2025年CFA《固定收益》利率模型卷考試時間：______分鐘總分：______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投資者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.試卷答案SectionA:MultipleChoiceQuestions1.b*解析思路：流動性偏好理論認(rèn)為投資者為了承擔(dān)長期債券的流動性風(fēng)險和利率風(fēng)險，會要求額外的流動性溢價，這會使得期限較長的債券收益率高于根據(jù)純預(yù)期理論計算的水平，從而影響收益率曲線的形狀。2.b*解析思路：校準(zhǔn)是指調(diào)整模型參數(shù)，使其預(yù)測的收益率或價格與市場觀察到的數(shù)據(jù)相匹配，確保模型符合無套利原則。Bootstrapping是利用短期零息債券價格估計零息收益率曲線的過程。Simulation是利用模型生成未來利率路徑的過程。Immunization是管理利率風(fēng)險的投資策略。3.c*解析思路：CIR模型的核心特征是短期利率服從一個隨機(jī)過程，該過程具有均值回歸性，即當(dāng)利率高于某個長期平均水平時會傾向于下降，低于該水平時會傾向于上升。4.b*解析思路：ModifiedDuration定義為債券價格變化的百分比與收益率變化的百分比之比，即(-%ΔP/%Δy)。它衡量了價格對收益率變化的敏感度。5.c*解析思路：CIR模型的關(guān)鍵假設(shè)之一是短期利率遵循一個帶有均值回歸項的隨機(jī)過程，即利率傾向于回歸到一個長期均值水平。6.d*解析思路：SABR模型（StochasticAlpha,Beta,Gamma）因其能夠較好地描述市場利率衍生品（如利率互換、國債期貨期權(quán)）的隱含波動率結(jié)構(gòu)而被廣泛應(yīng)用于這些產(chǎn)品的定價。7.d*解析思路：PresentValue(現(xiàn)值)是基于零息利率曲線計算得出的，代表了未來現(xiàn)金流的理論價值，是模型估值的基礎(chǔ)。8.b*解析思路：根據(jù)久期公式，價格變化百分比與久期和收益率變化率成反比。當(dāng)收益率（YTM）增加時，價格變化百分比減小，因此ModifiedDuration的衡量效果（即價格變化的百分比）會降低。9.b*解析思路：免疫策略的主要目標(biāo)是使投資組合的價值對利率變動不敏感，從而在一定的市場利率變動范圍內(nèi)，保護(hù)投資組合價值免受損失。10.c*解析思路：純預(yù)期理論假設(shè)長期債券的收益率等于市場對未來短期利率預(yù)期的平均值。該理論忽略了投資者對流動性、風(fēng)險和交易成本的需求，因此不包含流動性溢價。這是其主要的局限性。11.b*解析思路：根據(jù)ModifiedDuration的定義，價格變化百分比約為-ModifiedDuration*Δy。若Δy為100basispoints(0.01)，則價格變化約為-5*0.01=-0.05，即-5%。12.c*解析思路：Heath-Jarrow-Mercurey(HJM)模型是一個動態(tài)的、無套利的框架，它描述了如何從今天的零息利率曲線出發(fā)，通過模擬未來利率樹或過程，推導(dǎo)出所有未來時間的無套利價格。13.c*解析思路：Bootstrapping是通過已知的短期債券價格，逐步推算出期限更長的零息債券收益率的過程，從而構(gòu)建整個零息利率曲線。14.c*解析思路：雖然久期提供了價格變動的線性近似，但當(dāng)收益率變動較大時，這種近似不夠準(zhǔn)確。凸性可以修正久期估計的誤差，尤其在收益率變動幅度較大時，凸性有助于更準(zhǔn)確地估計價格變化，使曲線更彎曲。15.c*解析思路：在BGM模型中，不同期限利率的隨機(jī)過程是相關(guān)的，這種相關(guān)性由參數(shù)rho(ρ)來度量，它決定了不同期限利率變動之間的聯(lián)動程度。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：使用零息利率計算零息債券價格。零息債券價格等于其面值按相應(yīng)期限的零息利率折現(xiàn)后的現(xiàn)值。注意年化利率和折現(xiàn)期的對應(yīng)。此處假設(shè)年化利率是有效年利率，折現(xiàn)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：首先計算債券的當(dāng)前價格P_0。然后計算MacaulayDuration（年化的MacaulayDuration）。最后，使用MacaulayDuration和半年度YTM計算ModifiedDuration。注意所有時間單位和利率單位需匹配。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過程為：r_(t+Δt)=r_t+α*(μ-r_t)*Δt+σ*sqrt(Δt)*z，其中z是標(biāo)準(zhǔn)正態(tài)分布隨機(jī)變量。一年后的即期利率f1可以表示為從時間0到時間1的利率變動。如果假設(shè)z=0（沒有隨機(jī)沖擊），則f1=r0+α*(μ-r0)*Δt。如果考慮z=1，則f1=r0+α*(μ-r0)*Δt+σ*sqrt(Δt)。題目未明確z值，兩種常見處理方式給出不同結(jié)果。通常校準(zhǔn)過程會使用市場數(shù)據(jù)確定參數(shù)，包含z的期望影響。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進(jìn)行價格變化的近似估計。計算價格變化的百分比，然后乘以初始價格得到新的近似價格。注意Δy是收益率變化率。20.r_(0.5)=r0+σ*sqrt(Δt)*z1=0.03+0.01*sqrt(0.5)