2025年CFA二级公式与计算专项卷

2025年CFA二级公式与计算专项卷考试时间：______分钟总分：______分姓名：______试卷内容1.Aninvestmentmanagerhasaportfoliowiththefollowingcharacteristics:E(RP)=12%,σ(P)=18%,E(RM)=10%,σ(M)=15%,andβ(P)=0.8.Therisk-freerateis4%.CalculatetheexpectedreturnonthemarketportfoliousingtheCapitalAssetPricingModel(CAPM)formula,assumingthemarketportfolioistheonlyfactor.2.Astockisexpectedtopayadividendof$2.50nextyear(D1).Therequiredrateofreturnis9%,andtheexpectedconstantgrowthrateis5%.CalculatetheintrinsicvalueofthestockusingtheDividendDiscountModel(GordonGrowthModel).3.Acompanyhasthefollowingbalancesheetitems:TotalAssets=$500,000,TotalLiabilities=$200,000,PreferredStock=$50,000,andCommonEquity=$250,000.Calculatethedebt-to-equityratio.4.Abondwithafacevalueof$1,000paysanannualcouponof8%.Thebondhas10yearstomaturity,andthemarketrequiredyieldtomaturity(YTM)is6%.Calculatethecurrentpriceofthebond.5.Calculatethemodifiedduration(MacD)ofabondifthebond'sMacDureeis7.5yearsandtheannualyieldtomaturityis5%.6.Aninvestorpurchasesacalloptionwithastrikepriceof$50forapremiumof$3.Thestockpriceatexpirationis$65.Calculatetheinvestor'sprofitorlosspershare.7.Aninvestorpurchasesaputoptionwithastrikepriceof$45forapremiumof$2.Thestockpriceatexpirationis$40.Calculatetheinvestor'sprofitorlosspershare.8.Calculatetheyieldtomaturity(YTM)forabondwithacurrentpriceof$920,afacevalueof$1,000,anannualcouponrateof6%,and5yearsremainingtomaturity.9.Calculatethebeta(β)ofastockifthecovariancebetweenthestock'sreturnsandthemarketreturnsis0.014,thevarianceofthemarketreturnsis0.009,andthestandarddeviationofthemarketreturnsis3%.10.Aportfolioconsistsof60%StockAand40%StockB.TheexpectedreturnofStockAis14%,theexpectedreturnofStockBis8%,andthecorrelationcoefficientbetweenthereturnsofStockAandStockBis0.25.Calculatetheexpectedreturnoftheportfolio.11.Calculatethebeta(β)ofastockifthestock'sreturnshaveastandarddeviationof25%,themarketreturnshaveastandarddeviationof15%,andthecovariancebetweenthestock'sreturnsandthemarketreturnsis0.0375.12.AcompanyestimatesitsWACCtobe10%.Ithas$100millionindebtwithacostofdebt(beforetax)of5%andataxrateof30%.Ifthecompanyhasnopreferredstockandthemarketvalueofitsequityis$300million,calculatethecostofequityusingtheCAPM.Assumetherisk-freerateis3%andthemarketriskpremiumis7%.13.Aninvestorrequiresa12%rateofreturntoinvestinastock.Thestockpaysanannualdividendof$2pershare,andthegrowthrateofdividendsisexpectedtobe6%peryear.CalculatetheintrinsicvalueofthestockusingtheDividendDiscountModel(GordonGrowthModel).14.Calculatethecurrentyieldofabondwithafacevalueof$1,000,anannualcouponpaymentof$80,andacurrentmarketpriceof$950.15.Abondhasamodifieddurationof4.5years.Ifthemarketyieldtomaturitydecreasesby0.5percentagepoints,calculatetheapproximatepercentagechangeinthebond'sprice.16.CalculatethepriceofacalloptionusingtheBlack-Scholesmodelgiventhefollowinginputs:Stockprice(S)=$100,Strikeprice(K)=$110,Risk-freerate(r)=5%,Volatility(σ)=30%,andTimetoexpiration(T)=6months(0.5years).Assumethestockpaysnodividends.17.CalculatethepriceofaputoptionusingtheBlack-Scholesmodelgiventhefollowinginputs:Stockprice(S)=$75,Strikeprice(K)=$80,Risk-freerate(r)=4%,Volatility(σ)=20%,andTimetoexpiration(T)=9months(0.75years).Assumethestockpaysnodividends.18.Aninvestorislongastockwithabetaof1.2.Thecurrentmarketreturnis10%andtherisk-freerateis2%.CalculatetheexpectedreturnonthestockaccordingtotheCapitalAssetPricingModel(CAPM).19.Calculatetheannualgrowthrate(g)individendsifastock'scurrentpriceis$50,itsexpecteddividendnextyear(D1)is$3,andtherequiredrateofreturn(k)is12%.20.Acompanyhasforecastedthefollowingfinancialdatafornextyear:Sales=$1,000,000,OperatingMargin=10%,InterestExpense=$50,000,TaxRate=25%,andDividendPayoutRatio=40%.Calculatetheforecastedadditiontoretainedearnings.试卷答案1.E(RM)=Rf+β(P)*[E(RM)-Rf]=4%+0.8*(10%-4%)=4%+0.8*6%=4%+4.8%=8.8%*解析思路：直接应用CAPM公式E(RM)=Rf+β(P)*[E(RM)-Rf]，其中Rf是风险-freerate，β(P)是股票的beta，E(RM)是市场预期回报率。题目已知Rf=4%，β(P)=0.8，E(RM)是要求的值，但公式中E(RM)也出现在括号内，需解方程。更直接的公式是E(RM)=[E(RP)-Rf]/β(P)+Rf，代入E(RP)=12%,Rf=4%,β(P)=0.8计算得到E(RM)=[12%-4%]/0.8+4%=8%/0.8+4%=10%+4%=14%。此处题目表述可能略有混淆，按标准CAPM公式推导，市场预期回报率E(RM)应为14%。但若严格按照题目给出的公式形式E(RM)=Rf+β(P)*[E(RM)-Rf]代入求解，则E(RM)=4%+0.8*[E(RM)-4%]，解得E(RM)=8.8%。假设题目意图是考察应用公式，采用后者8.8%。2.P0=D1/[k-g]=2.50/(9%-5%)=2.50/4%=2.50/0.04=62.50*解析思路：应用戈登增长模型（股利贴现模型的一种），公式为P0=D1/(k-g)，其中P0是股票内在价值，D1是下一期预期股利，k是要求回报率，g是股利预期增长率。代入题目数据D1=2.50,k=9%=0.09,g=5%=0.05。3.Debt-to-EquityRatio=TotalLiabilities/CommonEquity=$200,000/$250,000=0.80or80%*解析思路：应用债务权益比公式，该比率衡量公司负债相对于股东权益的规模。公式为Debt-to-EquityRatio=总负债/股本权益。题目数据已给出TotalLiabilities=$200,000和CommonEquity=$250,000。4.P=C*[1-1/(1+y)^n]/y+M/(1+y)^n=80*[1-1/(1+6%)^10]/6%+1000/(1+6%)^10*解析思路：应用债券定价公式P=C*[1-1/(1+y)^n]/y+M/(1+y)^n，其中P是债券价格，C是年couponpayment，y是yieldtomaturity，n是yearstomaturity，M是facevalue。代入C=8%*1000=80,y=6%=0.06,n=10,M=1000。*P=80*[1-1/(1.06)^10]/0.06+1000/(1.06)^10*P=80*[1-1/1.7908477]/0.06+1000/1.7908477*P=80*[1-0.5583948]/0.06+558.3948*P=80*0.4416052/0.06+558.3948*P=35.3288267/0.06+558.3948*P=588.81378+558.3948*P≈$1147.21**注：计算器精确度可能导致微小差异。*5.MacD=MacDuree/(1+YTM)=7.5/(1+5%)=7.5/1.05≈7.1429*解析思路：应用修正久期（ModifiedDuration）公式MacD=MacDuree/(1+YTM)，其中MacDuree是MacDuree，YTM是yieldtomaturity。代入MacDuree=7.5,YTM=5%=0.05。6.Profit/Losspershare=Max(0,S_T-K)-Premium=Max(0,65-50)-3=Max(0,15)-3=15-3=$12.00*解析思路：计算看涨期权盈亏。盈亏=Max(0,股票到期价格S_T-行权价K)-期权费Premium。若S_T>K，则盈亏为S_T-K-Premium；若S_T<=K，则盈亏为-Premium。本题S_T=65,K=50,Premium=3，故盈亏为65-50-3=12。7.Profit/Losspershare=Max(0,K-S_T)-Premium=Max(0,45-40)-2=Max(0,5)-2=5-2=$3.00*解析思路：计算看跌期权盈亏。盈亏=Max(0,行权价K-股票到期价格S_T)-期权费Premium。若K>S_T，则盈亏为K-S_T-Premium；若K<=S_T，则盈亏为-Premium。本题K=45,S_T=40,Premium=2，故盈亏为45-40-2=3。8.YTMcalculationrequiresiterativemethodsorfinancialcalculator.Usingafinancialcalculator:*N=5,PV=-920,PMT=60(6%of1000),FV=1000*ComputeI/Y(YTM)≈7.41%*解析思路：计算到期收益率(YTM)需要用到债券定价公式，通常采用迭代法或财务计算器。公式为P=C*[1-1/(1+YTM)^n]/YTM+M/(1+YTM)^n。代入P=-920,C=60,M=1000,n=5。使用计算器输入相关数值并计算I/Y得到YTM。9.β=Cov(Ri,RM)/σ(M)^2=0.014/(0.009)=1.5556*解析思路：应用贝塔系数(Beta)公式β=Cov(Ri,RM)/σ(M)^2，其中Cov(Ri,RM)是股票回报率与市场回报率的协方差，σ(M)^2是市场回报率的方差。题目数据已给出Cov(Ri,RM)=0.014,σ(M)^2=0.009。10.E(RP)=w_A*E(RA)+w_B*E(RB)=0.60*14%+0.40*8%=8.4%+3.2%=11.6%*解析思路：应用投资组合预期回报率公式E(RP)=w_A*E(RA)+w_B*E(RB)，其中w_A和w_B是权重，E(RA)和E(RB)是个别资产预期回报率。题目数据w_A=60%,w_B=40%,E(RA)=14%,E(RB)=8%。11.β=Cov(Ri,RM)/σ(M)^2=0.0375/(0.15)^2=0.0375/0.0225=1.6667*解析思路：应用贝塔系数(Beta)公式β=Cov(Ri,RM)/σ(M)^2，其中Cov(Ri,RM)是股票回报率与市场回报率的协方差，σ(M)^2是市场回报率的方差。题目数据已给出Cov(Ri,RM)=0.0375,σ(M)=15%=0.15，故σ(M)^2=(0.15)^2=0.0225。12.First,checkWACCformulacomponents:WACC=(E/V*Re)+(D/V*Rd*(1-Tc)).GivenWACC=10%,D=$100M,Rd=5%,Tc=30%,E=$300M,V=E+D=$400M.*D/V=100M/400M=0.25*E/V=300M/400M=0.75*Rd*(1-Tc)=5%*(1-30%)=5%*0.70=3.5%*WACC=0.75*Re+0.25*3.5%=10%*0.75*Re+0.00875=0.10*0.75*Re=0.10-0.00875=0.09125*Re=0.09125/0.75=0.1216667or12.17%*解析思路：根据WACC公式WACC=(E/V*Re)+(D/V*Rd*(1-Tc))，已知WACC=10%,D=$100M,Rd=5%,Tc=30%,E=$300M,V=E+D=$400M。首先计算权重D/V=0.25和E/V=0.75。然后计算税后债务成本Rd*(1-Tc)=5%*(1-30%)=3.5%。将已知值代入WACC方程10%=0.75*Re+0.25*3.5%。解此方程求出Re(成本ofequity)。13.P0=D1/[k-g]=2/(12%-6%)=2/6%=2/0.06=33.3333or$33.33*解析思路：应用戈登增长模型（股利贴现模型的一种），公式为P0=D1/(k-g)，其中P0是股票内在价值，D1是下一期预期股利，k是要求回报率，g是股利预期增长率。题目数据D1=2,k=12%=0.12,g=6%=0.06。14.CurrentYield=AnnualCouponPayment/CurrentMarketPrice=$80/$950≈0.0842or8.42%*解析思路：应用当前收益率(CurrentYield)公式，该比率衡量债券年利息收入与其当前市场价格的比率。公式为CurrentYield=年票面利息/当前市场价格。题目数据AnnualCouponPayment=80(8%of$1000),CurrentMarketPrice=$950。15.%ChangeinPrice≈-MacD*Δy=-4.5*(-0.5%)=2.25%*解析思路：应用久期对价格敏感度近似公式%ChangeinPrice≈-MacD*Δy，其中MacD是ModifiedDuration，Δy是YTM的变化量。题目数据MacD=4.5years,Δy=-0.5percentagepoints(即-0.005)。注意负负得正，表示价格上升。16.UsingBlack-Scholesformulaforcalloption:*d1=[ln(S/K)+(r+σ^2/2)*T]/(σ*sqrt(T))*d2=d1-σ*sqrt(T)*CallPrice=S*N(d1)-K*e^(-rT)*N(d2)*WhereN(x)isthestandardnormalcumulativedistributionfunction.*Inputs:S=100,K=110,r=0.05,σ=0.30,T=0.5*d1=[ln(100/110)+(0.05+0.30^2/2)*0.5]/(0.30*sqrt(0.5))*d1=[ln(0.90909)+(0.05+0.045)*0.5]/(0.30*0.70711)*d1=[-0.09531+0.0475*0.5]/0.21213=[-0.09531+0.02375]/0.21213=-0.07156/0.21213≈-0.3374*d2=-0.3374-0.30*0.70711=-0.3374-0.21213≈-0.5495*N(d1)≈0.3679(fromstandardnormaltableorcalculator)*N(d2)≈0.2912(fromstandardnormaltableorcalculator)*CallPrice=100*0.3679-110*e^(-0.05*0.5)*0.2912*CallPrice=36.79-110*0.9753*0.2912*CallPrice=36.79-106.58*0.2912*CallPrice=36.79-31.017≈$5.77*解析思路：应用Black-Scholes模型计算看涨期权价格。公式为CallPrice=S*N(d1)-K*e^(-rT)*N(d2)，其中S是股票价格，K是行权价，r是无风险利率，σ是波动率，T是时间（年），N(x)是标准正态分布累积函数。首先计算d1和d2，然后查找N(d1)和N(d2)的值，最后代入公式计算期权价格。此处使用了计算器或标准正态分布表进行函数计算。17.UsingBlack-Scholesformulaforputoption:*PutPrice=K*e^(-rT)*N(-d2)-S*N(-d1)*WhereN(-x)=1-N(x)*Usinginputsandd1,d2frompreviouscalculation:*d1≈-0.3374*d2≈-0.5495*N(-d1)=1-N(d1)≈1-0.3679=0.6321*N(-d2)=1-N(d2)≈1-0.2912=0.7088*PutPrice=110*e^(-0.05*0.5)*0.7088-100*0.6321*PutPrice=110*0.9753*0.7088-63.21*PutPrice=76.348-63.21≈$13.14*解析思路：应用Black-Scholes模型计算看跌期权价格。公式为PutPrice=K*e^(-rT)*N(-d2)-S*N(-d1)，其中N(-x)=1-N(x)。使用与看涨期权相同的输入值S,K,r,σ,T，以及之前计算出的d1,d2和对应的N(d1),N(d2)。先计算N(-d1)和N(-d2)，然后代入公式计算期权价格。18.E(Rstock)=Rf