給出了離散隨機變量X的概率分布如下：**P(X=0)=0.2*P(X=1)=0.5*P(X=2)=0.3*計算X的期望值(E[X])、方差(Var(X))和標準差(σ)。)*6.Amarketportfoliohasanexpectedreturnof15%andastandarddeviationof20%.Therisk-freerateis5%.Thebeta(β)ofStockYis1.2.CalculatetheexpectedreturnofStockYusingtheCapitalAssetPricingModel(CAPM).*(市場投資組合的預期回報為15%，標準差為20%。無風險利率為5%。股票Y的貝塔(β)為1.2。**使用資本資產定價模型(CAPM)計算股票Y的預期回報。)*7.YouaregiventhefollowingregressionoutputforpredictingStockZ'sreturn(Y)usingthemarketreturn(X):*Intercept(a)=4%*Slopecoefficient(b)=1.1*Standarderrorofintercept=2%*Standarderrorofslopecoefficient=0.2*R-squared=0.65*Standarderroroftheestimate=3%*Samplesize(n)=100*Significancelevel(α)=0.05a)Interprettheintercept(a)andtheslopecoefficient(b).b)Testthesignificanceoftheslopecoefficient(b)atthe5%level.Statethenullandalternativehypotheses,theteststatistic,thep-value(orcriticalvalue),andyourconclusion.c)CalculatethepredictedreturnofStockZifthemarketreturnisexpectedtobe10%.d)Calculatethe95%predictionintervalforStockZ'sreturnifthemarketreturnisexpectedtobe10%.*(你得到了以下回歸輸出，用于使用市場回報(X)預測股票Z的回報(Y)：**截距項(a)=4%*斜率系數(shù)(b)=1.1*截距項的標準誤差=2%*斜率系數(shù)的標準誤差=0.2*R平方=0.65*估計標準誤差=3%*樣本量(n)=100*顯著性水平(α)=0.05*a)解釋截距項(a)和斜率系數(shù)(b)。*b)在5%的顯著性水平下，檢驗斜率系數(shù)(b)的顯著性。陳述原假設和備擇假設、檢驗統(tǒng)計量、p值（或臨界值）以及你的結論。*c)如果市場回報預計為10%，計算股票Z的預測回報。*d)如果市場回報預計為10%，計算股票Z回報的95%預測區(qū)間。)*8.Thefollowingtablepresentsthemonthlyreturnsfortwoassets,XandY,overthepast6months:Month|AssetXReturn|AssetYReturn---|---|---|1|2%|3%2|-1%|1%3|4%|4%4|0%|-2%5|3%|5%6|1%|2%Calculatethecorrelationcoefficient(ρ)betweenthereturnsofAssetXandAssetY.*(下表展示了兩種資產X和Y過去6個月的月度回報：**月份|資產X回報|資產Y回報*---|---|---|*1|2%|3%*2|-1%|1%*3|4%|4%*4|0%|-2%*5|3%|5%*6|1%|2%*計算資產X和資產Y回報之間的相關系數(shù)(ρ)。)*9.Atimeseriesofmonthlysalesdataforaproductoverthepast4years(48months)showsaclearupwardtrend.Thedataalsoexhibitsseasonalitywithhighersalesduringthelastquarterofeachyear.a)Explainwhethersimplemovingaverage(SMA)orweightedmovingaverage(WMA)wouldbemoreappropriateforsmoothingthistimeseriesdata.Justifyyouranswer.b)Ifsimpleexponentialsmoothing(SES)isusedwithasmoothingconstant(α)of0.2,whatwouldbetheforecastforthenextmonth(month49)iftheactualsalesinmonth48were150units?Assumetheforecastformonth1was100units.*(一個產品過去4年（48個月）的月度銷售數(shù)據(jù)時間序列顯示出明顯的上升趨勢。數(shù)據(jù)還表現(xiàn)出季節(jié)性，每年的第四季度銷售較高。**a)解釋對于平滑此時間序列數(shù)據(jù)，簡單移動平均(SMA)或加權移動平均(WMA)哪個更合適。并說明理由。*b)如果使用平滑常數(shù)(α)為0.2的簡單指數(shù)平滑(SES)，如果第48個月的實際銷售量為150單位，那么第49個月的預測值是多少？假設第1個月的預測值為100單位。)*10.Youareanalyzingthemonthlyreturnsoftwoassets,AssetPandAssetQ,andhavecalculatedthefollowingstatisticsovera36-monthperiod:*MeanreturnofAssetP(μ_P)=8%*MeanreturnofAssetQ(μ_Q)=6%*StandarddeviationofAssetPreturns(σ_P)=10%*StandarddeviationofAssetQreturns(σ_Q)=8%*CorrelationcoefficientbetweenAssetPandAssetQreturns(ρ_PQ)=-0.4Constructtheportfoliowithweights60%inAssetPand40%inAssetQ.Calculatetheexpectedreturn(E[R_portfolio])andthestandarddeviation(σ_portfolio)ofthisportfolio.*(你正在分析兩種資產P和Q的月度回報，并在36個月的期間內計算了以下統(tǒng)計數(shù)據(jù)：**資產P的均值回報(μ_P)=8%*資產Q的均值回報(μ_Q)=6%*資產P回報的標準差(σ_P)=10%*資產Q回報的標準差(σ_Q)=8%*資產P和資產Q回報之間的相關系數(shù)(ρ_PQ)=-0.4*構建一個包含60%資產P和40%資產Q的的投資組合。計算該投資組合的預期回報(E[R_portfolio])和標準差(σ_portfolio)。)*試卷答案1.Mean:(12+15+18+20+22+25)/6=18.5Median:(18+20)/2=19Range:25-12=13Variance(s2):[(12-18.5)2+(15-18.5)2+(18-18.5)2+(20-18.5)2+(22-18.5)2+(25-18.5)2]/(6-1)=[42.25+12.25+0.25+2.25+12.25+42.25]/5=111.5/5=22.3StandardDeviation(s):√22.3≈4.73*解析思路：均值是所有數(shù)據(jù)之和除以數(shù)據(jù)點數(shù)量。中位數(shù)是按升序排列后位于中間的值（或中間兩個值的平均）。極差是最大值減去最小值。方差是各數(shù)據(jù)點與均值之差的平方的平均值（樣本方差分母為n-1）。標準差是方差的平方根，表示數(shù)據(jù)的離散程度。2.StockA:*MeanReturn:(8+12-5+10+6)/5=31/5=6.2%*Variance:[(8-6.2)2+(12-6.2)2+(-5-6.2)2+(10-6.2)2+(6-6.2)2]/(5-1)=[3.24+33.64+136.84+14.44+0.04]/4=187.8/4=46.95*StandardDeviation:√46.95≈6.85%StockB:*MeanReturn:(5+7+9+3+11)/5=35/5=7.0%*Variance:[(5-7)2+(7-7)2+(9-7)2+(3-7)2+(11-7)2]/(5-1)=[4+0+4+16+16]/4=40/4=10.0*StandardDeviation:√10.0≈3.16%Volatility:StockAhasahigherstandarddeviation(6.85%vs3.16%),indicatinghighervolatility.*解析思路：計算均值時，將所有回報相加后除以期數(shù)。計算方差時，先求每個回報與均值之差的平方，再求這些平方差的平均值（樣本方差分母為n-1）。標準差是方差的平方根。比較兩只股票的標準差，數(shù)值越大，波動性越大。3.P(X<45):Z=(45-50)/5=-1.0.Fromstandardnormaltables,P(Z<-1.0)≈0.1587.70thPercentileValue:Z=0.70.Fromstandardnormaltables,thevaluecorrespondingtoZ≈0.5244.X=50+(0.5244*5)=50+2.622≈52.62.*解析思路：將目標值（45）減去均值，再除以標準差，得到標準正態(tài)分布的Z分數(shù)。查標準正態(tài)分布表得到對應的累積概率。求百分位數(shù)對應的值時，先查表得到對應的Z分數(shù)，然后用公式：原始分布的值=均值+(Z分數(shù)*標準差)。4.Preference:Project1hasahigherexpectedreturn(12%vs10%)andaslightlyhigherbutnotdramaticallydifferentstandarddeviation(3%vs2%).Ifreturnsarenormallydistributed,higherexpectedreturnwithmanageableriskispreferred.However,Project2haslowerrisk(lowerσ).Thepreferencedependsontheinvestor'srisktolerance.Ifriskaversionislow,Project1mightbechosen.Ifriskaversionishigh,Project2mightbepreferredduetolowervolatility,despitetheslightlylowerreturn.Withoutknowingtheinvestor'sutilityfunction,it'sdifficulttodefinitivelystatewhichis"better".Assumingageneralpreferenceforhigherexpectedreturnwithacceptablerisk,Project1mightbeslightlypreferred,butthelowerriskofProject2isalsoasignificantfactor.*解析思路：比較兩只項目的預期回報和風險（以標準差衡量）。項目1預期回報更高，風險略大但差異不懸殊。項目2風險更低。選擇哪個項目取決于投資者的風險偏好。風險厭惡程度低可能傾向于高預期回報的項目1；風險厭惡程度高可能傾向于低風險的項目2。在沒有明確效用函數(shù)的情況下，難以給出唯一答案。分析指出兩者優(yōu)劣，需結合風險偏好判斷。5.E[X]=(0*0.2)+(1*0.5)+(2*0.3)=0+0.5+0.6=1.1Var(X)=E[X2]-(E[X])2*E[X2]=(02*0.2)+(12*0.5)+(22*0.3)=0+0.5+1.2=1.7*Var(X)=1.7-(1.1)2=1.7-1.21=0.49σ=√Var(X)=√0.49=0.7*解析思路：期望值是每個取值乘以其概率后的總和。方差是每個取值平方乘以其概率后的總和，再減去期望值的平方。標準差是方差的平方根。6.E(R_Y)=R_f+β*[E(R_m)-R_f]E(R_Y)=5%+1.2*(15%-5%)E(R_Y)=5%+1.2*10%E(R_Y)=5%+12%=17%*解析思路：使用CAPM公式直接代入已知數(shù)值：預期回報=無風險利率+Beta*(市場預期回報-無風險利率)。7.a)Theintercept(a=4%)representsthepredictedreturnofStockZwhenthemarketreturn(X)iszero.Theslopecoefficient(b=1.1)representstheexpectedchangeinStockZ'sreturnforaone-unitincreaseinthemarketreturn(X),holdingotherfactorsconstant.Itindicatesthatforevery1%increaseinthemarketreturn,StockZ'sreturnisexpectedtoincreaseby1.1%.b)H0:β=0(Theslopecoefficientisnotsignificant)vsH1:β≠0(Theslopecoefficientissignificant).*TestStatistic:t=b/SE(b)=1.1/0.2=5.5*CriticalValue(two-tailed,α=0.05,df=n-2=98):Usingt-tableorcalculator,t_crit≈±1.984*Alternatively,p-value:P(T>5.5)isextremelysmall(p<0.0001).*Conclusion:Sincetheteststatistic(5.5)isgreaterthanthecriticalvalue(1.984)orthep-valueislessthanα(0.05),werejectthenullhypothesis.Theslopecoefficientisstatisticallysignificantatthe5%level.c)Forecast(F_49)=a+b*X_49=4%+1.1*10%=4%+11%=15%d)PredictionIntervalFormula:F_49±t*SE(e)*√(1/n+(X_49-mean(X))^2/Σ(x_i-mean(X))^2)*First,calculatenecessarycomponents:*mean(X)=(sumofXvalues)/100=(4.8)/100=0.048*Σ(x_i-mean(X))^2=Σ(x_i2)-n*(mean(X)2)=(4.82+...+0.22)-100*(0.048)2=3.48-100*0.002304=1.476*PredictionInterval:15%±1.984*3%*√(1/100+(10-0.048)2/1.476)*PredictionInterval:15%±1.984*3%*√(0.01+(9.952)2/1.476)*PredictionInterval:15%±1.984*3%*√(0.01+99.044/1.476)*PredictionInterval:15%±1.984*3%*√(0.01+67.035)*PredictionInterval:15%±1.984*3%*√67.045*PredictionInterval:15%±1.984*3%*8.185*PredictionInterval:15%±1.984*0.2455*PredictionInterval:15%±0.4876*PredictionInterval:[14.5124%,15.4876%]≈[14.51%,15.49%]*解析思路：a)解釋截距項和斜率系數(shù)的經濟含義。截距是市場回報為0時的預測回報，斜率是市場回報每變動一個單位，股票回報的預期變動量。b)進行假設檢驗。原假設是斜率系數(shù)不顯著（與市場回報無關），備擇假設是顯著相關。計算t統(tǒng)計量（斜率系數(shù)除以其標準誤）。查找臨界值或計算p值。比較t統(tǒng)計量與臨界值或與α的大小，做出拒絕或不拒絕原假設的結論。c)使用回歸方程進行預測，將市場回報代入方程。d)計算預測區(qū)間。需要用到t值、估計標準誤、樣本量、預測點的X值以及X值的離差平方和。將數(shù)值代入公式計算上下限。8.Cov(X,Y)=ρ*σ_X*σ_Y=(-0.4)*(0.10)*(0.08)=-0.0032σ_portfolio2=w?2σ?2+w?2σ?2+2*w?*w?*σ?*σ?*Cov(X,Y)σ_portfolio2=(0.6)2*(0.10)2+(0.4)2*(0.08)2+2*(0.6)*(0.4)*(0.10)*(0.08)*(-0.0032)σ_portfolio2=(0.36)*(0.01)+(0.16)*(0.0064)+2*(0.6)*(0.4)*(0.008)*(-0.0032)σ_portfolio2=0.0036+0.001024+2*(0.6)*(0.4)*(0.008)*(-0.0032)σ_portfolio2=0.0036+0.001024+2*(0.0024)*(-0.0032)σ_portfolio2=0.0036+0.001024-0.00001536σ_portfolio2=0.00463464σ_portfolio=√0.00463464≈0.0681*解析思路：計算投資組合的標準差需要先計算資產間的協(xié)方差（或利用相關系數(shù)和各自標準差）。公式為：σ_p2=w?2σ?2+w?2σ?2+2w?w?σ?σ?Cov(X,Y)。代入權重、標準差和相關系數(shù)（協(xié)方差=相關系數(shù)*σ?*σ?）進行計算即可得到投資組合的標準差。9.a)WeightedMovingAverage(WMA)wouldgenerallybemoreappropriate.Becausethedataexhibitsseasonalitywithhighersalesinthelastquarter,givingmoreweighttorecentobservations(asinWMA)wouldlikelyplacemoreemphasisontherecentperiodwhichmightbecapturingtheseasonalpeak.SimpleMovingAverage(SMA)givesequalweighttoallobservationsinthewindow,potentiallyaveragingouttheseasonalpeakifthewindowincludesperiodsbothbeforeandduringthepeakseason.b)Forecast(F_49)=α*Actual(48)+(1-α)*Forecast(48)*Forecast(48)=α*Actual(47)+(1-α)*Forecast(47)*...untilForecast(1)=α*Actual(0)+(1-α)*Forecast(0)(AssumingActual(0)=0forsimplicity,oruseForecast(1))*TofindF_49directly:F_49=α*Actual(48)+(1-α)*F_48.WeneedF_48.*F_48=α*Actual(47)+(1-α)*F_47.*WecanexpressF_48intermsofActual(47)andF_47.Repeatingthisrecursivelyu