CFA投资工具-Statistics-来自高顿财经CFA.ppt

Investment Tools StatisticsSASF CFA Quant.ReviewStatistical ConceptsPopulationisdefinedasallmembersofaspecifiedgroup.Sampleisasubsetofadefinedpopulation.Frequency Distribution:isatabulardisplayofdatasummarizedintoarelativelysmallnumberofintervals.Frequencydistributionisthelistofintervalstogetherwiththecorrespondingmeasuresoffrequencyforthevariableofinterest.Ahistogram-graphicalequivalentofafrequencydistribution;itisabarchartwherecontinuousdataonarandomvariablesobservationshavebeengroupedintointervals.Afrequency polygonisthelinegraphequivalentofafrequencydistribution;itisalinegraphthatjoinsthefrequencyforeachinterval,plottedatthemidpointofthatinterval.2Frequency Distribution TableRawData:24,26,24,21,27,27,30,41,32,38ClassFrequency15but25325but35535but4523Frequency Distn Table Steps1.DetermineRange2.SelectNumberofClassesUsuallyBetween5&15Inclusive3.ComputeClassIntervals(Width)4.DetermineClassBoundaries(Limits)5.ComputeClassMidpoints6.CountObservations&AssigntoClasses4012345HistogramFrequencyRelativeFrequencyPercent01525354555Lower BoundaryBarsTouchClassFreq.15but25325but35535but452Count5012345Frequency PolygonMidpointFictitiousClass0102030405060ClassFreq.15but25325but35535butMedianModeiii.Negatively-skeweddistribution:MeanMedianMode14Measures of ShapeAfrequencydistributionthatismoreorlesspeakedthanaNormaldistributionissaidtoexhibitkurtosis.IfthedistributionismorepeakedthanaNormal(i.e.exhibits“fattails”)itisleptokurtic.IfitislesspeakedthanaNormalitiscalledplatykurtic.Positiveexcesskurtosis,i.e.aleptokurticdistribution,meansthatlargepositiveandnegativedeviationsfromthemeanhavehigherprobabilitiesforoccurringthantheywouldunderaNormaldistribution.IfanportfoliosreturnsareleptokurticthenitstrueriskishigherthantherisksuggestedbyananalysisthatassumesreturnsareNormallydistributed.ThisisimportantforValueatRisk(VAR)calculationsthatmustassumedistributionsforassetreturnsinaportfolio.15Frequencies19.Ananalystgatheredthefollowingdata:63.596.9112.3134.166.498.3116.2138.575.699.5116.9139.877.5100.7118.3140.784.4102.0122.0143.087.6105.5122.2153.989.9108.4124.5155.5Fiveclassesasfollows:1.60 x80.2.80 x1003.100 x1204.120 x1405.140 x160Inconstructingafrequencydistributionusingfiveclasses,ifthefirstclassis60upto80,theclassfrequencyofthethirdclassis:A.4.B.5.C.6.D.8.Hencethereare8observationsinthethirdclass.Notethemisleadingwaythequestionisasked!Alwaysreadthequestioncarefully!16Geometric Mean21.Aportfolioofnon-dividend-payingstocksearnedageometricmeanreturnof5percentbetweenJanuary1,1995,andDecember31,2001.Thearithmeticmeanreturnforthesameperiodwas6percent.Ifthemarketvalueoftheportfolioatthebeginningof1995was$100,000,themarketvalueoftheportfolioattheendof2001wasclosestto:A.$135,000.B.$140,710.C.$142,000.D.$150,363.IdentifywhatyouarebeingaskedforPortfolioEndingvalueP12/31/2001Giventhefollowing:PortfolioBeginningvalue=P1/1/1995=$100,000Geometricmeanreturn=5%Arithmeticmeanreturn=6%Numberofperiods=7Non-dividendpayingstocksinportfolio.IdentifycorrectapproachusegeometricmeanreturnandformulaPt+7=(1+r)7PtPt+7=(1.05)7$100,000=$140,71017Other Questions23.WhichofthefollowingstatementsaboutstandarddeviationisTRUE?Standarddeviation:A.isthesquareofthevariance.B.canbeapositiveoranegativenumber.C.isdenominatedinthesameunitsastheoriginaldata.D.isthearithmeticmeanofthesquareddeviationsfromthemean.25.Astockwithacoefficientofvariationof0.50hasa(n):A.varianceequaltohalfthestocksexpectedreturn.B.expectedreturnequaltohalfthestocksvariance.C.expectedreturnequaltohalfthestocksstandarddeviation.D.standarddeviationequaltohalfthestocksexpectedreturn.Ifthen19Simple Linear Regression YY=mX+bb=Y-interceptXChangein YChange in Xm=SlopeLinear Equations&Regression1.AnswertoWhatIstheRelationshipBetweentheVariables?2.RegressionEquationRegressionEquationUsed1NumericalDependentDependent(Response)VariableVariabletobePredicted1orMoreNumericalorCategoricalIndependentIndependent(Explanatory)Variables3.UsedtoTestTheoriesandforPrediction21YXiii01Linear Regression ModelRelationship Between Variables Is a Linear FunctionDependent(Response)VariableIndependent(Explanatory)VariablePopulationSlopePopulationY-InterceptRandomError22Probabilistic ModelsHypothesize2Componentsinvolvedinexplainingbehaviorofavariableofinterest.DeterministicbasedonrelevanttheoryRandomErrorreflectsunknownelementsExample:Wanttoexplainthereturnonacompanysstock.Theory:ReturnonCompanyjis1.50TimesReturnonOverallStockMarketPlusRandomErrorProbabilisticModel:R Rj j=1.5 =1.5 R RMktMkt+j jRandomErrorMayBeDuetoCompany-specificFactors.2302040600102030XYScatter Diagram1.PlotofAll(Xi,Yi)Pairs2.SuggestsHowWellModelWillFitHow Would You Draw a Line Through the Points?How Do You Determine Which Line Fits Best?241.BestFitMeansDifferenceBetweenActualValues(Yi)&PredictedValues()AreaMinimumButPositiveDifferencesOff-SetNegativesousesquarederrorstodetermineclosestfittingline.2.LeastSquaresRegression(LS)MinimizestheSumoftheSquaredDifferences(orErrors).Ordinary Least Squares,LS YiYYeiiiniinej212125e2YXe1e3e4Least Squares(LS)GraphicallyYbb Xeiii01Ybb Xii01LSMinimizeseeeeeiin211222324226Interpretation of LS Coefficients1.Slope(b1)EstimatedYchangesbyb1foreach1unitchangeinXIfb1=2,thenCompanyReturn(Y)isexpectedtoincreaseby2foreach1unitincreaseinMarketsReturn(X)2.Y-Intercept(b0)AverageValueofYwhenX=0Ifb0=2,thenAverageCompanyReturn(Y)IsExpectedtoBe2WhenMarketReturn(X)Is027