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累积风险溢价(英)

累积风险溢价(英)

BISWorkingPapers No1128 Thecumulantriskpremium byAlbert(Pete)S.Kyle,KaramfilTodorov MonetaryandEconomicDepartment October2023 JELclassification:G1,G12,G13,G23 Keywords:Cumulants,leverage,ETF,factormodels,VIX,momentum,options BISWorkingPapersarewrittenbymembersoftheMonetaryandEconomicDepartmentoftheBankforInternationalSettlements,andfromtimetotimebyothereconomists,andarepublishedbytheBank.Thepapersareonsubjectsoftopicalinterestandaretechnicalincharacter.TheviewsexpressedinthemarethoseoftheirauthorsandnotnecessarilytheviewsoftheBIS. ThispublicationisavailableontheBISwebsite(www.bis.org). ©BankforInternationalSettlements2023.Allrightsreserved.Briefexcerptsmaybereproducedortranslatedprovidedthesourceisstated. ISSN1020-0959(print) ISSN1682-7678(online) ٨ TheCumulantRiskPremium٨ AlbertS.(Pete)Kylea,KaramfilTodorovb aUniversityofMaryland,RobertH.SmithSchoolofBusiness,askyle4@umd.edu bBankforInternationalSettlements,karamfil.todorov@bis.org Abstract Wedevelopanovelmethodologytomeasuretheriskpremiumofhigher-ordercumulants(closelyrelatedtothemomentsofadistribution)basedonleveragedETFs.Weshowthattheriskpre-miumontheseETFsreflectsthedifferencebetweenphysicalandrisk-neutralcumulants,whichwecallthecumulantriskpremium(CRP).WeshowthattheCRPisdifferentfromzeroacrossassetclasses(equities,bonds,commodities,currencies,andvolatility)andislargeintimesofstress.Weillustratethathighlyleveragedstrategiesareextremelyexposedtohigher-ordercu-mulants.Ourresultshaveimplicationsforhedgefunds,factormodels,momentumstrategies,andoptions. Keywords:Cumulants,leverage,ETF,factormodels,VIX,momentum,options JELclassification:G1,G12,G13,G23 TheviewsexpressedhereinarethoseoftheauthorsanddonotnecessarilyreflecttheviewsoftheBankforInternationalSettlements.WearegratefultoStevenHeston,SemyonMalamud,IanMartin,AndreasSchrimpf,TobiasSichert,NancyXu(discussant),andseminarparticipantsattheAFA,BIS,andUniversityofLausanneforhelpfulcommentsandsuggestions. 1.Introduction Manyepisodesofmarketturbulence,includingtheMarch2020COVID-19crisis,showthatassetreturnsarenotnormallydistributedandthathigher-ordermomentsplayanimportantroleinfinancialmarkets.Howcanwemeasuretheriskpremiumofhigher-ordermomentsacrossassetclassesinatractableway?Theclassicapproachintheexistingliteratureistouseoptionportfolios(e.g.,Bakshietal.(2003),SchneiderandZechner(2020)).However,theprob-lemwithimplementingthisapproachinpracticeisthatoptionsareoftenunavailableforallstrikesandareilliquid,especiallyforout-of-money(OTM)strikesandforlessliquidassetsthanequities.Withaveragebid-askspreadsabove74%,1itishardtoinferhigher-ordermomentsfromoptionsbecauseoptionpricesaremeasuredveryimprecisely.Inthispaper,wedevelopanovelmethodologytoquantifytheriskpremiumofhigher-ordermomentsbasedonleveragedETFs,whicharemuchmoreliquidthanoptionswithaveragebid-askspreadsofonly0.27%,morethan274timessmallerthanthoseofoptions.Weimplementournewmethodologyacrossseveralassetclasses:equities,bonds,commodities,currencies,andvolatility(VIX). = LeveragedETFs(whichwealsolabel“constant-betaassets”)areassetsthatmaintainacon-stantleverageβwithrespecttoagivenbenchmarkindex:e.g,adouble-leveragedETF(β2)ontheS&P500indexshoulddelivertwicetheperformanceoftheindexonagivenday.Inor-dertomaintainaconstantβ,theseETFsneedtorebalancewhentheindexmoves(seeChengandMadhavan(2009)andTodorov(2019)).2WeshowthatthisdynamicrebalancingbyETFsexposesthemtohigher-ordermoments.Thus,byobservingthereturnsonleveragedETFs,wecanquantifytheriskpremiumofhigher-ordermomentsontheindex. Weusecumulantstomeasuretheriskpremiumofhigher-ordermomentsinatractableway.Cumulantsareconvenienttosummarisethemaincharacteristicsofagivendistributionfunction,butaremoreintuitivetoworkwithcomparedtonon-centralmoments.Cumulantsarealsomoreconvenienttouseinthecaseoflog-returnsthatappearovermultipleperiodsin 1Basedontheaverageacrossequityindexes,Treasuries,currencies,commodities,fortheperiod2006–2020 andacrossstrikes,datafromIVolatility. 2Forexample,iftheindexrises,adouble-leveragedETFmakesmoneyandbecomeslessleveragedifitdoesnotrebalance.Tomaintaintheleverageconstant,theETFthenneedstoleverupandbuymoreoftheindex. oursetting,andtomodellinearcombinationsofrandomvariables.CornishandFisher(1938)describecumulantsinageneralsetting,whereasMartin(2013)isoneofthefirstresearcherstoapplycumulantsinfinance.Thefirstcumulantisthemeanofthedistribution;thesecondisvarianceσ2;thethirdandfourthcumulantsareskewnesstimesσ3andexcesskurtosistimesσ4.Higher-ordercumulantsaremorecomplicatedpolynomialfunctionsofthemoments.Foralognormaldistribution,thereareonlytwocumulants,meanandvariance,whereasforanyotherdistribution,therearealsohigher-ordercumulantsbeyondvariance. Inthefirstpartofthepaper,weshowhowtomeasuretheexposuretocumulantsfromconstant-betaassets.Weshowthattheriskpremiumonaconstant-betaassetisthesumofalineartermintheasset’sleverageβ,andanon-linearone,whichdependsonhigher-orderpowersofβ,weightedbydifferencesinhigher-orderphysicalandrisk-neutralcumulantsoftheindex.Wecallthesumofthesedifferencesthecumulantriskpremium(