FRM总复习计算题型汇总

1FinancialRiskManagement2009CopyRightbyDMC1FRM考试计算题型总复习20100424王刚2计算要求分布†风险管理基础†金融市场与产品†信用风险管理†操作风险管理†投资组合管理3一、风险管理基础1.CAPM模型zCML线E(Rp)=PMFMFRRERσσ⎥⎦⎤⎢⎣⎡−+)(zSML线（CAPM）E(Ri)=RF+[E(RM)-RF]βi42.PerformanceMeasurement指标公式不同点相同点Treynor1，反映超额收益对系统性风险的补偿2，展望未来3，更加适用于有效分散的组合比较Sharpe1，反映超额收益对总风险的补偿2，回顾过去3，标准差可用于所有组合，应用更加广泛Jensen’salpha1，反映超额收益取得情况2，绝对值指标，更加适用于相同β的组合比较1.值越大越好2.T和S均代表单位风险的超额收益⎥⎦⎤⎢⎣⎡−pFpRREβ)(⎥⎦⎤⎢⎣⎡−pFpRREσ)([]p)()(βFMFpRRERRE−−−5†例一：Foraportfolioof10stocks,theestimatesoftheindividualstocks,thattheportfolio’sexpectedreturnis14%withastandarddeviationof25%.Thebetaoftheportfoliois1.1.Theexpectedreturnofthemarketis12.5%withastandarddeviationof20.2%.Therisk-freerateis2.6%.†CalculatetheTreynor,Sharpe,andJensenmeasures.6†解析：†Portfolio†Treynormeasure=(0.14-0.026)/1.1=0.136†Sharpemeasure=(0.14-0.026)/0.25=0.456†Jensenmeasure=0.14–0.026–[0.125-0.026](1.1)=0.0051†Market†Treynormeasureofthemarket=(0.125-0.026)/1=0.099†Sharpemeasureofthemarket=(0.125-0.026)/0.202=†0.4901†Jensenmeasureofthemarket=0.125-0.026-(0.125-0.026)(1)†=0.0由上面的例子可以看出，这10支股票的投资组合绩效以Treynormeasure和Jensenmeasure来看是优于市场的。但如果用Sharpemeasure来看，结果则相反。【基金经理人最在意Sharpemeasure】这代表此投资组合相对于其系统风险而言，有较佳的绩效，但这投资组合比起市场而言，是较不风险分散的。这可由其标准差(25%)大于市场标准差(20.2%)看出。73.Trackingerror,Informationratio,Sortinoratio‡Trackingerror（跟踪误差）：组合偏离基准收益的标准差‡Informationratio(信息比率）：相对于基准的超额收益除以跟踪误差‡Sortinoratio:相对于基准（最低要求）的超额收益除以下方标准差83.Trackingerror,Informationratio,Sortinoratio例二：Overa10-yearperiod,amanagerusesacoveredcallstrategytoenhancethereturnoftheindexfundshemanages.Therecordofthefund’sreturnis:(0.095,0.08,-0.022,0.11,0.09,-0.05,-0.035,0.124,0.072,0.055).†Thecorrespondingbenchmarkreturnsrecordis:†(0.087,0.078,-0.034,0.124,0.10,-0.064,-0.042,0.131,0.062,†0.059).†Theminimumacceptablereturnis4%.Calculatethetrackingerror,theinformationratio,andtheSortinoratio.9†Trackingerror†首先要先计算投资组合和benchmark间的差异：†(0.008,0.002,0.012,-0.014,-0.01,0.014,0.007,-0.007,0.01,†-0.004)。†差异数值的平均数=0.0018,†Trackingerror就是差异数值的标准差=trackingerror=†0.00992†(样本标准差，分母是除以N–1=9)†Informationratio=差异数值平均数÷差异数值的标准差†=0.0018/0.00992=0.1815†Sortinoratio†10个投资组合报酬的平均数是0.0519†Sortinoratio分子=投资组合报酬平均数－minimum†acceptablereturn†=0.0519–4%=0.0119†Sortinoratio分母=0.00175690.0419minMSD==†其中，minMSD是(0,0,-0.062,0,0,-0.09,-0.075,0,0,0)的平方†和除以10，也就是0.0017569。†∴Sortinoratio=0.0119÷0.0419=0.2840104.APT模型之运用计算Excessreturn†Ananalystbelievesequitypricesarecompletelyexplainedbythemarketriskpremium,earningsgrowth,earningsyield(E/P),andthedividendpayoutratio(D/E).Theanalystusesbetatorepresentexposuretothemarketriskpremiumandstandardizesearningsgrowth,earningsyield,andthedividendpayoutratio.Theanalyst'sforecastedfactorreturnsareasfollows:†Marketriskpremium=3%†Earningsgrowth=-2%†Earningsyield=2%†Whatistheexpectedexcessreturnforastockwiththefollowingfactorexposures?†Beta=1†Earningsgrowth=-0.5†Earningsyield=0.5E(R)=(0.03×1)+(-0.02×-0.5)+(0.02×0.5)=0.05%11二、FinancialMarketsandProducts1.optimalhedgeratioSS,FFS,FS,FS,FSS,F2SFSFFFh,whichisalsothebetaofspotpriceswithrespecttofuturescontractpricessinceCovCovCovandσ=ρσσρ=×==βσσσσσσ12二、FinancialMarketsandProducts1.optimalhedgeratio†Supposeourcurrencytraderfromthepreviousexampleincreasedthesophisticationofhistradeandcomputedthecorrelationbetweenthespotandfuturestobe0.925,theannualstandarddeviationoftheeuroexchangeratetobe$0.1,andtheannualstandarddeviationoftheBritishpoundfuturestobe$0.125.Computethenumberoffuturestobesoldtohedgetheposition.†Answer:†h=0.925*0.1/0.125=0.74†numberofcontractstohedge=0.74*6,250,000/62500=74contracts132.运用股指期货避险†公式：†注意：本公式隐含条件是将现货头寸的风险完全对冲14二、FinancialMarketsandProducts2.Hedgingwithstockindexfutures†Youareaportfoliomanagerwitha$20milliongrowthportfoliothathasbetaof1.4,relativetotheS&P500.TheS&P500futuresaretradingat1,150,andthemultiplieris250.Youwouldliketohedgeyourexposuretomarketriskoverthenextfewmonths.Identifywhetheralongorshorthedgeisappropriate,anddeterminethenumberofS&P500contractsyouneedtoimplementthehedge.†Answer:†1.4×$20,000,000/（1150×250）≈97contracts15二、FinancialMarketsandProducts3.AdjustingthePortfolioBeta†Letβbeourportfoliobeta,β*beourtargetbetaafterweimplementthestrategywithindexfutures,Pbeourportfoliovalue,andAbethevalueoftheunderlyingasset.†Tocomputetheappropriatenumberoffutures:†Example:Adjustingportfoliobeta†Supposewehaveawell-diversified$100millionequityportfolio.TheportfoliobetarelativetotheS&P500is1.2.ThecurrentvalueoftheS&P500indexisl,080.TheportfoliomanagerwantstocompletelyhedgethesystematicriskoftheportfoliooverthenextthreemonthsusingS&P500indexfutures.⎟⎟⎠⎞⎜⎜⎝⎛⎟⎟⎠⎞⎜⎜⎝⎛−=)(PVcontractsofnumberFPFPtargetmultiplierβββ16二、FinancialMarketsandProducts4.基本利率换算17二、FinancialMarketsandProducts4.基本利率换算†Supposewehave5%ratethatiscompoundedsemiannually.Computethecorrespondingcontinuousrate.Repeatthisforquarterly,monthly,weeklyanddailycompounding.†Rc=mln(1+R/m)†Rc=2ln(1+0.05/2)=0.049385†Theresultsforothercompoundingfrequenciesareshownbelow.18二、FinancialMarketsandProducts5.Duration&ConvexityAdjustment†FromTaylorseriesexpansions,△B=dB/dy×△y+1/2d2B/dy2△y2△B/B=-D△y+1/2