AnalysisofFinancialTimeSeriesFanQingzhuPekingUniversityAnalysisofFinancialTimeSeriesFinancialtimeseries(FTS)analysisisconcernedwiththeoryandpracticeofassetvaluationovertime.ComparisonwithotherT.S.analysis?Highlyrelated,butwithsomeaddeduncertainty.FTSmustdealwiththeever-changingbusiness&economicenvironmentandthefactthatvolatilityisnotdirectlyobserved.Objectiveofthecoursetolearnwaystogetfinancialinformationfromwebdirectlyandtoprocesstheinformation.toprovidesomebasicknowledgeoffinancialtimeseriesdatasuchasskewness,heavytails,andmeasureofdependencebetweenassetreturnstointroducesomestatisticaltools&econometricmodelsusefulforanalyzingtheseseries.togainexperienceinanalyzingFTStointroducerecentdevelopmentsinnancialeconometricsandtheirapplications,e.g.,high-frequencyfinanceObjectiveofthecoursetostudymethodsforassessingmarketrisk,creditrisk,andexpectedloss.ThemethodsdiscussedincludeValueatRisk,expectedshortfall,andtaildependence.toanalyzehigh-dimensionalassetreturns,includingco-movementExamplesoffinancialtimeseries1.DailylogreturnsofApplestock:2004to2013(10years)2.TheVIXindex3.CDSspreads:Daily3-yearCDSspreadsofJPMorganfromJuly20,2004toSeptember19,2014.4.QuarterlyearningsofCoca-ColaCompany:1983-2009Seasonaltimeseriesusefulinearningforecasts/pricingweatherrelatedderivatives(e.g.energy)/modelingintradaybehaviorofassetreturns.Examplesoffinancialtimeseries5.USmonthlyinterestrates(3m&6mTreasurybills)Relationsbetweenthetwoseries?Termstructureofinterestrates6.ExchangeratebetweenUSDollarvsEuroFixedincome,hedging,carrytrade7.SizeofinsuranceclaimsValuesofreinsuranceclaims(1000Krone)thatexceeded500from1972to1992.8.High-frequencynancialdata:Tick-by-tickdataofCaterpillarsstock:January04,2010.OutlineofthecourseReturns&theircharacteristics:empiricalanalysis(summarystatistics)Simplelineartimeseriesmodels&theirapplicationsUnivariatevolatilitymodels&theirimplicationsNonlinearityinlevelandvolatilityNeuralnetwork&non-parametricmethodsHigh-frequencynancialdataandmarketmicro-structureContinuous-timemodelsandderivativepricingOutlineofthecourseValueatRisk,extremevaluetheoryandexpectedshortfall(alsoknownasconditionalVaR)Analysisofmultipleassetreturns:factormodels,dynamicandcrossdependence1.1AssetReturnsLetPtbethepriceofanassetattimet,andassumenodividend.One-periodsimplereturn:GrossreturnSimplereturn:Multiperiodsimplereturn:Grossreturn1.1AssetReturnsThek-periodsimplenetreturnisExample:TablebelowgivesfivedailyclosingpricesofApplestockinDecember2011.The1-daygrossreturnofholdingthestockfrom12/8to12/91+Rt=393.62/390.66=1.0076sothatthedailysimplereturnis0.76%,whichis(393.62-390.66)/390.66.Timeintervalisimportant!Defaultisoneyear.1.1AssetReturnsAnnualized(average)return:Anapproximation:Continuouslycompounding:Illustrationofthepowerofcompounding(int.rate10%perannum)1.1AssetReturnsGenerally,thenetvalueofcontinuouslycompounding

whereristheinterestrateperannum,Cistheinitialcapital,nisthenumberofyears,andexpistheexponentialfunction.1.1AssetReturnsPresentvalue:Continuouslycompounded(orlog)returnWhereMultiperiodlogreturn:1.1AssetReturnsExample:ConsideragaintheApplestockprice.1.Whatisthelogreturnfrom12/8to12/9:A:rt=ln(393.62)-

ln(390.66)=7.5%.2.Whatisthelogreturnfromday12/2to12/9?A:rt(4)=ln(393.62)-ln(389.7)=1%.Portfolioreturn:Nassets

1.1AssetReturnsExample:AninvestorholdsstocksofIBM,MicrosoftandCiti-Group.Assumethathercapitalallocationis30%,30%and40%.UsethemonthlysimplereturnsinTable1.2ofthetext.Whatisthemeansimplereturnofherstockportfolio?Answer:E(Rt)=0.3*1.35+0:3*2.62+0:4*1.17=1.66.Dividendpayment:Excessreturn:(adjustingforrisk)wherer0tdenotesthelogreturnofareferenceasset(e.g.risk-freeinterestrate).

1.1AssetReturnsRelationship:Ifthereturnsareinpercentage,thenTemporalaggregationofthereturnsproducesThesetworelationsareimportantinpractice,e.g.obtainannualreturnsfrommonthlyreturns.

1.1AssetReturnsExample:Ifthemonthlylogreturnsofanassetare4.46%,-7.34%and10.77%,thenwhatisthecorrespondingquarterlylogreturn?Answer:

4.46-

7.34+10.77=7.89%.Example:Ifthemonthlysimplereturnsofanassetare4.46%,-7.34%and10.77%,thenwhatisthecorrespondingquarterlysimplereturn?Answer:R=(1+0.0446)(1-0.0734)(1+0.1077)-1=1.0721-1=0.0721=7.21%

1.2DistributionalpropertiesofreturnsKey:WhatisthedistributionofSometheoreticalproperties:MomentsofarandomvariableXwithdensityf(x):l-thmomentFirstmoment:meanorexpectationofX.l-thcentralmoment2ndc.m.:VarianceofX.1.2DistributionalpropertiesofreturnsSkewness(symmetry)andkurtosis(fat-tails)K(x)-

3:Excesskurtosis.Whyarethemeanandvarianceofreturnsimportant?Theyareconcernedwithlong-termreturnandrisk,respectively.Whyisreturnsymmetryofinterestinfinancialstudy?Symmetryhasimportantimplicationsinholdingshortorlongfinancialpositionsandinriskmanagement.1.2DistributionalpropertiesofreturnsWhyiskurtosisimportant?Relatedtovolatilityforecasting,eciencyinestimationandtests,etc.Highkurtosisimpliesheavy(orlong)tailsindistribution.Estimation:Data:samplemean:samplevariance:sampleskewness:samplekurtosis:1.2DistributionalpropertiesofreturnsUndernormalityassumption,Somesimpletestsfornormality(forlargeT).1.Testforsymmetry:ifnormalityholds.Decisionrule:RejectHoofasymmetricdistributionif|S*|>Za/2orp-valueislessthana.2.Testfortailthickness:1.2Distributionalpropertiesofreturnsifnormalityholds.Decisionrule:RejectHoofnormaltailsif|K*|>Za/2orp-valueislessthana.3.Ajointtest(Jarque-Beratest):ifnormalityholds,wherechi-squre(df=2)denotesachi-squareddistributionwith2degreesoffreedomDecisionrule:RejectHoofnormalityiforp-valueislessthana.EmpiricalpropertiesofreturnsDatasources:1.2DistributionalpropertiesofreturnsCourseweb:/ruey.tsay/teaching/bs41202/sp2015/CRSP:CenterforResearchinSecurityPrices(WhartonWRDS)/Variouswebsites,e.g.FederalReserveBankatSt.Louis/fred2/Datasetsofthetextbook:/ruey.tsay/teaching/fts3/1.2DistributionalpropertiesofreturnsEmpiricaldistofassetreturnstendstobeskewedtotheleftwithheavytailsandhasahigherpeakthannormaldist.SeeTable1.2ofthetext.DemonstrationofDataAnalysis1.2DistributionalpropertiesofreturnsNormalandlognormaldists

:YislognormalifX=ln(Y)isnormal.IfthenY=exp(X)islognormalwithmeanandvarianceConversely,ifYislognormalwithmeanandvariance,thenX=ln(Y)isnormalwithmeanandvariance1.2DistributionalpropertiesofreturnsApplication:Ifthelogreturnofanassetisnormallydistributedwithmean0.0119andstandarddeviation0.0663,thenwhatisthemeanandstandarddeviationofitssimplereturn?Answer:Solvethisproblemintwosteps.Step1:Basedonthepriorresults,themeanandvarianceofYt=exp(rt)areStep2:SimplereturnisTherefore,1.2DistributionalpropertiesofreturnsRemark:SeethemonthlyIBMstockreturnsinTable1.2.Likelihoodfunction(forselfstudy)Finally,itpaystostudythelikelihoodfunctionofreturnsdiscussedinChapter1.Basicconcept:Jointdist=Conditionaldist*Marginaldist,i.e.1.2DistributionalpropertiesofreturnsFortwoconsecutivereturnsr1andr2,wehaveForthreereturnsr1,r2andr3,byrepeatedapplicationIngeneral,wehave1.2DistributionalpropertiesofreturnsFortwoconsecutivereturnsr1andr2,wehaveForthreereturnsr1,r2andr3,byrepeatedapplicationIngeneral,wehave1.2DistributionalpropertiesofreturnsIfisnormalwithmeanmutandvariancesigmatsqure,

thenlikelihoodfunctionbecomesForsimplicity,iff(r1)isignored,thenthelikelihoodfunctionbecomesThisistheconditionallikelihoodfunctionofthereturnsundernormality.1.2DistributionalpropertiesofreturnsQuantifyingdependence:ConsidertwovariablesX