2025年CFA《固定收益》利率模型卷

2025年CFA《固定收益》利率模型卷考试时间：______分钟总分：______分姓名：______SectionA:MultipleChoiceQuestions1.Whichofthefollowingtheoriessuggeststhattheshapeoftheyieldcurveisdeterminedbyinvestors'preferencesforliquidityandriskpremiums?a)TheExpectationsTheoryb)TheLiquidityPreferenceTheoryc)TheMarketSegmentationTheoryd)ThePreferredHabitatTheory2.Theprocessoffittingamodeltoobservedmarketdata,typicallyzero-couponyields,toensurenoarbitrageopportunitiesexist,isknownas:a)Bootstrappingb)Calibrationc)Simulationd)Immunization3.Whichofthefollowinginterestratemodelsexplicitlyassumesthattheshortratehasamean-revertingproperty?a)TheBlack-Derman-Toy(BDT)modelb)TheBlack-Karasinskimodelc)TheHeath-Jarrow-Mercurey(HJM)modeld)TheSquared-Brownian-Motion(BGM)model4.Ameasureofthesensitivityofabond'spricetochangesininterestrates,calculatedasthepercentagechangeinpricefora100basispoint(1%)changeinyield,isbestdescribedas:a)MacaulayDurationb)ModifiedDurationc)Convexityd)EffectiveDuration5.WhichofthefollowingisakeyassumptionoftheCox-Ingersoll-Ross(CIR)model?a)Thevolatilityoftheshortrateisconstantovertime.b)TheshortratefollowsageometricBrownianmotion.c)Theshortrateismean-revertingaroundalong-termmean.d)Themodelassumesaflatyieldcurve.6.TheSABRmodelisoftenusedinpracticeforpricingderivativeson:a)Equityoptionsb)Foreignexchangeforwardsc)Interestrateswapsd)Creditdefaultswaps7.Whichofthefollowingtermsreferstothetheoreticalvalueofabondderivedfromazero-couponyieldcurve?a)MarketPriceb)FaceValuec)BookValued)PresentValue8.Iftheyieldtomaturityofabondincreases,thebond'smodifieddurationwill:a)Increaseb)Decreasec)Remainunchangedd)Becomenegative9.Theprimarygoalofimmunizationinfixedincomeportfoliomanagementisto:a)Maximizeportfolioyieldb)Minimizetheimpactofinterestrateriskonportfoliovaluec)Increasethedurationoftheportfoliod)Reducetheconvexityoftheportfolio10.WhichofthefollowingisalimitationofthePureExpectationsTheoryofthetermstructure?a)Itassumes投资者arerisk-averse.b)Itsuggeststhattheyieldcurvealwaysreflectsfutureshortrates.c)Itdoesnotaccountforliquiditypremiums.d)Itisdifficulttocalibratetomarketdata.11.Abondwithadurationof5yearswillexperienceapproximatelya_______changeinpriceifitsyieldtomaturitychangesby100basispoints.a)0.5%b)5%c)50%d)500%12.Whichofthefollowingmodelsisconsideredano-arbitragemodelforpricinginterestratederivatives?a)Theexpectationshypothesismodelb)Theliquiditypremiumtheorymodelc)TheHeath-Jarrow-Mercurey(HJM)modeld)Themarketsegmentationtheorymodel13.Theprocessofestimatingthezero-couponyieldcurvefrommarketpricesofcouponbondsisknownas:a)Immunizationb)Calibrationc)Bootstrappingd)Convexityadjustment14.WhichofthefollowingstatementsisTRUEregardingtherelationshipbetweendurationandconvexity?a)Durationaloneissufficienttoaccuratelyestimatethepricechangeforlargeyieldchanges.b)Convexityisgenerallynegativeformostfixedincomesecurities.c)Convexityhelpstoprovideamoreaccurateestimateofpricechange,especiallyforlargeryieldmovements.d)Thehighertheduration,thelowertheconvexityofabond.15.TheBGMmodelassumesthattheshortratefollowsageometricBrownianmotionandthatthevolatilitiesofdifferentmaturitiesarecorrelated.Whatparametercapturesthiscorrelationstructure?a)Alpha(α)b)Beta(β)c)rho(ρ)d)Sigma(σ)SectionB:CalculationQuestions16.Youaregiventhefollowingspotrates(annual,compoundedannually):*1-yearspotrate(S1)=2.00%*2-yearspotrate(S2)=2.50%*3-yearspotrate(S3)=2.75%Usingbootstrapping,calculatethetheoreticalpriceofa3-yearzero-couponbondwithafacevalueof$100.17.Considera5-yearbondwithafacevalueof$100,acouponrateof4%paidsemi-annually,andayieldtomaturity(YTM)of5%.Calculatethebond'smodifiedduration(assumingsemi-annualcouponpayments).18.YouaregiventhefollowingparametersforaCIRmodel:*Shortrate(r0)=2.00%(annual)*Meanreversionfactor(α)=0.1*Volatility(σ)=0.02*Timetomaturity(T)=10yearsCalculatetheone-yearforwardrate(f1)impliedbytheCIRmodel.19.A10-yearbondwithafacevalueof$1,000andacouponrateof6%(annualpayments)hasamodifieddurationof7.5yearsandaconvexityof120.Iftheyieldtomaturityincreasesby50basispoints,whatistheapproximatenewpriceofthebond?(AssumetheinitialpriceisthepresentvaluecalculatedattheoriginalYTM).20.YouareusingaBGMmodelwiththefollowingparameters:*Shortrate(r0)=3.00%(annual)*Volatility(σ)=0.01*Correlationparameter(ρ)=0.5(betweenratesatdifferentmaturities)*Timesteps(Δt)=0.5yearsSimulatetheshortrateforthenext2years(twotimesteps)startingfromr0.SectionC:ShortAnswerQuestions21.ExplainthedifferencebetweenMacaulaydurationandModifiedduration.InwhichscenariowouldModifieddurationbemoreappropriateformanaginginterestraterisk?22.DescribethekeyassumptionsoftheMarketSegmentationTheoryofthetermstructure.Whatarethemainimplicationsofthistheory?23.Whyisitimportantforaninterestratemodeltobe"no-arbitrage"?Provideanexampleofhowmodelcalibrationensuresno-arbitrageconditions.24.Discusstheroleofconvexityinbondportfoliomanagement.Howdoesconvexitybenefitaportfoliowheninterestrateschange?25.CompareandcontrasttheCIRmodelandtheBGMmodel.Identifyatleasttwokeydifferencesintheirassumptionsandapplications.试卷答案SectionA:MultipleChoiceQuestions1.b*解析思路：流动性偏好理论认为投资者为了承担长期债券的流动性风险和利率风险，会要求额外的流动性溢价，这会使得期限较长的债券收益率高于根据纯预期理论计算的水平，从而影响收益率曲线的形状。2.b*解析思路：校准是指调整模型参数，使其预测的收益率或价格与市场观察到的数据相匹配，确保模型符合无套利原则。Bootstrapping是利用短期零息债券价格估计零息收益率曲线的过程。Simulation是利用模型生成未来利率路径的过程。Immunization是管理利率风险的投资策略。3.c*解析思路：CIR模型的核心特征是短期利率服从一个随机过程，该过程具有均值回归性，即当利率高于某个长期平均水平时会倾向于下降，低于该水平时会倾向于上升。4.b*解析思路：ModifiedDuration定义为债券价格变化的百分比与收益率变化的百分比之比，即(-%ΔP/%Δy)。它衡量了价格对收益率变化的敏感度。5.c*解析思路：CIR模型的关键假设之一是短期利率遵循一个带有均值回归项的随机过程，即利率倾向于回归到一个长期均值水平。6.d*解析思路：SABR模型（StochasticAlpha,Beta,Gamma）因其能够较好地描述市场利率衍生品（如利率互换、国债期货期权）的隐含波动率结构而被广泛应用于这些产品的定价。7.d*解析思路：PresentValue(现值)是基于零息利率曲线计算得出的，代表了未来现金流的理论价值，是模型估值的基础。8.b*解析思路：根据久期公式，价格变化百分比与久期和收益率变化率成反比。当收益率（YTM）增加时，价格变化百分比减小，因此ModifiedDuration的衡量效果（即价格变化的百分比）会降低。9.b*解析思路：免疫策略的主要目标是使投资组合的价值对利率变动不敏感，从而在一定的市场利率变动范围内，保护投资组合价值免受损失。10.c*解析思路：纯预期理论假设长期债券的收益率等于市场对未来短期利率预期的平均值。该理论忽略了投资者对流动性、风险和交易成本的需求，因此不包含流动性溢价。这是其主要的局限性。11.b*解析思路：根据ModifiedDuration的定义，价格变化百分比约为-ModifiedDuration*Δy。若Δy为100basispoints(0.01)，则价格变化约为-5*0.01=-0.05，即-5%。12.c*解析思路：Heath-Jarrow-Mercurey(HJM)模型是一个动态的、无套利的框架，它描述了如何从今天的零息利率曲线出发，通过模拟未来利率树或过程，推导出所有未来时间的无套利价格。13.c*解析思路：Bootstrapping是通过已知的短期债券价格，逐步推算出期限更长的零息债券收益率的过程，从而构建整个零息利率曲线。14.c*解析思路：虽然久期提供了价格变动的线性近似，但当收益率变动较大时，这种近似不够准确。凸性可以修正久期估计的误差，尤其在收益率变动幅度较大时，凸性有助于更准确地估计价格变化，使曲线更弯曲。15.c*解析思路：在BGM模型中，不同期限利率的随机过程是相关的，这种相关性由参数rho(ρ)来度量，它决定了不同期限利率变动之间的联动程度。SectionB:CalculationQuestions16.$100*e^{-(2.00%*1)+(2.50%*1)+(2.75%*1)}=$100*e^{0.0100+0.0250+0.0275}=$100*e^{0.0625}=$100*1.064537=$106.45(roundedtotwodecimalplaces)*解析思路：使用零息利率计算零息债券价格。零息债券价格等于其面值按相应期限的零息利率折现后的现值。注意年化利率和折现期的对应。此处假设年化利率是有效年利率，折现也是一年一次。17.ModifiedDuration=MacaulayDuration/(1+(YTM/n))=(N*(T-t)*CF_t/P_0)/(1+(YTM/n))+(T-t)*CF_t/P_0*其中:N=10(semi-annualperiods),T=5years,t=0,CF_t=2(semi-annualcoupon),YTM=5%/2=2.5%(semi-annualYTM),P_0=C*[1-1/(1+y)^n]/y+F/(1+y)^n=2*[1-1/(1+0.025)^10]/0.025+100/(1+0.025)^10=2*[1-1/1.2800845]/0.025+100/1.2800845=2*[1-0.7811947]/0.025+78.11947=2*8.367736/0.025+78.11947=673.4218+78.11947=$751.54127*MacaulayDuration=10*(5-0)*2/751.54127+5*2/751.54127=100/751.54127+10/751.54127=0.132975+0.013297=0.146272*ModifiedDuration=0.146272/(1+(0.025/2))=0.146272/1.0125=0.1448years*解析思路：首先计算债券的当前价格P_0。然后计算MacaulayDuration（年化的MacaulayDuration）。最后，使用MacaulayDuration和半年度YTM计算ModifiedDuration。注意所有时间单位和利率单位需匹配。18.f1=r0+α*(r0-μ)*Δt+σ*sqrt(Δt)*z*其中:r0=0.02,α=0.1,Δt=1year,μ(long-termmean,oftenapproximatedbyr0forsimpleproblems)=0.02,σ=0.02.z~N(0,1).Assumingz=0forasinglestepsimplecalculationorusingastandardnormaldeviate(e.g.,z=0orz=1).*Ifz=0:f1=0.02+0.1*(0.02-0.02)*1+0.02*sqrt(1)*0=0.02+0+0=0.02or2.00%*Ifz=1:f1=0.02+0.1*(0.02-0.02)*1+0.02*sqrt(1)*1=0.02+0+0.02=0.04or4.00%*解析思路：CIR模型的短期利率向前一期的过程为：r_(t+Δt)=r_t+α*(μ-r_t)*Δt+σ*sqrt(Δt)*z，其中z是标准正态分布随机变量。一年后的即期利率f1可以表示为从时间0到时间1的利率变动。如果假设z=0（没有随机冲击），则f1=r0+α*(μ-r0)*Δt。如果考虑z=1，则f1=r0+α*(μ-r0)*Δt+σ*sqrt(Δt)。题目未明确z值，两种常见处理方式给出不同结果。通常校准过程会使用市场数据确定参数，包含z的期望影响。19.%ΔP≈-ModifiedDuration*Δy+0.5*Convexity*(Δy)^2*其中:ModifiedDuration=7.5years,Convexity=120,Δy=0.005(50basispoints).*%ΔP≈-7.5*0.005+0.5*120*(0.005)^2=-0.0375+0.5*120*0.000025=-0.0375+0.5*0.003=-0.0375+0.0015=-0.036*NewPrice≈InitialPrice*(1+%ΔP)=InitialPrice*(1-0.036)=InitialPrice*0.964*解析思路：使用修正久期和凸性进行价格变化的近似估计。计算价格变化的百分比，然后乘以初始价格得到新的近似价格。注意Δy是收益率变化率。20.r_(0.5)=r0+σ*sqrt(Δt)*z1=0.03+0.01*sqrt(0.5)