2025年CFA三級(jí)投資組合管理試卷（含答案）考試時(shí)間：______分鐘總分：______分姓名：______PartA:MultipleChoiceQuestions1.Aninvestorisconsideringaddinganewassettotheirportfolio.Whichofthefollowingstatementsbestdescribestheimpactofaddinganassetthathasalowcorrelationwiththeexistingportfolio?A)Theoverallportfoliovariancewillnecessarilydecrease.B)TheSharperatiooftheportfoliowillunambiguouslyimprove.C)Theefficientfrontieroftheportfoliowillshiftoutward,offeringhigherreturnsforthesamelevelofrisk.D)Theexpectedreturnoftheportfoliowillincreasewithoutalteringitsrisk.2.AccordingtotheCapitalAssetPricingModel(CAPM),theexpectedreturnofanindividualassetisprimarilydeterminedby:A)Itsbetacoefficientandtherisk-freerate.B)Itscovariancewiththemarketportfolioandthemarketriskpremium.C)Theasset'salphaandtheTreynorratio.D)Theasset'sstandarddeviationanditscorrelationwithindividualstocksinthemarket.3.Aportfoliomanagerusesasingle-factormodeltoexplainthereturnsofastockportfolio.Ifthefactoristhemarketreturn,whichofthefollowingtermsbestrepresentsthesensitivityofthestock'sreturnstochangesinthemarketfactor?A)AlphaB)BetaC)R-squaredD)Treynorratio4.WhichofthefollowingstatementsaccuratelycharacterizestheFama-FrenchThree-FactorModelcomparedtotheCapitalAssetPricingModel?A)TheFama-Frenchmodelexplicitlyincorporatesthesizeandvaluefactors,whiletheCAPMdoesnot.B)TheFama-Frenchmodelreliessolelyonhistoricaldataforestimation,whereastheCAPMisbasedontheoreticalassumptions.C)TheFama-FrenchmodelisconsideredamorerobustexplanationofreturnsthantheCAPMinmanyempiricalstudies.D)TheFama-Frenchmodelrequirestheestimationofthreebetas,whiletheCAPMrequiresonlyone.5.Aninvestorrequiresaportfoliowithanexpectedreturnof12%andastandarddeviationof8%.Therisk-freerateis3%.AccordingtotheCapitalMarketLine(CML),whatistheproportionoftheinvestor'swealththatshouldbeinvestedintheriskymarketportfoliotoachievethisrequirement?A)44.44%B)66.67%C)100%D)155.56%6.AportfoliomanagerisconductingaperformanceattributionanalysisusingtheFama-FrenchThree-FactorModel.Whichofthefollowingcomponentsrepresentstheportionoftheportfolio'sexcessreturnattributabletothemanager'sabilitytoselectstocksthatoutperformthemarket,afteradjustingforthemarketfactor?A)AlphaB)MarketfactorpremiumC)SizefactorpremiumD)Industryfactorpremium7.Whichofthefollowingriskmeasuresisgenerallyconsideredtobemoreconservativethanthestandarddeviationofportfolioreturns,especiallyforlarge,negativedeviations?A)ValueatRisk(VaR)B)ConditionalValueatRisk(CVaR)C)BetaD)Sharperatio8.Aportfoliomanagerusesaprotectiveputstrategybybuyingputoptionsonastockportfolio.Whichofthefollowingbestdescribesthepotentialoutcomeofthisstrategy?A)Theportfolio'smaximumpotentiallossislimitedtothepremiumpaidfortheputoptions.B)Theportfolio'sexpectedreturnwillincreaseduetotheleverageprovidedbytheoptions.C)Theportfolio'svolatilitywilldecreaseastheoptionsprovidedownsideprotection.D)Theportfoliowillbenefitfrombothrisingstockpricesandthepotentialgainfromsellingtheputoptions.9.WhichofthefollowingstatementsismostaccurateregardingtheuseofMonteCarlosimulationsinportfoliomanagement?A)Theyprovideaprecisecalculationoftheportfolio'sValueatRisk(VaR).B)Theyareprimarilyusedforestimatingthefairvalueofoptionscontracts.C)Theyallowforthemodelingofcomplexdependenciesandcorrelationsbetweenassets.D)Theyrelyontheassumptionthatfuturereturnswillexactlyreplicatehistoricalreturns.10.Aclienthasarisktoleranceof15%andanexpectedreturnrequirementof10%.Themanagerconstructsaportfoliowithanexpectedreturnof12%andastandarddeviationof10%.Whichofthefollowingstatementsismostlikelytrueaboutthisportfolioinrelationtotheclient'srequirements?A)Theportfoliomeetstheclient'sexpectedreturnrequirementbutexceedstheirrisktolerance.B)Theportfolioexceedsboththeclient'sexpectedreturnrequirementandtheirrisktolerance.C)Theportfoliofallsshortoftheclient'sexpectedreturnrequirementandalsoexceedstheirrisktolerance.D)Theportfoliomeetsboththeclient'sexpectedreturnrequirementandtheirrisktolerance.11.Whichofthefollowingstatementsbestdescribesthedifferencebetweenriskparityandtraditionalassetallocationstrategies?A)Riskparityallocatescapitalbasedonhistoricalreturns,whiletraditionalallocationfocusesonriskmeasures.B)Riskparitytypicallyrequiresahigherproportionoffixedincomeassetstoachieveequalcontributiontoportfoliorisk,whiletraditionalallocationdoesnot.C)Riskparityaimstoequalizethevolatilitycontributionofeachassetclass,whiletraditionalallocationaimstomaximizeportfolioSharperatio.D)Riskparityisprimarilyusedforhedgingpurposes,whiletraditionalallocationisfocusedonlong-termgrowth.12.AportfoliomanagerusestheBlack-Scholes-MertonmodeltopriceaEuropeancalloptiononanon-dividendpayingstock.Whichofthefollowingfactors,ifincreased,wouldleadtoahigherpriceforthecalloption?A)Thestock'svolatilityB)ThetimetoexpirationC)Theoption'sstrikepriceD)Therisk-freeinterestrate13.Whichofthefollowingstatementsaccuratelydescribestheconceptofdeltahedginginoptionstrading?A)Itinvolvesbuyingcalloptionsandsellingputoptionstoincreaseportfolioleverage.B)Itinvolvesdynamicallyadjustingtheholdingsoftheunderlyingassettoneutralizetheportfolio'ssensitivitytosmallchangesintheoption'sprice.C)Itisastrategywheretheinvestorholdsaportfolioofassetsthatperfectlyreplicatesthepayoffoftheoption.D)ItistheprocessofcalculatingthefairvalueofanoptionusingtheBlack-Scholes-Mertonmodel.14.Aninvestorisconsideringaninterestrateswaptoconvertthefixedinterestpaymentsonafloating-ratebondintofloatingpayments.Whichofthefollowingbestdescribestheprimarypurposeofthisswap?A)Toeliminatethecreditriskassociatedwiththebondissuer.B)Tochangethecouponpaymentstructureofthebondtobettermatchtheinvestor'sinterestrateexpectations.C)Toincreasetheyieldonthebondbytakingadvantageofalowerfixedinterestrateenvironment.D)Tohedgeagainsttheriskofdefaultonthebond.15.Whichofthefollowingbehavioralfinanceconceptssuggeststhatinvestorsmaybemorewillingtosellalosinginvestmentthanawinningone,leadingtosuboptimalportfolioperformance?A)OverconfidenceB)HerdingC)DispositionEffectD)AnchoringPartB:EssayQuestions1.Aninvestorhas$1,000,000toinvest.Theyhavearisktolerancecorrespondingtoa10%toleranceforportfoliolossoveroneyear.Thecurrentrisk-freerateis2%.Theinvestor'sutilityfunctionisU=E(R)-0.5*A*σ^2,whereE(R)istheexpectedreturn,σ^2isthevarianceofreturns,andArepresentsthecoefficientofriskaversion.AssumeAisconstantforthisinvestor.Themanageridentifiestworiskyassets,AssetAandAssetB,withthefollowingcharacteristics:*AssetA:ExpectedReturn=12%,StandardDeviation=15%*AssetB:ExpectedReturn=18%,StandardDeviation=25%,CorrelationwithAssetA=0.4ConstructanefficientportfolioofAssetAandAssetBforthisinvestor.Clearlyexplainyoursteps,includinganycalculations.Determinetheexpectedreturn,standarddeviation,andSharperatiooftheoptimalportfolio.Discussanyassumptionsmadeandthelimitationsofthemodelused.2.Aportfoliomanagerisevaluatingtheperformanceofalarge-capequityportfolio.Theportfolioreturned15%overthepastyear,whilethebenchmarkindexreturned10%.ThemanagerusestheFama-FrenchThree-FactorModelforattribution.Thefactorreturnswereasfollows:*MarketFactor=8%*SizeFactor=-2%*ValueFactor=3%Theportfolio'sfactorexposures(betas)were:*MarketBeta=1.2*SizeBeta=-0.5*ValueBeta=0.8Theportfolio'sturnoverwas30%.Calculatetheportfolio'salpha,thecontributiontoreturnsfromeachfactor,andthecontributionfromstockselection(securitytiming).Discussthemanager'sperformance,highlightingstrengthsandweaknessesbasedontheattributionanalysis.Assumetherisk-freeratewas2%duringtheperiod.3.Aclientholdsawell-diversifiedportfoliowithanexpectedreturnof8%andastandarddeviationof12%.Theclientisconcernedaboutpotentialmarketdownturnsandhasexpressedadesiretolimitthemaximumpotentiallossoverthenextyeartonomorethan5%.Thecurrentrisk-freerateis1%.Themanagerconsidersusingoptionstocreateaprotectedportfolio.Discussthepotentialuseofoptions(suchasbuyingputoptionsonthemarketindex)toachievetheclient'sobjective.Calculatetheapproximatenumberofputoptions(assumingtheyhaveastrikepriceequaltothecurrentindexlevelandoneyeartomaturity)thatwouldneedtobepurchasedtoprotecttheportfolio,basedonaBlack-Scholes-Mertonvaluation(youmayusesimplifiedassumptionsorapproximations).Analyzethecosts,benefits,andpotentialdrawbacksofthisstrategy,includingtheimpactontheportfolio'sexpectedreturnandvolatility.試卷答案PartA:MultipleChoiceQuestions1.A解析：添加低相關(guān)性資產(chǎn)通常會(huì)降低投資組合的整體方差（協(xié)方差項(xiàng)減少），從而可能降低風(fēng)險(xiǎn)。雖然不能保證Sharpe比率一定提高（取決于預(yù)期回報(bào)的變化程度），但降低風(fēng)險(xiǎn)通常有利于提高比率。選項(xiàng)B過于絕對(duì)，因?yàn)轭A(yù)期回報(bào)是否增加取決于資產(chǎn)的具體參數(shù)。選項(xiàng)C不準(zhǔn)確，因?yàn)樾是把氐男螤钊Q于資產(chǎn)的相關(guān)性和收益關(guān)系，不一定向外移動(dòng)。選項(xiàng)D不正確，添加資產(chǎn)通常會(huì)影響風(fēng)險(xiǎn)和預(yù)期回報(bào)。2.A解析：CAPM的核心公式是E(Ri)=Rf+βi*[E(Rm)-Rf]。其中，E(Ri)是資產(chǎn)i的預(yù)期回報(bào)，Rf是風(fēng)險(xiǎn)-freerate，βi是資產(chǎn)i的beta系數(shù)，E(Rm)是市場(chǎng)組合的預(yù)期回報(bào)。因此，預(yù)期回報(bào)由beta和市場(chǎng)風(fēng)險(xiǎn)溢價(jià)（E(Rm)-Rf）決定。選項(xiàng)B錯(cuò)誤，covariance是其中的一個(gè)組成部分，但不是主要決定因素。選項(xiàng)C涉及alpha和Treynorratio，但不是CAPM的核心。選項(xiàng)D錯(cuò)誤，標(biāo)準(zhǔn)差和相關(guān)性是計(jì)算beta和相關(guān)模型的基礎(chǔ)，但不是CAPM直接決定預(yù)期回報(bào)的因子。3.B解析：在單因子模型中，股票回報(bào)與因子（如市場(chǎng)回報(bào)）之間的關(guān)系通常表示為股票回報(bào)=α+β*因子回報(bào)+ε。這里的β（Beta）衡量了股票對(duì)因子變化的敏感度或敏感性。4.A解析：Fama-French三因子模型在CAPM的基礎(chǔ)上增加了公司規(guī)模（Size）因子和價(jià)值（Value）因子，試圖更好地解釋股票的橫截面回報(bào)差異。CAPM只考慮市場(chǎng)因子。選項(xiàng)B錯(cuò)誤，兩種模型都使用歷史數(shù)據(jù)。選項(xiàng)C部分正確，F(xiàn)ama-French在許多實(shí)證研究中解釋力更強(qiáng)，但不是絕對(duì)“更robust”。選項(xiàng)D錯(cuò)誤，CAPM只需要一個(gè)beta，F(xiàn)ama-French需要三個(gè)因子（包括市場(chǎng)beta）的估計(jì)。5.B解析：根據(jù)CML公式，最優(yōu)投資比例w=[E(Rp)-Rf]/[σp*sqrt(T)]，其中E(Rp)是目標(biāo)組合預(yù)期回報(bào)（12%），Rf是風(fēng)險(xiǎn)-freerate（3%），σp是目標(biāo)組合標(biāo)準(zhǔn)差（8%）。將數(shù)值代入，w=(12%-3%)/(8%*sqrt(1))=9%/8%=1.125=0.6667或66.67%。6.A解析：股票選擇能力（StockPickingAbility）或Alpha，衡量的是在控制了市場(chǎng)、大小和價(jià)值等宏觀因素之后，經(jīng)理通過主動(dòng)選擇股票所獲得的超額回報(bào)。這是對(duì)因子回報(bào)調(diào)整后的剩余部分。7.B解析：CVaR（條件價(jià)值在風(fēng)險(xiǎn)）衡量的是在VaR（價(jià)值在風(fēng)險(xiǎn)）損失發(fā)生的前提下，額外的平均損失。因此，CVaR通常小于VaR，尤其是在分布有長(zhǎng)尾的情況下，它被認(rèn)為比VaR更保守，因?yàn)樗紤]了更嚴(yán)重的損失情景。8.A解析：保護(hù)性看跌期權(quán)策略通過支付期權(quán)費(fèi)獲得在股價(jià)下跌至特定行使價(jià)以下時(shí)的補(bǔ)償。因此，該策略將限制投資組合（或股票）的最大潛在損失，這個(gè)損失上限大約是股票當(dāng)前價(jià)值減去期權(quán)費(fèi)。期權(quán)費(fèi)是執(zhí)行該策略的成本。9.C解析：蒙特卡洛模擬通過生成大量隨機(jī)情景來模擬資產(chǎn)未來可能的表現(xiàn)，特別適用于模擬復(fù)雜金融工具（如衍生品）或資產(chǎn)組合在不同經(jīng)濟(jì)環(huán)境下的行為，以及處理資產(chǎn)間復(fù)雜的依賴關(guān)系（可以通過copulas等方法建模）。選項(xiàng)A錯(cuò)誤，它不能精確計(jì)算VaR，只能近似。選項(xiàng)B錯(cuò)誤，雖然可以用于衍生品定價(jià)，但不是其主要用途。選項(xiàng)D錯(cuò)誤，它基于隨機(jī)模擬，不假設(shè)未來完全復(fù)制歷史。10.A解析：預(yù)期回報(bào)（12%）高于要求（10%），但標(biāo)準(zhǔn)差（10%）等于風(fēng)險(xiǎn)容忍度（15%的損失容忍度對(duì)應(yīng)的波動(dòng)性可能近似為10%，這里直接給出）。這意味著雖然滿足了回報(bào)要求，但風(fēng)險(xiǎn)也達(dá)到了或可能超過了投資者的容忍度上限。Sharpe比率=(12%-2%)/10%=1.0，這是一個(gè)合理的比率，但關(guān)鍵在于風(fēng)險(xiǎn)容忍度是否被滿足。11.B解析：風(fēng)險(xiǎn)平價(jià)策略的核心思想是，在投資組合中，每個(gè)資產(chǎn)類別對(duì)總風(fēng)險(xiǎn)（通常是波動(dòng)率或跟蹤誤差）的貢獻(xiàn)應(yīng)該是相等的，而不是按資本市場(chǎng)的權(quán)重分配。這通常導(dǎo)致傳統(tǒng)資產(chǎn)配置中股權(quán)的比重被提高，而固定收益（尤其是高信用等級(jí)債券）的比重被降低，因?yàn)楣潭ㄊ找娴牟▌?dòng)率通常較低。12.AB解析：根據(jù)Black-Scholes-Merton模型，calloptionprice=S*N(d1)-K*e^(-rT)*N(d2)，其中d1=(ln(S/K)+(r+σ^2/2)T)/(σ*sqrt(T))，d2=d1-σ*sqrt(T)。S是標(biāo)的資產(chǎn)價(jià)格，K是行使價(jià)，r是無風(fēng)險(xiǎn)利率，σ是波動(dòng)率，T是時(shí)間到到期。當(dāng)波動(dòng)率σ增加時(shí)，d1和d2都會(huì)增加，導(dǎo)致N(d1)和N(d2)的值增加，從而calloptionprice上升。時(shí)間到到期T增加也會(huì)使N(d2)增加（因?yàn)閟qrt(T)在分母），同樣使價(jià)格上升。行使價(jià)K增加會(huì)使價(jià)格下降（因?yàn)镵在公式中是乘積項(xiàng)）。無風(fēng)險(xiǎn)利率r增加會(huì)使K*e^(-rT)項(xiàng)減少，從而使價(jià)格上升。13.B解析：Deltahedging是指通過持有與期權(quán)頭寸相反的、數(shù)量等于期權(quán)Delta的標(biāo)的資產(chǎn)多頭（或空頭），來對(duì)沖期權(quán)價(jià)格因標(biāo)的資產(chǎn)價(jià)格微小變動(dòng)而產(chǎn)生的風(fēng)險(xiǎn)。這是一個(gè)動(dòng)態(tài)調(diào)整的過程，因?yàn)镈elta本身會(huì)隨著標(biāo)的資產(chǎn)價(jià)格的變化而變化。14.B解析：利率互換的核心目的是讓一方（通常是浮動(dòng)利率收方）將其支付的利率從浮動(dòng)利率轉(zhuǎn)換為固定利率，或反之。在這個(gè)例子中，投資者（或債券持有人）希望將支付給債券發(fā)行人的浮動(dòng)利率（例如基于LIBOR）轉(zhuǎn)換為固定利率（例如基于某個(gè)固定基準(zhǔn)），以對(duì)沖浮動(dòng)利率上升帶來的風(fēng)險(xiǎn)或簡(jiǎn)化現(xiàn)金流管理。15.C解析：處置效應(yīng)（DispositionEffect）是行為金融學(xué)中的一個(gè)概念，指的是投資者傾向于持有盈利的資產(chǎn)過久（希望獲得更多利潤(rùn)），而較早賣出虧損的資產(chǎn)（希望避免進(jìn)一步損失）。這與優(yōu)化投資組合的原則相悖。PartB:EssayQuestions1.Toconstructtheefficientportfolio,weneedtofindthecombinationofAssetAandAssetBthatoffersthehighestSharperatioforagivenlevelofrisk(orthelowestriskforagivenlevelofexpectedreturn),giventheclient'srisktoleranceandutilityfunction.Therisktoleranceof10%peryearsuggestsautilitythresholdofU=10%-0.5*A*(10%)^2=10%-0.5*A*0.01=10%-0.005A.Weneedtofindtheportfolioweights(wAandwB)thatmaximizestheSharperatio,whichis(E(Rp)-Rf)/σp,subjecttotheconstraintthattheportfoliovarianceσp^2=wA^2*σA^2+wB^2*σB^2+2*wA*wB*ρAB*σA*σB,andtheweightssumto1(wA+wB=1).Let'sassumeaconstantriskaversioncoefficientAfortheclient.First,wecalculatetheSharperatiofora100%investmentintherisk-freeasset(0%)and100%intheriskyassets(wA=1,wB=0orwA=0,wB=1).Risk-free:Sharpe=(2%-2%)/0=undefined(bututilityis2%).RiskyAssets:IfwA=1:σp=15%,E(Rp)=12%,Sharpe=(12%-2%)/15%=0.6667.IfwB=1:σp=25%,E(Rp)=18%,Sharpe=(18%-2%)/25%=0.70.Bothriskyassetsofferpositiveutility(U=12%-0.5*A*0.15^2andU=18%-0.5*A*0.25^2,both>10%-0.005AforsufficientlysmallA).SincetheSharperatioofinvesting100%inAssetB(0.70)ishigherthanthatofinvesting100%inAssetA(0.6667),andbotharehigherthantherisk-freerate,theclientshouldinvest100%intheriskyassets.BetweenAssetAandAssetB,AssetBhasahigherexpectedreturnandahigherSharperatio.Therefore,theoptimalportfolio,tomaximizetheSharperatio,istoinvest100%inAssetB.ExpectedReturn=18%,StandardDeviation=25%,SharpeRatio=(18%-2%)/25%=0.70.Assumptions:Assumestheclient'sutilityfunctionisstrictlyconcave(representingriskaversion),therisk-freerateisconstant,thereturnsofAandBarenormallydistributed(forvariancecalculationsandSharperatiovalidity),andthecorrelationbetweenAandBisconstant.Limitations:The2-factormodelissimplified;real-worldportfoliosmightneedmorefactors.Assumesnotransactioncostsortaxes.Thesingle-periodframeworkignoresthetimevalueofmoneybeyondtheutilityfunctionsimplification.ThechoicebetweenAandBisbasedpurelyonSharperatio,ignoringotherfactorsliketheclient'sspecifictimehorizonorliquidityneeds.2.Tocalculatealpha,weusetheformula:Alpha=Rp-[Rf+βm*(Rm-Rf)+βs*(Rs-Rf)+βv*(Rv-Rf)].WhereRp=15%,Rf=2%,Rm=8%,Rs=-2%,Rv=3%,βm=1.2,βs=-0.5,βv=0.8.Alpha=15%-[2%+1.2*(8%-2%)+(-0.5)*(-2%-2%)+0.8*(3%-2%)]Alpha=15%-[2%+1.2*6%+0.5*4%+0.8*1%]Alpha=15%-[2%+7.2%+2.0%+0.8%]Alpha=15%-11.0%=4.0%.ContributionfromMarketFactor=βm*(Rm-Rf)=1.2*(8%-2%)=1.2*6%=7.2%.ContributionfromSizeFactor=βs*(Rs-Rf)=-0.5*(-2%-2%)=-0.5*(-4%)=2.0%.ContributionfromValueFactor=βv*(Rv-Rf)=0.8*(3%-2%)=0.8*1%=0.8%.TotalFactorContribution=7.2%+2.0%+0.8%=10.0%.StockSelection(SecurityTiming)Contribution=Alpha=4.0%.Analysis:Themanagerachievedaportfolioreturnof15%,whichincludesafactorcontributionof10.0%andanalphaof4.0%.Thepositivealphasuggeststhemanageraddedvaluethroughstockselection,beyondwhatcouldbeexplainedbymarketmovements,size,andvaluefactors.Themarketcontribution(7.2%)ispositive,indicatinggoodperformancerelativetothemarketbenchmark.Thesizecontribution(2.0%)isalsopositive,suggestinggoodperformanceinsmall-capstocks.Thevaluecontribution(0.8%)ispositive,indicatinggoodperformanceinvaluestocks.Strengths:Overallpositiveperformance,significantalphageneration,goodperformanceacrossmultiplefactors(especiallymarketandsize).Weaknesses:Thetotalreturn(15%)isdrivensignificantlybyfactorreturns(10%),withonly4%attributedtostockselection.Thissuggeststhemanager'sactivestockpickingcontributionmightbemodestrelativetopassivefactormovementsduringthisperiod.Themanagershouldaimtoincreasethealphacomponentinfutureperiods.3.Theclientdesirestolimitthemaximumpotentiallossto5%.Thecurrentportfoliohasanexpectedreturnof8%andastandarddeviationof12%.Usingputoptionscanprovidedownsideprotection.Thestrategyinvolvesbuyingputoptionsonabroadmarketindexthatrepresentstheportfolio'sexposure(e.g.,SPXorE-miniSPX).Thestrikeprice(K)shouldbesetatorslightlyabovethecurrentindexleveltoensuretheoptionhasrealvalue.Thetimetoexpiration(T)shouldbeatleastoneyeartomatchtheclient'shorizon.Thecostoftheputoptionsisthepremiumpaid.Toestimatethenumberofputsneeded,wecanuseasimplifiedapproximationbasedontheratioofthedesiredprotectiontothepremium.Themaximumlosswithoutprotectionis12%.Thedesiredprotectionis5%oftheportfoliovalue.Let'sassumetheportfoliovalueis$1,000,000(the$1Mfigureisusedinthepromptbutnotexplicitlystatedinthequestionitself,sowe'llassumeitforcalculation).Desiredprotectionamount=5%*$1,000,000=$50,000.Thepremium(P)ofoneputoptiondependsonthestrikeprice(K),timetomaturity(T),risk-freerate(r),andvolatility(σ).Foraputontheindex,thepremiumcanbeapproximatedusingtheBlack-Scholesmodel:P≈SN(-d1)-Ke^(-rT)N(-d2),whereSistheindexlevel,Kisthestrikeprice,ristherisk-freerate,Tisthetime,σisthevolatility,andN()isthecumulativestandardnormaldistribution.d1=(ln(S/K)+(r+σ^2/2)T)/(σ*sqrt(T)),d2=d1-σ*sqrt(T).Foraprotectiveput,KisclosetoS.Let'sassumeK=S,T=1year,r=1%(0.01),σ=12%(0.12),S=100(representingtheindexlevel).d1≈(ln(100/100)+(0.01+0.12^2/2)*1)/(0.12*sqrt(1))=(0+(0.01+0.0072))*1/0.12=0.0172/0.12≈0.1433.d2≈0.1433-0.12*1=0.0233.N(-d1)≈N(-0.1433)≈0.4429.N(-d2)≈N(-0.0233)≈0.4901.P≈100*0.4429-100*e^(-0.01*1)*0.4901≈44.29-100*0.9901*0.4901≈44.29-49.04≈-4.75.Thisnegativepremiumsuggestsanerrorinsignorassumptions,ortheoptionisdeepout-of-the-money.Let'stryastrikepriceslightlybelowthecurrentlevel,sayK=95.d1≈(ln(100/95)+(0.01+0.0072)*1)/(0.12*1)≈(0.0513+0.0172)/0.12≈0.0685/0