paper: “restrictions on riskprices in dynamicterm ... § risk-price restrictions change our...

Paper:“Restrictions onRisk PricesinDynamic Term Structure Models ”

byMichaelD.Bauer

March2016

Student:FrancescaCaturano

AdvancedFinancialEconometricsIII

March2020

Introduction

2

Policymakersandresearchesareinterestedinestimatingtheexpectations andtermpremiumcomponentsinlong-terminterestrates.Dynamictermstructuremodels(DTSM)whichimposeabsenceofarbitrageareusedforthispurpose

Paper:“Restrictions onRisk Prices inDynamic Term Structure Models”

§ Theno-arbitrageassumptioncanbepowerfulifitcreatesalink betweenthecross-sectionalvariationofinterestratesandtheirtime-seriesvariation,butitonlydoessoiftheriskadjustmentisrestricted.

§ ThispaperprovidesaneconometricframeworkforestimatingDTSMsunderrestrictionsonriskprices

§ Estimationoftermpremiaamountstoestimationofexpectationsoffutureshort-termrate.o DoingsowithonlytimeseriesinformationisextremelydifRicult,becausetheveryhighpersistenceofinterestratesleadstolargestatisticaluncertaintyandsmall-samplebias

§ ThenoarbitrageassumptioninDTSMcanalleviatetheseproblems,becauseitrequiresthatcrosssectionofinterestratesreRlectsforecastsoffutureoffutureshortrates,allowingforariskadjustment,thecrosssectionalinformationcanhelptopindowntheunobservedexpectations

Introduction

3

Policymakersandresearchesareinterestedinestimatingtheexpectations andtermpremiumcomponentsinlong-terminterestrates.Dynamictermstructuremodels(DTSM)whichimposeabsenceofarbitrageareusedforthispurpose


§ Theno-arbitrageassumptioncanbepowerfulifitcreatesalinkbetweenthecross-sectionalvariationofinterestratesandtheirtime-seriesvariation,butitonlydoessoiftheriskadjustmentisrestricted.

§ ThispaperprovidesaneconometricframeworkforestimatingDTSMsunderrestrictionsonriskprices

§ Estimationoftermpremiaamountstoestimationofexpectationsoffutureshort-termrate.o Doingsowithonlytimeseriesinformationisextremelydifficult,becausetheveryhighpersistenceofinterestratesleadstolargestatisticaluncertaintyandsmall-samplebias

§ ThenoarbitrageassumptioninDTSMcanalleviatetheseproblems,becauseitrequiresthatcrosssectionofinterestratesreflectsforecastsoffutureoffutureshortrates,allowingforariskadjustment,thecrosssectionalinformationcanhelptopindowntheunobservedexpectations

Introduction

4Paper:“Restrictions onRisk Prices inDynamic Term Structure Models”

§ Choosingrestrictionsontheparametersthatdeterminetheriskadjustmentisdifficult:

o modelselectioniscomplicatedbythelargenumberofpossiblerestrictions

o choiceofrestrictionsentailsmodeluncertaintyà equallyplausiblemodels,whichdifferonlylittleintermsofrisk-pricerestrictions,oftenrevealdramaticallydifferentshort-rateexpectationsandtermpremia

§ ThispaperintroducesaBayesianeconometricframeworkthatovercomesthesechallengesà TheframeworkreliesonMarkovchainMonteCarlo(MCMC)methodstoestimateaffineGaussianDTSMswithriskpricerestrictionso Modelselection doesnotrequireseparateestimationofeverysinglepossiblemodelspecificationbecausetheMCMCsamplersvisitonlyplausiblemodelsanddonotwastetimeinotherareasofthemodelspace.

o ModeluncertaintyisdealtwithbymeansofBayesianModelAveraging(BMA).

Introduction


§ Choosingrestrictionsontheparametersthatdeterminetheriskadjustmentisdifficult:

o modelselectioniscomplicatedbythelargenumberofpossiblerestrictions

o choiceofrestrictionsentailsmodeluncertaintyà equallyplausiblemodels,whichdifferonlylittleintermsofrisk-pricerestrictions,oftenrevealdramaticallydifferentshort-rateexpectationsandtermpremia

§ ThispaperintroducesaBayesianeconometricframeworkthatovercomesthesechallengesà TheframeworkreliesonMarkovchainMonteCarlo (MCMC)methodstoestimateaffineGaussianDTSMswithriskpricerestrictionso Modelselection doesnotrequireseparateestimationofeverysinglepossiblemodelspecificationbecausetheMCMCsamplersvisitonlyplausiblemodelsanddonotwastetimeinotherareasofthemodelspace.

o ModeluncertaintyisdealtwithbymeansofBayesianModelAveraging(BMA).

Introduction


§ Risk-pricerestrictionschange ourinterpretation oftheevolutionofinterestratesovercertainhistoricalepisodes,inparticularoftheseculardeclineinlongratesoverthelasttwodecadeso ConventionalDTSMsexplainthisbysubstantialdeclinesintermpremia andimplyonlyasmallroleforshort-rateexpectations

o Restrictedmodelsattributeamoreimportantroletoadecliningexpectationscomponent.o Thefindingthatexpectationsofshort-terminterestrateshavedecreasedoverthe1990’sand2000’sisconsistentwiththesizabledeclinesinsurvey-basedexpectationsofinflationandpolicyrates

Econometricframework:DTSM

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§ ThemodelsusedinthispaperbelongtotheclassofafRineGaussianDynamictermstructuremodel(DTSM)

§ 𝑋" isthe(𝑁×1) vectorofriskfactors,whichrepresentsthenewinformationthatmarketparticipantsobtainattime𝑡o assumethat𝑋" followsa1st orderVARunderthephysical(real-world) probabilitymeasure ℙ

o 𝑋" = 𝜇 + Φ𝑋"./ + Σ𝜀"o where𝜀"~𝑁(0, 𝐼6),Σ lowertriangularand𝐸 𝜀8𝜀9: = 0, 𝑟 ≠ 𝑠

§ Theoneperiodinterestrate𝑟"isanafRinefunctionofthefactors,§ 𝑟"= 𝛿? + 𝛿/:𝑋" (usingmonthlydata,oneperiodisonemonth)

§ Assumingabsenceofarbitrage,thereexistarisk-neutralprobabilitymeasureℚwhichpricesallRinancialassets



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§ Thestochasticdiscountfactor(SDF),whichdeRinesthechangeofprobabilitymeasurebetweenℙ andℚ isspeciRiedasexponentiallyafRine

− log 𝑀"F/ = 𝑟" +12𝜆": 𝜆" + 𝜆": 𝜀"F/

§ whitthe(𝑁×1) vector𝝀𝒕,themarketpriceofrisk,beinganafRinefunctionofthefactors,

§ 𝜆" = Σ./(𝜆? + 𝜆/𝑋"),𝜆" isavectorwithtimevaryingpricesofriskasafunctionofthestate

§ 𝝀𝒕 à istheriskpricesthatmeasuretheadditionalexpectedreturnrequiredforunitofriskineachoftheshocksin𝜀"

§ Considerhowtheexpectedexcessreturnofann-periodbondratedependsonriskprices:

§ 𝐸" 𝑟𝑥"F/L + /

M𝑉𝑎𝑟" 𝑟𝑥"F/

L = 𝜆": 𝐶𝑜𝑣 𝜀"F/, 𝑟𝑥"F/§ i.e.𝑅𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚 = 𝑝𝑟𝑖𝑐𝑒𝑠 𝑜𝑓 𝑟𝑖𝑠𝑘 ∗ 𝑞𝑢𝑎𝑛𝑡𝑖𝑡𝑖𝑒𝑠 𝑜𝑓 𝑟𝑖𝑠𝑘 𝑐𝑜𝑣𝑎𝑟𝑖𝑎𝑛𝑐𝑒𝑠§ InaGaussianmodel,thecovariancesareconstantà onlysourceoftime-

variationintermpremiaarechangesinthemarketpricesofriskPaper:“Restrictions onRisk Prices inDynamic Term Structure Models”


9

§ Startfromamodelunderℙ (𝑋" = 𝜇 + Φ𝑋"./ + Σ𝜀" )andmovetowardtherisk-neutralframework

§ Therisk-neutral dynamicsaregivenby:§ 𝑋" = 𝜇ℚ + Φℚ𝑋"./ + Σ𝜀"

ℚ,where𝜀"ℚ~ 𝑁(0, 𝐼_),𝐸ℚ 𝜀8ℚ𝜀9

ℚ` = 0, 𝑟 ≠ 𝑠

§ Theparametersdescribingthephysicalandrisk-neutral dynamicsarerelated inthefollowingway:

§ 𝜇ℚ = 𝜇 − 𝜆?, Φℚ = Φ− 𝜆/§ Note:𝜆? and𝜆/representtheriskpremianeededtomovefromℙ

dynamicstoℚ§ Inthismodel,yieldareaffineinthestatevariables.§ DenotingtheJmodel-implied (fitted)yields by a𝑌",authorwrites

a𝑌" = 𝐴 + 𝐵𝑋",wheretheJ-vectorAandthe𝐽×𝑁-matrix𝐵 containthemodel-implied

loadingofyieldsonriskfactors.Thesearedeterminedbyparameters𝛿?,𝛿/,𝜇ℚ,Φℚ andΣ




§ RiskFactorsarelinearcombinationofyields:§ 𝑋" = 𝑊 a𝑌",fora 𝑁×𝐽matrix𝑊

§ Sinceinterestliesininferenceaboutthepricesofriskassociatedwithshocksto𝑋",itisconvenientthat𝑋" isaspecificlinearcombinationsofyieldso Taketheriskfactors𝑋" asthefirstthreeprincipalcomponentsoftheobservedyields.

o 𝑊 containstheloadingsofthefirstthreePCsofobservedyields:level,slope,andcurvatureoftheyieldcurve(theyaresufficienttocapturemostofthevariationintheyieldcurve)

§ Theobservedbondyieldsusedforestimationare:§ 𝑌" = a𝑌" + 𝑒",

§ 𝑒" isavectorofmeasurementerrorsthatisiid normalo MeasurementerrorsareincludedbecauseanN-dimensionfactormodelcannotperfectlypriceJ>Nyields.Assume𝑋" isobservable

Econometricframework:restrictionsonriskprices

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Absenceofarbitragerequirestheconsistencyoftime-seriesdynamicsofinterestrateswiththeircrosssectionalbehavior,allowingforariskadjust.


§ Therisk-priceparameters𝜆? and𝜆/determinethisriskadjustmentandthebehaviorofriskpremia(𝜇ℚ = 𝜇 − 𝜆?, Φℚ= Φ − 𝜆/)

§ WeakEHà 𝜆/ = 0 andtermpremiaareconstant§ Maximally-=lexiblemodelà allelementsof𝜆? and𝜆/areunrestricted§ Author thinksà thetruthliessomewherebetweenthe2extremes

o focusonzerorestrictionon𝜆? and𝜆/§ Therisk-priceparameters𝜆? and𝜆/determinethisriskadjustmentandthebehaviorofriskpremia(𝜇ℚ = 𝜇 − 𝜆?, Φℚ= Φ − 𝜆/)

§ Authorprovidesasystematicframeworktoselectrestrictions.§ Let𝛾 beavectorofindicatorvariables,eachofwhichcorrespondstoanelementofλ ≡(𝜆?,vec(𝜆/)).

§ Ifanelementof𝛾 isequalto0,thecorrespondingparameterisrestrictedto0,anditisunrestrictedotherwise.

§ 𝛾 cantakeon26F6hdifferentvalues- ina3-factormodel,thereare4096candidatespecifications.

Econometricframework:restrictionsonriskprices

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Absenceofarbitragerequirestheconsistencyoftime-seriesdynamicsofinterestrateswiththeircrosssectionalbehavior,allowingforariskadjust.


§ Therisk-priceparameters𝜆? and𝜆/determinethisriskadjustmentandthebehaviorofriskpremia(𝜇ℚ = 𝜇 − 𝜆?, Φℚ= Φ − 𝜆/)

§ WeakEHà 𝜆/ = 0 andtermpremiaareconstant§ Maximally-=lexiblemodelà allelementsof𝜆? and𝜆/areunrestricted§ Author thinksà thetruthliessomewherebetweenthe2extremes

o focusonzerorestrictionon𝜆? and𝜆/§ Therisk-priceparameters𝜆? and𝜆/determinethisriskadjustmentandthebehaviorofriskpremia(𝜇ℚ = 𝜇 − 𝜆?, Φℚ= Φ − 𝜆/)

§ Authorprovidesasystematicframeworktoselectrestrictions.§ Let𝛾 beavectorofindicatorvariables,eachofwhichcorrespondstoanelementofλ ≡(𝜆?,vec(𝜆/)).

§ Ifanelementof𝛾 isequalto0,thecorrespondingparameterisrestrictedto0,anditisunrestrictedotherwise.

§ 𝛾 cantakeon26F6hdifferentvalues- ina3-factormodel,thereare4096candidatespecifications.

Econometricframework:MCMCalgorithms


ModelestimationandmodelselectionwillbecarriedoutusingMarkovchainMonteCarloalgorithms.

§ ForagivenmodelspeciRication𝛾,authorrequiresanMCMCsamplersthatdrawsfromthejointposteriordistributionoftheparameters

§ UsingMCMCalgorithmsonecaninsteadsamplejointlyacrossmodelsandparameterstoidentifyasmallersetofplausiblemodelspeciRications,sothatthereisnoneedtoestimateallcandidatemodels.o Themostinterestingmodels(theoneswithhighposteriorprobability)willbevisitedmorefrequentlybysuchsamplers

§ Thetaskofchoosingzerorestrictionsonrisk-priceparametersinaDTSMcloselyparallelstheproblemofselectingvariablesinmultivariateregressions

§ Theauthorusesdifferentexistingapproachestovariableselection:o Gibbsvariableselection(GVS)o StochasticSearchVariableSelection(SSVS)o Reversible-JumpMCMC(RJMCMC)



ModelestimationandmodelselectionwillbecarriedoutusingMarkovchainMonteCarloalgorithms.

§ Foragivenmodelspecification𝛾,authorrequiresanMCMCsamplersthatdrawsfromthejointposteriordistributionoftheparameters

§ UsingMCMCalgorithmsonecaninsteadsamplejointlyacrossmodelsandparameterstoidentifyasmallersetofplausiblemodelspecifications,sothatthereisnoneedtoestimateallcandidatemodels.o Themostinterestingmodels(theoneswithhighposteriorprobability)willbevisitedmorefrequentlybysuchsamplers

§ Thetaskofchoosingzerorestrictionsonrisk-priceparametersinaDTSMcloselyparallelstheproblemofselectingvariablesinmultivariateregressions

§ Theauthorusesdifferentexistingapproachestovariableselection:o Gibbsvariableselection(GVS)o StochasticSearchVariableSelection(SSVS)o Reversible-JumpMCMC(RJMCMC)



§ Usingthedrawsfromamodel-selectionsampler,onecaneasilyaccountformodeluncertaintyusingBayesianModelAveraging(BMA).

o InBMA,estimatesofmodelparametersandanyobjectsofinterest—interestratepersistence,volatilities,short-rateexpectations,andtermpremia—arecalculatedasaveragesacrossspecifications,usingposteriormodelprobabilitiesasweights.

o TheresultingBMAposteriordistributionsnaturallyaccountforthestatisticaluncertaintyaboutthemodelspecification.

§ Inthiswayonecanavoidafalsesenseofconfidencewhichmayresultfromconditioningononespecificrestrictedmodeldespitethepresenceofmodeluncertainty.

Simulationstudy


Becauseofthenoveltyoftheeconometricframework,itisimportanttoassessitsreliabilityandeffectivenessinasimulationstudy§ Tothisend,authorappliestheestimationmethodtodatathatissimulatedfromaknownmodel

§ Simulateyielddatafromadata-generatingprocess(DGP)whichisa2-factorDTSM

§ Todetermineplausibleparametersandrestrictionsauthorusesmaximumlikelihoodestimatesinactualyielddata,whichleadstoamodelwithonlyonesignificantriskpriceparameters.

§ Generate100samplesofsizeT=300andforeachsample:o estimatethemaximally-flexible modelusingMCMCo usethemodel-selectionsamplersSSVS,GVS,andRJMCMCtocarryoutestimationunderrisk-pricerestrictions

Simulationstudy


Thetableshowshowwellthesedifferentapproachesfareinrecoveringthetruemodel

Truemodel

Model-selectionsamplers

Unrestricted model

only1non-zerorisk-priceparameterà

reportsforeachparameterhowoftenthecredibilityintervalsdonotstraddlezero

averageposteriorprobabilitiesofinclusion

Simulationstudy


Theposteriorprobabilityofinclusionislargestforthatparameterwhichisnon-zerointheDGP(inclusionprobabilityisabove60%)whereasfortheparametersthataretrulyzerothisprobabilityisalwaysbelow50%.

Thenon-zeroparameterissignificantinonly26%ofthesamples,andtheparameterswhicharezerointheDGPareoftenfoundtobesignificantIfonechoosesamodelbasedonwhichparametersaresignificant,thentheDGPmodeliscorrectlyidentifiedinonly13%ofthesimulatedsamples

Simulationstudy


Thepercentageofsamplesinwhichthemodalmodel(themodelwiththehighestposteriorprobability)correspondstotheDGPmodel,reportedinthelastcolumn,isnearorabove60%

§ Insum,model-selectionsamplersdo quitewellinrecoveringthetrueDGPmodel,inparticularforaplausibleDGPinformedbyestimatesonactualyielddata.

Simulationstudy


Cantheestimationmethodsuggestedinthispapermoreaccuratelyrecovershort-rateexpectationsandtermpremiathanestimationofamaximally-flexibleDTSM?

§ Persistence measuredby:- thelargesteigenvalueofΦ- theimpulse-responsefunctionforthelevelfactorinresponsetolevel

shocksatthefive-yearhorizon§ Themodel-impliedvolatilitiesareforchangesin5-to-10yrisk-neutral

forwardrates(i.e.short-rateexpectations),andintheforwardtermpremium,inannualizedpercentagepoints.

ThisTablecompares theestimatedinterestrate

persistence andvolatilitiestothetruevaluesintheDGP

Simulationstudy


§ Thehighpersistencecauseslong-horizonexpectationsofshortratestobequitevolatile:Thevolatilityofmonthlychangesinfive-to-ten-yearrisk-neutralforwardishigherthanthevolatilityoftheforwardtermpremium

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MCMCoftheunrestrictedmodelleadsto- persistence thatisconsiderablylower,reflectingtheusualdownwardbiasinestimatedpersistence

- impliesmuchtoostableexpectationsandtoovolatileforwardtermpremia.

<

Simulationstudy


Estimationunderrisk-pricerestrictionsleadsto:- estimatesofpersistence thataremuchclosertothetruevalues,andit

accuratelyrecoversvolatilitiesoftheexpectationsandtermpremiumcomponentsinlong-horizonforwardrates.

- Inthiscase,the95%-CIsforpersistenceandvolatilitiescontainthetrueDGPvalueinalmostallofthesimulatedsamples.

- thelong-horizonforwardtermpremiumtobelessvolatilethanshort-rateexpectations.

§ Bayesianestimationunderrestrictionsonriskpricesissuccessfulinrecoveringthetruerestrictions,thepersistenceofinterestrates,andthevolatilitiesofshort-rateexpectationsandtermpremia.

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Simulationstudy


Estimationunderrisk-pricerestrictionsleadsto:- estimatesofpersistencethataremuchclosertothetruevalues,andit

accuratelyrecoversvolatilitiesoftheexpectationsandtermpremiumcomponentsinlong-horizonforwardrates.

- Inthiscase,the95%-CIsforpersistenceandvolatilitiescontainthetrueDGPvalueinalmostallofthesimulatedsamples.

- thelong-horizonforwardtermpremiumtobelessvolatilethanshort-rateexpectations.

§ Bayesianestimationunderrestrictions onriskpricesissuccessfulinrecoveringthetruerestrictions,thepersistence ofinterestrates,andthevolatilitiesofshort-rateexpectationsandtermpremia.

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§ Theeconometricframeworkperformswellinrecoveringthezerorestrictionsonriskpricesandtheestimatedrisk-priceparameters

§ Thestudyshowsthatestimationunderrisk-pricerestrictionsaccuratelyinfersthetruepersistence ofinterestratesandthevolatilityofshort-rateexpectationsandtermpremia.

§ Incontrast,estimationofanunrestrictedmodelleadsto:o persistencethatistoolow,o short-rateexpectationsthataretoostableo termpremiumestimatesthatareexcessivelyvolatile.

Simulationstudy

Economicimplications


Authordiscussestheeconomicimplicationsofrestrictionsonriskprices.Hecomparestheresultsfortheunrestrictedmodel𝓜𝟎,therestrictedmodels𝓜𝟏,𝓜𝟐,and𝓜𝟑 aswellastheBMAresultsusingtheGVSsample.

§ Sincerestrictionsonriskpricesaffectmainlythetime-seriespropertiesofaDTSMandleavethecross-sectionalRitessentiallyunchanged,thefocuswillbeonshort-rateexpectationsandtermpremia

§ Theestimatedpersistenceofriskfactorsandinterestratescruciallydeterminesthepropertiesofshort-rateexpectationsandtermpremia.

§ Applytheeconometricframeworkinreal-worlddata§ Usemonthlyobservationsofnominalzero-couponU.S.Treasuryyields,withmaturitiesof1through5,7,and10years

§ Sampleperiod:Jan.1990–Dec.2007(T=216monthlyobs.)o 𝓜𝟎 à estimatesoftheunrestricted,maximally-flexible DTSMàwillserveasabenchmark againstwhichtocomparesubsequentresults

o 𝓜𝟏àmodelwithonlythisoneelementofλ non-0o 𝓜𝟐àmodelwith2elementsofλ non-0o 𝓜𝟑àmodelwith3elementsofλ non-0

Economicimplications:persistenceandvolatilities


Authordiscussestheeconomicimplicationsofrestrictionsonriskprices.Hecomparestheresultsfortheunrestrictedmodel𝓜𝟎,therestrictedmodels𝓜𝟏,𝓜𝟐,and𝓜𝟑 aswellastheBMAresultsusingtheGVSsample.§ Thistablereports:

o thepersistenceunderbothprobabilitymeasures,ℚ andℙ ,measuredbythelargesteigenvaluesofΦℚ andΦ

o model-impliedvolatilities ofmonthlychangesin5-to-10-yearforwardrates,inrisk-neutralforwardrates,andinthecorrespondingforwardtermpremium.

o Foreachstatistic,posteriormeansand95%-CIsarereported



5-to-10-yforwardrates

risk-neutralforwardrates:shortrateexpectations

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§ ℚ-persistenceà verysimilaracrossmodels,sincetheℚ-dynamicsarelargelyunaffectedbyrisk-pricerestrictions.Consequently,thevolatilityofRittedforwardratesdoesnotvaryacrossmodels.

§ Undertheℙmeasure interestratesaremuchlesspersistentthanunderℚ,andthisistrueforallmodels.Consequently,short-rateexpectations arelessvariablethanforwardrates.

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forwardtermpremium

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5-to-10-yforwardrates

risk-neutralforwardrates

forwardtermpremium

>§ There are important differences across models.§ The restricted models (with the exception of𝓜𝟐) generally exhibithigher ℙ−persistence than𝓜𝟎.

§ The intuition is that risk−price restrictions tighten the connectionbetween cross section and time series, and “pull up”the ℙ−persistence toward toℚ−persistence.

§ All restricted models imply more volatile short−rate expectationsthan the maximally−Rlexible model𝓜𝟎.



>§ SinceconventionalDTSMstypicallyimplyverystablelong-horizonshort-rateexpectations,theyattributealargeroletothetermpremiumforexplainingmovementsinlongrates,whichisapuzzlingshort-coming ofthesemodels.

§ Risk-pricerestrictionsoften,lowerthevolatilityoftermpremia.Thepuzzleofanimplausiblylargerolefortermpremiainexplainingvariationinlongratesissomewhatalleviatedwhenplausiblerestrictions areimposedonanotherwisestandardDTSM.



>§ LargeCIsfortheunrestrictedmodelà itisdifficulttoestimatethedynamicpropertiesofinterestratesusingonlytime-seriesinformation.

§ CIs generallynarrower fortherestrictedmodelsà absenceofarbitragemakesinformation inthecrosssection useful forestimatingthetime-seriespropertiesofinterestrates.

§ Imposingaspecificsetofrestrictionsleadstotighterinferenceaboutℙ -dynamics,incorporationofmodeluncertaintynaturallymakestheinferencelessprecise.

Historicalevolution:shortrateexp&termpremia


§ Tablesummarizesthemodels’implicationsfordecomposingthedecline.Forbothactualandrisk-neutralyields,itreportsthelevelsin1990and2007,calculatedasaveragesovereachyear,andthechangesoverthisperiod.Alsoshownare95%-CIsforlevelsandchangesinrisk-neutralforwardrates

Long-terminterestrateshavedeclinedbyasigniRicantamountoverthesampleperiod.Towhichextentwasthisduetochangesinmonetarypolicyexpectations andmovementsintermpremia?



§ Theten- yearyielddeclinedby3.8ppso Unrestrictedmodel𝓜𝟎 impliesonlyasmall

shareofthisdecline,lessthan1/6,isduetodecliningshort-rateexpectations,andCIforthedeclineinexpectationsstraddleszero.

o Therestrictedmodels,withtheexceptionofℳM ,implyadeclineofshort-rateexpectationsthatismuchmorepronouncedandsignificantlydifferentfrom0;

o BMA attributesmorethan½oftheyielddeclinetofallingshort-rateexpectations.Thedeclineinexpectationsisestimatedmoreprecisely,eventhoughthisaccountsformodeluncertainty.

§ Thissuggeststhattheseculardeclineinlong-terminterestrateswasnotonlycausedbyalowertermpremium,butalsotoasignificantextentbyadownwardshiftinexpectationsoffuturenominalinterestrates,inlinewiththesizabledecreasesinsurvey-basedexpectationsofinflationandpolicyrates

Predictabilityofbondreturns


ItiswellknownthatreturnsonU.S.TreasurybondsarepredictableusingcurrentinterestrateandmaximallyflexibleaffineGaussianDTSMshavebeenshowntosuccessfullycapturethisfeatureofinterestratedata.Intherestrictedmodels,termpremiaandexpectedreturnaremorestablethaninunrestrictedmodelssoreturnpredictabilityismorelimited.CanDTSMswithtightrestrictionsonriskpricingstillmatchthereturnpredictabilitythatweseeinthedata?

§ Runapredictiveregressionforexcessreturnsonlong-termbondsandcheckwhethertheestimated𝑅M arematchedbythoseimpliedbythemodels,bothinpopulationandsmallsamples.

§ 𝑟𝑥","F/M(L) = 𝛼(L) + 𝛽(L)𝑋" + 𝑣"

(L),where§ 𝑟𝑥","F/M

(L) istheannualholdingperiodreturns,inexcessofthe1yyield,onabondwithmaturityn;𝑣"

(L) isthepredictiveerror.Thepredictorsarethefirst3PCsoftheyieldcurveandhencecorrespondstotheriskfactorsofthemodels.

§ Regressionisestimatedforbondswithmaturitiesof2,5,7and10years,using204monthlyobservations



ItiswellknownthatreturnsonU.S.TreasurybondsarepredictableusingcurrentinterestrateandmaximallyflexibleaffineGaussianDTSMshavebeenshowntosuccessfullycapturethisfeatureofinterestratedata.Intherestrictedmodels,termpremiaandexpectedreturnaremorestablethaninunrestrictedmodelssoreturnpredictabilityismorelimited.CanDTSMswithtightrestrictionsonriskpricingstillmatchthereturnpredictabilitythatweseeinthedata?

§ Runapredictiveregressionforexcessreturnsonlong-termbondsandcheckwhethertheestimated𝑅M arematchedbythoseimpliedbythemodels,bothinpopulationandsmallsamples.

§ 𝑟𝑥","F/M(L) = 𝛼(L) + 𝛽(L)𝑋" + 𝑣"

(L),where§ 𝑟𝑥","F/M

(L) istheannualholdingperiodreturns,inexcessofthe1yyield,onabondwithmaturityn;𝑣"

(L) isthepredictiveerror.Thepredictorsarethefirst3PCsoftheyieldcurveandhencecorrespondstotheriskfactorsofthemodels.

§ Regressionisestimatedforbondswithmaturitiesof2,5,7and10years,using204monthlyobservations



§ The𝑅M fortheactualyielddataarereportedinthistable.Annualexcessreturnsarestronglypredictable,with35%oftheirvariationexplainedbylevel,slopeandcurvatureoftheyieldcurve

§ Themodel-impliedpopulation 𝑅M aretypicallybelowthevaluesofthedata,andthediscrepancyismorepronouncedfortherestrictedmodelswithlessvariabletermpremia,suchas𝓜/.

§ TheBMA estimatesimplypopulation𝑅M thatarequitesubstantiallybelowthoseinthedata.

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§ Small-sampleissuesplayanimportantroleforthedistributionof𝑅M inpredictiveregressionsà necessarytoconsiderthesmall-sampledistributionof𝑅M impliedbythemodels.

o Authorobtainsitbysimulating,foreachmodel,1000yielddatasetsofthesamelengthastheoriginaldata(T=216),usingtheposteriormeansofthemodelparameters,andthenrunningthesameregressionsinthesimulatedasintheactualdata.

§ Tablereportsmeansandstand.dev.forthedistributionsofsmall-sample𝑅Màmeansarehigherthanthepopulation𝑅M,areclosetothe𝑅M inthedata.

o Insmallsamplesallmodels,includingthosewithtightlyrestrictedriskprices,areconsistentwiththeempiricalevidenceonbondreturnpredictability.

Conclusion


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§ ThispaperhasintroducedanoveleconometricframeworktoestimateDTSMsunderrestrictionsonriskpricing.

§ Itallowsforasystematicmodelchoiceamongalargenumberofrestrictionsandforparsimonyinotherwiseoverparameterizedmodels.

§ Empirically,theresultsusingU.S.Treasuryyieldsshowthatthedatasupporttightrestrictionsonriskprices.

o Thisstandsincontrasttothecommonpracticeofleavingmostoralloftherisk-priceparametersunrestricted.

o Therestrictionschangetheeconomicimplications,becausetheyincreasetheestimatedpersistenceofinterestratesandthereforemakeshort-rateexpectations(i.e.,risk-neutralrates)significantlymorevariable.

o Thisresolvesthepuzzleofimplausiblystableshort-rateexpectationssharedbymostconventionalDTSMmodels.

APPENDIX


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LikelihoodandPriors

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§ TheapproachchosenfollowstheobjectiveBayesiantraditionà Littlepriorinformationisimposedinordertoletdataspeakforitselfà applylargelyuninformativepriorformulations

§ Authorassumesauniformpriordistributionovermodels:eachelementof𝛾 isindependentlyBernoullidistributedwithsuccessprobability0.5

§ LikemostotherstudiesthatuseBayesianvariableselectionmethods,authorassumesconditionalpriorindependenceoftheelementsof𝜆.Thisassumptionsubstantiallysimplifiesthemodelselectionproblem.


§ Theconditionallikelihoodfunctionof𝑌" is:

“ℚ-likelihood”.Capturesthecrosssectional

dependenceofyieldsonriskfactors

“ℙ-likelihood”.Capturesthetimeseriesdynamicsofrisk

factors

§ Note:riskprice𝜆 affectsonlytheℙ-likelihood

Estimationresults



§ Sampleperiod:Jan.1990–Dec.2007(T=216monthlyobs.)§ 𝓜𝟎 à estimatesoftheunrestricted,maximally-flexible DTSMàwillserveasabenchmark againstwhichtocomparesubsequentresultso thiscomparisonwillrevealhowrisk-pricerestrictionschangetheeconomicimplicationsofatypicalaffineGaussianDTSMs.

§ Tocarryoutposteriorinferenceaboutrisk-pricerestrictions,authorobtainsmodel-selectionresultsusingtheGVS,SSVS,andRJMCMCsamplers.

§ Usingthesedata,theGVSalgorithmemergesasthefavouredmodel-selectionsampler,becauseitconvergesquicklyandrestrictstheexcludedparameterstobeexactlyzeroà TherestofthepaperfocusesontheGVSresults

Estimationresults



§ Sampleperiod:Jan.1990–Dec.2007(T=216monthlyobs.)§ 𝓜𝟎 à estimatesoftheunrestricted,maximally-flexibleDTSMàwillserveasabenchmarkagainstwhichtocomparesubsequentresultso thiscomparisonwillrevealhowrisk-pricerestrictionschangetheeconomicimplicationsofatypicalaffineGaussianDTSMs.

§ Tocarryoutposteriorinferenceaboutrisk-pricerestrictions,authorobtainsmodel-selectionresultsusingtheGVS,SSVS,andRJMCMCsamplers.

§ Usingthesedata,theGVSalgorithmemergesasthefavouredmodel-selectionsampler,becauseitconvergesquicklyandrestrictstheexcludedparameterstobeexactlyzeroà TherestofthepaperfocusesontheGVSresults

Estimationresults


§ Theevidence stronglyfavourstightlyrestrictedmodels,withonlyveryfewfreerisk-priceparameters.

§ Thistableshowsposteriormeans(theposteriorprobabilitiesofinclusionforthecorrespondingelementsofλ)

o onlyoneelementofλ hasahighposteriorprobabilityforinclusion,whichisabove95%.Forallotherparameters,theinclusionprobabilitiesarebelow50%,andformostoftheparameterstheyarenearzero.

Indicatorsrestricting𝜆?

Indicatorsrestricting𝜆/

Estimationresults


§ Thistableshowsthatallofthe10mostplausiblemodelsleaveonlyonetothreerisk-priceparametersunrestricted.

§ Thepriormeanforthenumberofunrestrictedparametersis6,whichcontrastswiththeposteriormeanofonly2.2.

§ Notation:o 𝓜𝟏àmodelwithonlythisoneelementofλ non-0

o 𝓜𝟐àmodelwith2elementsofλ non-0o 𝓜𝟑àmodelwith3elementsofλ non-0

o Posteriormodelprobabilitiesrelativetothemodalmodelforthe10mostfrequentlyvisitedmodels.

o Modelsaredenotedbytheindicesoftheunrestrictedelementsinλ.

Estimationresults


§ Theevidenceisalsoquiteclearonwhichrestrictionsarefavouredbydata.

§ 𝓜𝟏 hasbyfarthehighestposteriormodelprobabilityà datacallfortightrestrictions onthemarketpricesofriskandfavouramodelinwhichonlyoneoutoftwelverisk-priceparametersisunrestricted

§ Asarealitycheckforthemodel-selectionresults,lookatSBICà SBICisconsistentwiththerankingbasedonBayesianmodelselectionà results areactuallydrivenbyinformationinthedata,andnotbythechoiceofpriorsorsomefeatureofthesamplingalgorithms.



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TounderstandthedynamicpropertiesofaDTSM,itisinstructivetoconsidervolatilitiesacrossmaturities.Focusingonvolatilitiesofforwardrateshelpstoisolatethebehaviourofexpectationsatspecifichorizons.

§ Thefiguredisplaysthetermstructureofvolatility.§ Posteriormeansofvolatilitiesofchangesinfittedforwardrates(thinsolidline)andrisk-neutralforwardrates (thicksolidline),aswellas95%-credibilityintervalsforrisk-neutralvolatilities(dashedlines).

§ Itshowsmodel-impliedvolatilitiesofmonthlychangesinforwardratesandrisk-neutralforwardratesformaturitiesfromonemonthtotenyears



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§ Theforwardratevolatilitiesaresimilarforthetwomodels,decliningonlyslowlyandalmostlevellingoutforhorizonslongerthan5years.

§ Therisk-neutralvolatilitycurvesdiffersubstantially.

o 𝓜𝟎 à Exceptfortheveryshortestmaturities,risk-neutralvolatilitiesaremuchlowerthanforwardratevolatilities,implyingonlyalimitedroleforchangesinexpectationstoaccountformovementsininterestrates.

o BMAà risk-neutralvolatilitiesstaymuchclosertoforwardratevolatilitiesforhorizonsuptofiveyears,andonlyforlongermaturitiesdotheydropbelow



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Overall,BMA attributesalargerroletoshort-rateexpectationsforexplaininginterestratevolatilitiesacrossmaturities,duetotherestrictionsonriskprices.

§ Figurealsoshowsthatitishard toestimaterisk-neutralvolatilities-theCIsarequitewideinbothcases.Whileforanyindividualrestrictedmodel,thesearemuchnarrower(notshown)thanforthemaximallyflexiblemodel,itisshownthattakingintoaccountthemodeluncertaintysignificantlywidenstherangeofplausiblevolatilityestimates.



Figureshowshow𝓜𝟎 andBMA differintheirdecompositionoftheten-yearyieldintoexpectationsandtermpremiumcomponents.

§ Thisshowsforwardratesandestimatesofrisk-neutralforwardrates(RN)acrossmodels

§ RNareestimatesoftherisk-neutralyield,i.e.,oftheexpectationscomponentoftheten-yearyield

§ Thisshowsforwardratesandestimatesofforwardtermpremium(TP)acrossmodels.

§ TParecalculatedasthedifferencebetweenfittedandrisk-neutralyield.

§ TheRittedyieldsisobtainedfrom𝓜𝟎



§ ForBMA,theestimateoftherisk-neutralyield exhibitspronouncedvariation,fallingverysignificantlyaroundthe2001recession,andwiththeonsetoftheGreatRecession(2007–2009).

§ Theexpectationscomponentestimatedfrommodel𝓜𝟎 isverystable,andthemovementsaroundtherecessionsaremoremuted.

§ TheyieldtermpremiumisnoticeablymorestableforBMA relativeto𝓜𝟎,andmorecounter-cyclicalasitrisesbeforeandduringrecessionsandfallsduringexpansions.

§ Thisisappealinginlightofmuchtheoreticalandempiricalworksuggestingthattermpremiaareslow-movingandbehaveinacounter-cyclicalfashion



§ ForBMA,theestimateoftherisk-neutralyieldexhibitspronouncedvariation,fallingverysignificantlyaroundthe2001recession,andwiththeonsetoftheGreatRecession(2007–2009).

§ Theexpectationscomponentestimatedfrommodel𝓜𝟎 isverystable,andthemovementsaroundtherecessionsaremoremuted.

§ TheyieldtermpremiumisnoticeablymorestableforBMA relativeto𝓜𝟎,andmorecounter-cyclicalasitrisesbeforeandduringrecessionsandfallsduringexpansions.

§ Thisisappealinginlightofmuchtheoreticalandempiricalworksuggestingthattermpremiaareslow-movingandbehaveinacounter-cyclicalfashion

paper: “restrictions on riskprices in dynamicterm ... § risk-price restrictions change our...

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