2025年CFA考试《数量方法》真题演练

2025年CFA考试《数量方法》真题演练考试时间：______分钟总分：______分姓名：______第一部分1.Aportfolioconsistsoftwoassets,AssetAandAssetB.TheexpectedreturnofAssetAis12%,theexpectedreturnofAssetBis8%,thevarianceofAssetA'sreturnis0.04,thevarianceofAssetB'sreturnis0.01,andthecovariancebetweenthereturnsofAssetAandAssetBis-0.005.Whatistheexpectedreturnofaportfoliothatconsistsof60%AssetAand40%AssetB?2.Thedailyreturnsofastockareapproximatelynormallydistributedwithameanof0.1%andastandarddeviationof1.5%.Whatistheprobabilitythatthestock'sreturnwillbenegativeinagivenday?3.Asampleof30observationsistakenfromapopulation.Thesamplemeanis50,andthesamplestandarddeviationis5.Whatisthe95%confidenceintervalforthepopulationmean?4.Aregressionanalysisyieldsthefollowingoutput:Intercept=10,Slope=2,R-squared=0.64,Standarderroroftheestimate=3.ThedependentvariableisY,andtheindependentvariableisX.AresearcherpredictsthevalueofYwhenX=5.WhatisthepredictedvalueofY,andwhatisthe95%predictionintervalforYgivenX=5?Assumethesamplesizeislarge.5.Thefollowingdatarepresentsthemonthlyreturnsoftwoassetsoverthepast6months:AssetA:5%,3%,-2%,7%,4%,1%AssetB:6%,4%,-1%,8%,5%,2%CalculatethecorrelationcoefficientbetweenthereturnsofAssetAandAssetB.6.Afinancialanalystwantstotestwhethertheaveragereturnofaparticularstockisdifferentfrom10%.Shecollectsasampleof25days'returnsandfindsasamplemeanof9.5%andasamplestandarddeviationof2%.Assumingthereturnsarenormallydistributed,whatistheteststatisticforthishypothesistestatthe5%significancelevel?第二部分7.Acompany'ssalesdataforthepast4quartersareasfollows:100,120,130,150.Usingamovingaveragemethodwithaspanof3quarters,whatistheforecastforthenextquarter?8.Thefirst-orderautoregressive(AR(1))modelisgivenbyY_t=0.8Y_{t-1}+ε_t,whereε_tisawhitenoiseerrorterm.Whatisthevalueoftheautoregressivecoefficient(ρ),andisthisprocessstationaryornon-stationary?9.Aportfoliomanagerwantstoconstructaportfoliousingtwoassets,XandY.Theexpectedreturns,standarddeviations,andcorrelationcoefficientbetweentheassetsareasfollows:ExpectedReturn:E(R_X)=12%,E(R_Y)=10%StandardDeviation:σ_X=15%,σ_Y=10%Correlation:ρ_{XY}=0.4Whatistheminimumvarianceportfolio'sweightinAssetX?10.Aresearcherisconductingachi-squaretestforindependenceonacontingencytablewith3rowsand4columns.Whatisthedegreesoffreedomforthistest?11.Thefollowingisapartialprintoutfromaregressionanalysis:VariableCoefficientStandardErrort-statisticX4.51.23.75Constant20.82.5R-squared=0.72Whatisthep-valueforthetestofthesignificanceoftheslopecoefficient(X)atthe5%significancelevel?12.Astock'sreturnsfollowageometricBrownianmotionwithdrift.Thedriftrateis8%,thevolatilityis20%,andtheinitialstockpriceis$50.Whatistheexpectedstockpriceafter1year?13.Aninvestorwantstoinvestinaportfoliooftwoassets,AandB.Theexpectedreturnsandstandarddeviationsoftheassetsareasfollows:E(R_A)=14%,σ_A=12%E(R_B)=10%,σ_B=8%ThecorrelationcoefficientbetweenAandBis0.2.Whatisthevarianceofaportfoliothatinvests60%inAssetAand40%inAssetB?14.Asampleof100observationsistakenfromapopulation.Thesamplemeanis100,andthesamplestandarddeviationis15.Whatisthe90%confidenceintervalforthepopulationproportion,assumingthesampleproportionis0.3?15.Thefollowingarethefirst-orderpartialderivativesofafunctionf(x,y)withrespecttoxandy:∂f/∂x=3x^2+2y∂f/∂y=2x+4y^3Atwhatpoint(s)(x,y)isthefunctionf(x,y)atacriticalpoint?第三部分16.Aninvestorrequiresa7%returnonariskyasset.Therisk-freerateis2%.Theexpectedreturnofthemarketportfoliois10%,andthestandarddeviationofthemarketportfoliois15%.Thebetaoftheassetis1.2.AccordingtotheCapitalAssetPricingModel(CAPM),whatistheexpectedreturnofthemarketportfolio?17.Thefollowingdatarepresentstheannualreturnsoftwostocksoverthepast5years:StockX:12%,15%,8%,10%,14%StockY:10%,8%,12%,6%,9%CalculatethevarianceofthereturnsforStockX.18.Afinancialanalystwantstotestwhetherthemeanweightofaparticularproductis500grams.Shetakesasampleof50productsandfindsasamplemeanof495gramsandasamplestandarddeviationof10grams.Assumingtheweightsarenormallydistributed,whatisthep-valueforthehypothesistestatthe1%significancelevel?19.Thefollowingisapartialprintoutfromaregressionanalysis:R-squared=0.65,Standarderroroftheestimate=5ThedependentvariableisY,andtheindependentvariableisX.Thesamplesizeis30.Whatisthecoefficientofdetermination(R-squared)interpretedas?20.Aportfolioconsistsofthreeassets,A,B,andC,withweightsof30%,40%,and30%respectively.Theexpectedreturnsoftheassetsare12%,10%,and8%.Thevariancesoftheassetsare0.04,0.02,and0.01,andthecovariancesbetweentheassetsareasfollows:Cov(A,B)=0.006,Cov(A,C)=0.004,Cov(B,C)=-0.005Whatistheexpectedreturnandvarianceoftheportfolio?21.Thefollowingisatimeseriesplotofavariable.Describethepatternobservedinthedata.22.Aresearcherwantstotestwhetherthereisasignificantdifferenceinthemeanscoresofthreedifferentgroups.Shecollectsdatafrom30individualsineachgroup.Whattypeofstatisticaltestshouldsheuse?23.ThefollowingistheprobabilitydensityfunctionofacontinuousrandomvariableX:f(x)=2xfor0≤x≤1f(x)=0otherwiseWhatisthemeanofX?24.Afinancialanalystwantstoconstructaportfoliowithaminimumvariance.Whichofthefollowingstatementsistrue?a)Theportfolioshouldinvest100%intheassetwiththeloweststandarddeviation.b)Theportfolioshouldinvestinassetsthatareperfectlynegativelycorrelated.c)Theportfolioshouldinvestinassetssuchthattheweightedsumoftheircovariancesisminimized.25.Asampleof200observationsistakenfromapopulation.Thesamplemeanis50,andthesamplestandarddeviationis10.Whatisthe95%confidenceintervalforthepopulationvariance?---试卷答案第一部分1.0.096or9.6%2.0.4332or43.32%3.(49.06,50.94)4.PredictedY=17;PredictionInterval=(10.54,23.46)5.-0.56.-1.25第二部分7.1258.ρ=0.8;Stationary9.0.5or50%10.711.Between0.01and0.02512.$53.9613.0.0176or1.76%14.(0.267,0.333)15.(x,y)=(0,0)第三部分16.12%17.9.618.Between0.005and0.0119.Theproportionofthetotalvariationinthedependentvariablethatisexplainedbytheindependentvariable(s).20.ExpectedReturn=11.4%;Variance=0.0164or1.64%21.[Assumeaplotshowinganupwardtrendwithincreasingvariabilityovertime]Describedpattern:Thedataexhibitsanupwardtrendovertime,butthevariabilityorvolatilityofthevariableappearstobeincreasing.22.One-wayANOVA23.2/3orapproximately0.666724.c25.(84.96,115.04)---解析第一部分1.解析思路：使用加权平均公式计算组合预期收益率：E(R_p)=w_A*E(R_A)+w_B*E(R_B)。其中，w_A=60%=0.6,w_B=40%=0.4。2.解析思路：根据正态分布性质，计算P(R<0)=P(Z<(0-0.1)/1.5)=P(Z<-0.0667)。查标准正态分布表或使用计算器得到概率值。3.解析思路：使用样本均值和标准差计算总体均值的95%置信区间：样本均值±(tcriticalvalue*samplestandarddeviation/sqrt(samplesize))。对于n=30，df=29，95%置信区间的tcriticalvalue约为2.045。计算上下限。4.解析思路：预测值Y=Intercept+Slope*X=10+2*5=20。95%预测区间：Y±(tcriticalvalue*StandardErroroftheEstimate*sqrt(1+1/n+(X-mean(X))^2))。由于n大，可近似为Y±(Zcriticalvalue*StandardErroroftheEstimate)=20±(1.96*3)。计算上下限。此处假设X的样本均值mean(X)已知或在题目中隐含，若未给出，则无法精确计算预测区间。5.解析思路：分别计算A和B的样本均值（mean_A,mean_B）。计算A和B的样本协方差（cov(A,B)=sum((A_i-mean_A)*(B_i-mean_B))/(n-1)）。计算A和B的样本标准差（std_A,std_B）。相关系数ρ=cov(A,B)/(std_A*std_B)。6.解析思路：计算检验统计量t=(samplemean-hypothesizedmean)/(samplestandarddeviation/sqrt(samplesize))=(9.5-10)/(2/sqrt(25))=-1.25。该统计量用于检验样本均值是否显著不同于假设值。第二部分7.解析思路：移动平均法（3期）：计算前3期的平均值(100+120+130)/3=120。将此平均值作为第4期（即下一期）的预测值。8.解析思路：AR(1)模型Y_t=ρY_{t-1}+ε_t。比较系数，得到autoregressivecoefficientρ=0.8。判断平稳性：若|ρ|<1，则过程是平稳的。由于|0.8|<1，该过程是平稳的。9.解析思路：使用最小方差组合公式：w_A=[σ_B^2-ρ_{XY}*σ_X*σ_Y]/[σ_X^2+σ_Y^2-2*ρ_{XY}*σ_X*σ_Y]。代入数据计算。10.解析思路：卡方检验自由度df=(numberofrows-1)*(numberofcolumns-1)=(3-1)*(4-1)=6。11.解析思路：查t分布表或使用计算器，根据自由度（df=n-2=30-2=28）和t统计量（3.75）查找双尾p值。p值位于0.005和0.01之间。12.解析思路：几何布朗运动模型：S_t=S_0*exp((μ-0.5*σ^2)*t+σ*Z*sqrt(t))。其中，μ=8%,σ=20%,S_0=50,t=1。由于题目要求期望值，使用期望的性质E(exp(a+bZ))=exp(a+0.5*b^2)。期望价格E(S_1)=50*exp((0.08-0.5*0.2^2)*1)=50*exp(0.076)≈53.96。13.解析思路：计算组合方差σ_p^2=w_A^2*σ_A^2+w_B^2*σ_B^2+2*w_A*w_B*Cov(A,B)。注意协方差Cov(A,B)=ρ_{XY}*σ_A*σ_B。代入数据计算。14.解析思路：使用样本比例和样本量计算总体比例的90%置信区间：样本比例±(Zcriticalvalue*sqrt(sampleproportion*(1-sampleproportion)/samplesize))。对于90%置信区间，Zcriticalvalue约为1.645。计算上下限。15.解析思路：找到函数f(x,y)的一阶偏导数∂f/∂x和∂f/∂y。令这两个偏导数同时等于零，解联立方程组：3x^2+2y=0和2x+4y^3=0。解得x=0,y=0是唯一的临界点。第三部分16.解析思路：CAPM公式E(R_i)=R_f+β_i*[E(R_m)-R_f]。已知E(R_i)=?,R_f=2%,E(R_m)=10%,β_i=1.2。代入公式求解E(R_m)。17.解析思路：计算StockX的样本方差s_X^2=sum((X_i-mean(X))^2)/(n-1)。其中，mean(X)=