2025年CFA《數(shù)量方法》案例分析試卷考試時間：______分鐘總分：______分姓名：______CaseStudy1Youareanalyzingthefinancialperformanceoftwocompanies,CompanyAandCompanyB,withinthesameindustry.Thefollowingdatarepresentsthemonthlyreturnsofthetwocompaniesandamarketindexoverthepast36months:*CompanyAReturns:Mean=1.2%,StandardDeviation=8.5%*CompanyBReturns:Mean=0.9%,StandardDeviation=6.2%*MarketIndexReturns:Mean=1.0%,StandardDeviation=7.0%*CorrelationbetweenCompanyAReturnsandMarketIndexReturns:0.65*CorrelationbetweenCompanyBReturnsandMarketIndexReturns:0.55*CorrelationbetweenCompanyAReturnsandCompanyBReturns:0.30CompanyAusesasimplelinearregressionmodeltorelateitsreturnstothemarketindexreturnsoverthepast36months.Themodelestimatedinterceptis0.5%andtheslope(beta)is0.7.Themodel'sR-squaredvalueis0.42.CompanyBisconsideringimplementingasimplemovingaverage(SMA)of3monthsforitsmonthlyreturnsasaforecastingtoolforthenextmonth'sreturn.Required:1.CalculatethevarianceofthereturnsforCompanyAandCompanyBbasedontheprovidedstandarddeviations.2.Usingthegivencorrelationsandstandarddeviations,calculatethecovariancebetweenthereturnsofCompanyAandCompanyB.3.Assumingtherisk-freerateis0.2%permonth,calculatetheSharpeRatioforCompanyAandCompanyBbasedontheirrespectivemeansandstandarddeviations.4.Interprettheestimatedintercept(alpha)andslope(beta)forCompanyA'sregressionmodel.Whatdoesthebetaof0.7implyaboutCompanyA'svolatilitycomparedtothemarket?5.Describethepotentialadvantagesanddisadvantagesofusingasimplemovingaverage(SMA)of3monthsforCompanyB'sreturnforecasting.DiscussonealternativemethodfortimeseriesforecastingthatCompanyBcouldconsider.6.IfaninvestorholdsaportfolioequallyinvestedinCompanyAandCompanyB,whatistheexpectedreturnandstandarddeviationoftheportfolio,assumingnocorrelationbetweenCompanyAandCompanyBreturns?(Forthiscalculation,youmayassumethemarketindexreturnistheonlysourceofsystematicrisk,andignoretherisk-freerateforportfoliovariancecalculation).CaseStudy2Aninvestmentmanagerisanalyzingthepotentialinvestmentinastock,StockX.ThemanagerhascollectedhistoricaldataonStockX'sreturnsandbelievesthatStockX'sreturnsmightfollowanormaldistribution.Thefollowinginformationisavailable:*HistoricalStockXReturns:Thesamplemeanreturnoverthepast50periodsis12.0%.Thesamplestandarddeviationofreturnsis18.0%.*HypothesisTesting:ThemanagerwantstotestifthetruemeanreturnofStockXisgreaterthan10.0%perperiodatthe5%significancelevel.Themanagercollectsoneadditionalmonth'sreturndataforStockX,whichis15.0%.Required:1.Statethenullhypothesis(H0)andthealternativehypothesis(H1)forthemanager'stest.2.Calculatetheteststatistic(t-score)forthesamplemeanreturnof12.0%basedonthesampledata(50periods).Assumethepopulationstandarddeviationisunknown.3.Determinethecriticalt-valueforaone-tailedtestatthe5%significancelevelwith49degreesoffreedom.Basedonthecalculatedteststatisticandthecriticalvalue,shouldthemanagerrejectthenullhypothesis?4.Explaintheconceptofap-value.Ifthemanagercalculatedthep-valueforthesamplemeanof12.0%tobe0.008,interpretthisp-valueinthecontextofthehypothesistest.5.Consideringtheadditionalmonth'sreturnof15.0%,discusshowthisdatapointmightinfluencethemanager'sdecisionregardingthehypothesistest.Wouldthissingledatapointprovidestrongevidenceagainstthenullhypothesis?6.ThemanagerisalsointerestedinunderstandingthevariabilityofStockX'sreturns.Theywanttoconstructa95%confidenceintervalforthetruepopulationmeanreturnofStockX.Usingthesampledataandassuminganormaldistribution,calculatethelowerandupperboundsofthisconfidenceinterval.試卷答案CaseStudy11.VarianceofCompanyAReturns:8.5%*8.5%=0.7225or72.25%**解析思路：*方差是標(biāo)準(zhǔn)差的平方。計算公式為σ2=(StandardDeviation)2。**解析思路：*方差是標(biāo)準(zhǔn)差的平方。計算公式為σ2=(StandardDeviation)2。2.CovariancebetweenCompanyAandCompanyBReturns:0.30*8.5%*6.2%**解析思路：*協(xié)方差的計算公式為Cov(X,Y)=ρ(X,Y)*σ(X)*σ(Y)，其中ρ(X,Y)是X和Y的相關(guān)系數(shù)，σ(X)和σ(Y)分別是X和Y的標(biāo)準(zhǔn)差。3.SharpeRatioforCompanyA:(1.2%-0.2%)/8.5%**解析思路：*夏普比率(SharpeRatio)的計算公式為(Rp-Rf)/σp，其中Rp是投資組合的預(yù)期回報率，Rf是無風(fēng)險利率，σp是投資組合的標(biāo)準(zhǔn)差。在此題中，用公司本身的平均回報率代替預(yù)期回報率，用標(biāo)準(zhǔn)差代替投資組合標(biāo)準(zhǔn)差進行計算。CompanyB:(0.9%-0.2%)/6.2%**解析思路：*夏普比率(SharpeRatio)的計算公式為(Rp-Rf)/σp，其中Rp是投資組合的預(yù)期回報率，Rf是無風(fēng)險利率，σp是投資組合的標(biāo)準(zhǔn)差。在此題中，用公司本身的平均回報率代替預(yù)期回報率，用標(biāo)準(zhǔn)差代替投資組合標(biāo)準(zhǔn)差進行計算。4.InterpretationofAlpha(0.5%)andBeta(0.7):**InterpretationofAlpha:*Theintercept(alpha)of0.5%suggeststhatevenwhenthemarketreturnis0%,CompanyAisexpectedtogenerateamonthlyreturnof0.5%.ItrepresentstheexpectedreturnofCompanyArelativetothemarketwhenthemarkethasnoreturn.**InterpretationofBeta:*Thebeta(β)of0.7indicatesthatCompanyA'sreturnsareexpectedtobe0.7timesasvolatileasthemarketindex.Ifthemarketindexincreasesby1%,CompanyA'sreturnsareexpectedtoincreaseby0.7%.Abetagreaterthan1suggestshighervolatilitycomparedtothemarket.**解析思路：*回歸分析中，截距（α）代表當(dāng)自變量（市場回報率）為0時因變量（公司回報率）的期望值。斜率（β）代表自變量每變動一個單位，因變量期望變動的幅度。β>1表示該資產(chǎn)的波動性高于市場。5.SMAAdvantages/Disadvantages&Alternative:**Advantages:*Simpletocalculateandunderstand;smoothsoutshort-termfluctuations;easytoimplement.**Disadvantages:*Doesnotreactquicklytorecentchangesinthedata(laggingindicator);assumespastpatternswillcontinue;canbelessaccurateduringperiodsofsignificanttrendchanges.**Alternative:*AnalternativemethodistheExponentialSmoothing(ES)model,particularlytheHolt-Wintersmethodifseasonalityispresent,whichassignsexponentiallydecreasingweightstoolderobservationsandgivesmoreimportancetorecentdata.**解析思路：*SMA的優(yōu)缺點是相對直接的。優(yōu)點在于簡單、平滑。缺點在于滯后、假設(shè)過去持續(xù)、對趨勢變化反應(yīng)慢。替代方法需要考慮賦予近期數(shù)據(jù)更高權(quán)重的模型，如指數(shù)平滑法（ES）。6.PortfolioExpectedReturnandStandardDeviation:**ExpectedReturn(Rp):*(1.2%+0.9%)/2=1.05%**StandardDeviation(σp):*sqrt(((0.65*8.5%)2+(0.55*6.2%)2+2*0.30*0.65*8.5%*6.2%)/2)**解析思路：*對于等權(quán)重（各50%）且資產(chǎn)間相關(guān)系數(shù)已知的投資組合，預(yù)期回報率是各資產(chǎn)回報率的加權(quán)平均。投資組合標(biāo)準(zhǔn)差的公式為σp=sqrt(w?2σ?2+w?2σ?2+2w?w?σ?σ?ρ??)，其中w是權(quán)重，σ是標(biāo)準(zhǔn)差，ρ是相關(guān)系數(shù)。在此題中，無風(fēng)險利率和投資組合回報率計算被簡化。CaseStudy21.Hypotheses:*H0:μ≤10.0%(Thetruemeanreturnislessthanorequalto10.0%)*H1:μ>10.0%(Thetruemeanreturnisgreaterthan10.0%)(One-tailedtest)**解析思路：*檢驗?zāi)繕?biāo)是判斷均值是否大于10%，因此是單尾檢驗。原假設(shè)（H0）通常表示“無效應(yīng)”或“不大于”某個值，備擇假設(shè)（H1）表示研究者的假設(shè)（大于、小于或等于）。2.TestStatistic(t-score)forsamplemean:*t=(SampleMean-HypothesizedMean)/(SampleSD/sqrtSampleSize)*t=(12.0%-10.0%)/(18.0%/sqrt(50))**解析思路：*當(dāng)總體標(biāo)準(zhǔn)差未知且樣本量較大（n≥30）或中等時，使用t分布。檢驗統(tǒng)計量t的計算公式為t=(樣本均值-假設(shè)均值)/(樣本標(biāo)準(zhǔn)差/樣本量平方根)。3.Criticalt-valueandDecision:**Criticalt-value:*Usingat-distributiontablewith49degreesoffreedom(df=n-1=50-1)andaone-tailedtestatthe5%significancelevel(α=0.05),thecriticalt-valueisapproximately1.676.**Decision:*Sincethecalculatedt-score(approximately1.77)isgreaterthanthecriticalt-value(1.676),themanagershouldrejectthenullhypothesis(H0).**解析思路：*比較計算出的t統(tǒng)計量與臨界t值。如果t統(tǒng)計量落在拒絕域（大于正臨界值或小于負臨界值），則拒絕原假設(shè)。4.Interpretationofp-value(0.008):**Interpretation:*Ap-valueof0.008meansthatifthenullhypothesis(μ≤10.0%)istrue,thereisonlya0.8%probabilityofobservingasamplemeanasextremeas12.0%(ormoreextreme)bychancealone.Sincethisp-valueislessthanthesignificancelevelof0.05,itprovidesstrongevidenceagainstthenullhypothesis.**解析思路：*p值是在原假設(shè)為真的前提下，觀察到當(dāng)前樣本結(jié)果或更極端結(jié)果的概率。如果p值小于設(shè)定的顯著性水平α，則認為結(jié)果具有統(tǒng)計顯著性，傾向于拒絕原假設(shè)。5.Influenceofadditionalreturn(15.0%):**Discussion:*Theadditionalmonth'sreturnof15.0%issignificantlyhigherthanboththesamplemean(12.0%)andthehypothesizedmean(10.0%).Thisdatapointwouldlikelypulltheoverallsamplemeanupwardsandincreasethesamplestandarddeviation.Itpr