A No-Arbitrage Analysis of Economic Determinants of the Credit Spread Term Structure

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. A No-Arbitrage Analysis of Economic Determinants of the Credit Spread Term Structure Liuren Wu and Frank Xiaoling Zhang 2005-59 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

A No-Arbitrage Analysis of Economic Determinants of ∗ the Credit Spread Term Structure LIUREN WU † ZicklinSchoolofBusiness,BaruchCollege FRANK XIAOLING ZHANG ‡ FederalReserveBoard Firstdraft: October20,2004 Thisversion: May18,2005 ∗WethankLindaAllen,GurdipBakshi,TuranBali,Ren-rawChen,GregoryDuffee,MichaelGordy,DianaHancock,Paul Harrison,ArmenHovakimian,DonKim,PatrickMcCabe,JohnMerrick,TilSchuermann,MinWei,andseminarparticipants atBaruchCollegeandtheFederalReserveBoardforcomments.WealsothankBradHowellsandLauraKawanoforexcellent researchassistance. Anyremainingerrorsareours. Theviewsexpressedhereinaretheauthorsownanddonotnecessarily reflectthoseoftheFederalReserveBoardoritsstaff. †OneBernardBaruchWay,BoxB10-225,NewYork,NY10010-5585;Tel:(646)312-3509;Fax:(646)312-3451;Email: Liuren Wu@baruch.cuny.edu;Homepage:http://faculty.baruch.cuny.edu/lwu/. ‡DivisionofResearchandStatistics,FederalReserveBoard,Washington,DC20551; Tel: (202)452-2581; Fax: (202) 452-5295;Email:Xiaoling.Zhang@frb.gov.

A No-Arbitrage Analysis of Economic Determinants of the Credit Spread Term Structure ABSTRACT This paper presents an internally consistent analysis of the economic determinants of the term structureofcreditspreadsacrossdifferentcreditratingclassesandindustrysectors. Ouranalysis proceedsintwosteps.First,weextractthreeeconomicfactorsfrom13timeseriesthatcapturethree major dimensions of the economy: inflation pressure, real output growth, and financial market volatility. In the second step, we build a no-arbitrage model that links the dynamics and market pricesofthesefundamentalsourcesofeconomicriskstothetermstructureofTreasuryyieldsand corporatebondcreditspreads.Viamodelestimation,weinferthemarketpricingoftheseeconomic factorsandtheirimpactsonthewholetermstructureofTreasuryyieldsandcreditspreads. EstimationshowsthatpositiveinflationshocksincreasebothTreasuryyieldsandcreditspreads across all maturities and credit rating classes. Positive shocks on the real output growth also increasetheTreasuryyields,moresoatshortmaturitiesthanatlongmaturities. Theimpactsonthe creditspreadsarepositiveforhighcreditratingclasses,butbecomenegativeandincreasinglysoat lowercreditratingclasses. Thefinancialmarketvolatilityfactorhassmallpositiveimpactsonthe Treasuryyieldcurve,buttheimpactsarestronglypositiveonthecreditspreads,andincreasingly soatlongermaturitiesandlowercreditratingclasses. Finally,whenwedivideeachratingclassintotwoindustrysectors: financialandcorporate,we find that within each rating class, the credit spreads in the financial sector are on average wider andmorevolatilethanthespreadsinthecorporatesector. Estimationfurthershowsthattheterm structure of credit spreads in the financial sector is more responsive to shocks in the economic factors. JELClassification: E43;G12;G13. Keywords: Credit spreads; term structure; interest rates; macroeconomic factors; financial leverage; volatility;dynamicfactormodel;Kalmanfilter.

A No-Arbitrage Analysis of Economic Determinants of the Credit Spread Term Structure Numerous empirical studies, mostly based on regression analysis, show that the frequency of credit eventsandtheexpectedlossfromsucheventsdependcruciallyonthestateofthemacroeconomyand the financial market. In this paper, we quantify in an internally consistent manner the link between the dynamics and market prices of the fundamental economic factors on the one hand and the term structureofcreditspreadsontheother. The task at hand is important but challenging. First, many macroeconomic numbers and financial marketvariablesareavailable. Eachseriescontainssomeinformation,andalsopossiblyatremendous amountofnoise,aboutthestateoftheeconomy. Itisinefficienttofocusmerelyononeorafewofthese variableswhilediscardingmanyothers. Meanwhile,itisunrealistictoincorporateallofthemasstate variables into a formal model of credit spreads. Therefore, how to identify the systematic movements fromthemanynoiseseriesposesthefirstchallenge. Second, the pricing of credit risk, which is embedded in the prices of many corporate bonds, is likelytobedifferentforbondsatdifferentmaturities. Pickingbondsatanyonematurityorafewmaturitieswouldnotrevealthecompletepictureacrossthewholetermstructure. Hence,howtoconsistently summarizeandquantifythepricingofdifferentrisksacrossthewholetermstructureofcreditspreads posesanotherchallenge. In this paper, we handle both challenges by building a dynamic factor model of interest rates and creditspreads. First,weextractasmallnumberofdynamicfactorsfromawidearrayofmacroeconomic and financial series to capture the key dimensions of the economy. Thus, through a dynamic factor structure,wesuccinctlysummarizetheinformationcontentinmanynoisyseries. Second,wepropose flexiblespecificationsonhowthesedynamicfactorsarepricedandhowtheinstantaneousinterestrates andcreditspreadsrespondtothesefactors. Giventhesespecifications,weuseno-arbitragearguments asinDuffieandSingleton(1999)andDuffie,Pedersen,andSingleton(2003)toderivethewholeterm structuresofinterestratesandcreditspreadsasfunctionsofthesedynamicfactors. Therefore,vianoarbitrage arguments, we are able to make an internally consistent analysis of the impacts of the large 1

number of macroeconomic and financial variables on the interest rates and credit spreads across the wholespectrumofmaturities. Furthermore,theestimationnotonlyshowshowthevariablesaffectthe termstructure,butalsorevealsthereasonsbehindit. Inparticular,themodelestimationillustrateshow differentdimensionsofriskintheaggregateeconomyvaryovertimeandhowthemarketpricesthem differently. Risk dynamics and pricing jointly determine how the factors impact the term structure of interestratesandcreditspreads. Our estimation involves two sequential steps. In the first step, we decompose the economy into threekeydimensions: theinflationpressure,therealoutputgrowth,andthefinancialmarketvolatility. Inflationandrealoutputgrowthrepresentafirst-orderdecompositionofthetwosidesofthemacroeconomy,whereasfinancialmarketvolatilityrepresentsanaggregateriskmeasure,asecond-ordermoment thathasbeengeneratingfirst-orderimpactsonthefinancialmarketanditsoperations(Engle(2004)). Wecapturethesethreedimensionsoftheeconomyusingthreesystematicdynamicfactors, which weextractfrom13observedseriesusingmaximumlikelihoodmethodjointwithKalmanfilter. Thefirst twofactorsareextractedfrom11macroeconomicseries,includingyear-over-yearpercentagechanges on the consumer price index (CPI), the core CPI, the producer price index (PPI), the core PPI, the personal consumption expenditure (PCE) deflator, the core PCE deflator, and the gross domestic production (GDP) deflator, the real GDP, industrial production, non-farm payrolls, and the real PCE. We make structural constraints on the factor loadings to align the first factor with inflation and the second factor with real output growth. We extract the third factor from two volatility indices: the VXO volatilityindexcomputedfromoptionsonS&P100index,andtheVIXvolatilityindexcomputedfrom optionsonS&P500index. Thisvolatilityfactorrepresentsacompoundmeasureoftheeconomy-wide business risk and financial leverage, both of which impact the credit risk and credit spreads (Merton (1974)). In the second step, we use the three economic factors to explain the term structure of Treasury yieldsandcorporatebondcreditspreadsatdifferentcreditratingclasses. Inlinkingthethreedynamic factors to the term structure of Treasury yields and corporate bond credit spreads, we build a noarbitrage dynamic term structure model, under which the whole term structure of Treasury interest ratesisdeterminedbythefactordynamics,marketpricesoffactorrisks,andtheinstantaneousriskfree 2

interest rate as a function of the factors. A further specification for the instantaneous credit spread functionateachcreditratinggroupdeterminesthewholetermstructureofcreditspreadsatthatrating group. Thus,byestimatingthemodelparameters,welearntheimpactsofthesystematicfactorsonthe wholetermstructureofTreasuryyieldsandcreditspreads. Wealsolearnthedynamicsoftheeconomic factorsandhowthesefactorsarepricedindifferentmarkets. Estimation shows that positive inflation shocks increase Treasury yields and credit spreads across all maturities and credit rating classes. The impacts on the Treasury yields are the strongest. The impacts on the credit spreads are weaker and increasingly so at lower credit ratings. Thus, a positive inflationshockmovesupboththebenchmarkyieldcurveandthecreditspread,moresofortheTreasury andcreditspreadsonhighcreditratingcompaniesthanforspreadsonlowratingcompanies. PositiveshocksontherealoutputgrowthalsoincreasetheTreasuryyields,moresoatshortmaturitiesthanatlongmaturities. Theimpactsonthecreditspreadsarepositiveforhighcreditratingclasses, but become negative and increasingly so as the credit rating moves to a lower class. Furthermore, the impacts are more negative at shorter maturities than at longer maturities. Thus, a positive shock to the real side of the economy increases the benchmark interest-rate level, flattens an otherwise upward slopingyieldcurve,butnarrowsthecreditspread,particularlyatshortmaturitiesandlowcreditrating classes. ThevolatilityfactorhasonlysmallpositiveimpactsontheTreasuryyieldcurve,butitsimpactsare stronglypositiveonthecreditspreads,andincreasinglysoatlongermaturitiesandlowercreditrating classes. We also divide each credit rating group into two broad industry sectors: financial and corporate, andstudywhethertheimpactsdiffersignificantlyacrossthesetwoindustrysectors. Wefindthatcredit spreads on the financial sector are on average wider and more volatile than the spreads on corporate sector, especially at lower credit rating classes. Estimation further shows that the term structure of creditspreadsinthefinancialsectorismoreresponsivetoshocksintheeconomicfactors. Inparticular, wefindthatforthefinancialsector,theimpactsoftheinflationfactoraremorepositiveacrossallrating classes,thattheimpactsoftherealoutputfactoraremorenegativefortheBBBclass,andtheimpacts ofthevolatilityfactormorepositiveforAA,A,andBBBclasses. 3

Ourworkinthispaperintegratestwostrandsofextantliterature. Thefirststrandusesregressionsto analyzethedeterminantsofcreditspreads. ProminentexamplesincludeBevanandGarzarelli(2000), Frye (2000), Carey (1998), Pedrosa and Roll (1998), Collin-Dufresne, Goldstein, and Martin (2001), Elton,Gruber,Agrawal,andMann(2001),andAltman,Brady,Resti,andSironi(2004). Thoughrichin economicintuition,theresultsofthesestudiesoftendependonthespecificchoicesoftheexplanatory variables, as well as the choices of the maturity and credit rating of the credit spreads used as the dependent variable. Given the correlations among the many commonly used explanatory variables, the regression coefficient estimates often change depending on the choice of the other explanatory variables. Furthermore, the estimates also change when the credit spreads used in the analysis switch maturities or credit rating classes. It asks for a dramatic leap of faith to extend the regression results fromonematurityand/orratingclasstoothermaturitiesandratingclasses. The second strand of literature uses a small number of dynamic factors to summarize the variation on the term structure of interest rates and credit spreads via no-arbitrage arguments. Important contributionsincludeJones,Mason,andRosenfeld(1984),LongstaffandSchwartz(1995),Duffieand Singleton (1997), Duffee(1999), Nickell, Perraudin, and Varotto (2000), Liu, Longstaff, and Mandell (2000), Delianedis and Geske (2001), Bangia, Diebold, Kronimus, Schagen, and Schuermann (2002), Collin-Dufresne,Goldstein,andHelwege(2003),HuangandHuang(2003),Bakshi,Madan,andZhang (2004),Longstaff,Mithal,andNeis(2004),Eom,Helwege,andHuang(2003),andLongstaff,Mithal, andNeis(2004). Differentfromtheregressionanalysis,thesestudiescanderivetheimpactsofthedynamicfactorsonthewholetermstructureofinterestratesandcreditspreadsinaninternallyconsistent manner. However,mostofthesestudiesrelyonlatentfactors,directlyderivedfromtheyieldcurveand credit spread term structure. The economic meanings of these latent factors are not clear. In the few studies that try to incorporate economic variables, often only a small number of observable variables areincludedfortractabilityreasons,andothervaluableeconomicvariablesareconspicuouslyleftout. In this paper, we exploit the advantages of both strands of studies. On the one hand, we use a few dynamic factors to summarize the information and suppress the noises in many observed macroeconomic and financial time series. On the other hand, we exploit the no-arbitrage framework to provide aninternallyconsistentanalysisontheimpactsofthesemacroeconomicandfinancialseriesacrossthe 4

whole term structure of credit spreads. We also rely on the no-arbitrage model to provide insights on riskdynamicsandriskpricingthatformthemaindrivingforcebehindthetermstructureimpacts. The rest of the paper is organized as follows. Section I describes the procedure for extracting the dynamic economic factors. Section II presents a no-arbitrage model that links the dynamic economic factors to the whole term structure of Treasury and corporate bond yields. Section III describes the construction of Treasury and corporate yields and our estimation strategy. Section IV discusses the estimationresultsoftheno-arbitragemodels,andexaminestherelationbetweencreditspreadsacross different maturities and rating classes and the extracted economic factors. Section V further divides the corporate bond data at each rating class into two broad industry sectors and analyze how the term structureofcreditspreadsdifferandwhethertheeconomicfactorsimpactthetermstructuredifferently acrossthetwosectors. SectionVIconcludes. I. Extracting Dynamic Economic Factors We observe many economic and financial time series, yet many of them contain similar information, mingled with a significant portion of noises either due to measurement errors or idiosyncratic movements. Weuseadynamicfactormodeltosuccinctlysummarizetheinformationandsuppressthe noiseinmanyobservedmacroeconomicandfinancialseries. A. ImportantDimensionsoftheAggregateEconomy In applying a dynamic factor model, we first need to specify the dimensions of the factor space. Basedonbothempiricalevidenceandeconomicrationale,wedecomposetheaggregateeconomyinto threebroaddimensions: (1)thenominalsideoftheeconomy,(2)therealsideoftheeconomy,and(3) thevolatilityofthefinancialmarket. Macroeconomists often decompose the economy into the nominal and real sides and argue that shocks to the two sides of the economy should be separated and treated differently. For example, Woodford(2003)arguesfromtheperspectiveofmonetarypolicythatnominalshocksshouldbeminimizedwhereasrealshocksshouldnotbeintervened. EarlystudiesbySargentandSims(1977),Sargent 5

(1989),andStockandWatson(1989)alsosuggestthatanominalandarealfactorcanaccountformuch oftheobservedvariationinmajoreconomicaggregates. In addition to these two dimensions, we also incorporate a financial market volatility dimension, which captures the compound effect of economy-wide business risk and financial leverage, both of whichimpactthecreditriskaccordingtotheclassicstructuralmodelofMerton(1974). Furthermore, the 2004 Nobel price in economics manifests the first-order importance of modeling the time-varying dynamicsofthissecond-ordermoment(Engle(2004)). Thus, we use three systematic factors to capture the dynamic variation in the first and second momentsoftheaggregateeconomy. Regressionanalysisintheliteratureoftenincorporatesmanymore explanatoryvariables,butintheaggregatelevelmostofthesevariablescanbeclassifiedintooneofthe threedimensions. Atafirmlevel,financialleveragedirectlyimpactscreditriskandcreditspreads,but itismoreacapitalstructuredecision(control)variablethanaseparatedimensionofexogenousshocks. B. EstimatingDynamicFactorModelswithMaximumLikelihoodandKalmanFilter We describe the economy by fixing a filtered probability space {W ,F ,P,(F ) }, with some t 0≤t≤T fixedlonghorizonT . WeuseX ∈Rntodenoteann-dimensionalvectorMarkovprocessthatrepresents thesystematicstateoftheeconomy. Asdiscussedabove,wesetn=3. Wefurtherassumethatthestate vectorX followssimpleVAR(1)dynamics. Undercontinuous-timenotation,X followsamulti-variate Ornstein-UhlenbeckprocessunderthestatisticalmeasureP, dX =−k Xdt+dW, (1) t t t whereW denotesann-dimensionalstandardBrownianmotionandk controlsthemean-reversionspeed t ofthestates. Foridentificationpurpose,wenormalizethelong-runmeanofthestatesX tozeroandthe instantaneous covariance matrix to be an identity matrix. We also constrain k to be a lower triangular matrix. Next, let y∈RN denote a set of macroeconomic and financial time series. The dimension N can be very large, and much larger than the dimension of the state of the economy, N ≫n. In this paper, 6

we choose N =13, which includes seven inflation-related series, four output-related series, and two financial market volatility indices constructed from stock index options. We summarize the systematicmovementsunderlyingthe13macroeconomicandfinancialseriesusingthreedynamiceconomic factorsviathefollowinglinearfactorstructure, y =HX +e , (2) t t t whereH isan(N×n)matrixoffactorloadingcoefficientsande denotesan(N×1)vectorofmeasuret menterrorsofthedataseries. WeuseR y=E[e e⊤]todenotethecovariancematrixofthemeasurement t t errors. We assume that the measurement errors are independent of the state vector. In our estimation, we further assume that the measurement errors are mutually independent, but with distinct variance: R y =s 2,i=1,···,N,andR y =0fori6= j. ii i ij If we know the parameters that govern the factor dynamics (k ), the factor loadings (H), and the measurementerrorvariance(R y),wecaninferthesystematicstatesoftheeconomyfromtheobserved data series, with the technique of Kalman filtering. For this purpose, we rewrite the economic factor dynamicsinitsdiscrete-timeanalog, X =F X + Q e , (3) t t−1 t p whereF =exp(−Dk t),Q =ID t,e denotesan(n×1)iidstandardnormalrandomvector,D t denotes t the discrete time interval, and I denotes an identity matrix of the relevant dimension. With monthly timeinterval,wesetD t =1/12. ForKalmanfiltering,weregardequation(3)asourstate-propagationequationandequation(2)as our measurement equation. Let X ,V ,y ,A denote the time-(t−1) ex ante forecasts of time-t values t t t t of the systematic economic factors, the covariance matrix of the economic factors, the measurement series, and the covariance matrix of the measurement series. Let X andV denote the ex post update, t t orfiltering,ontheeconomicfactorsandtheircovarianceattimetbbasedobnobservations(y )attimet. t 7

The Kalman filter provides the efficient updates on these quantities. Specifically, we have the ex ante predictionsas X = F X ; (4) t t−1 V = F V b F ⊤+Q ; (5) t t−1 b y = HX ; (6) t t A = HV H⊤+R y. (7) t t Theexpostfilteringupdatesare, X = X +K (y −y ); (8) t+1 t+1 t+1 t+1 t+1 V b = V −K A K⊤ , (9) t+1 t+1 t+1 t+1 t+1 b whereK =V H⊤ A −1 istheKalmangain. t+1 t+1 t+1 (cid:0) (cid:1) Thus,wecanobtainatimeseriesoftheexanteforecastsandexpostupdatesonboththemeanand covarianceoftheeconomicfactorsandthedataseries,viatheiterativeproceduredefinedbyequations (4) to (9). To estimate model parameters Q ≡[k ,H,R y] that govern the factor dynamics and factor loading, we define the monthly log likelihood function by assuming that the forecasting errors on the observedtimeseriesarenormallydistributed, 1 1 l (Q )=− log A − (y −y )⊤ A −1 (y −y ) . (10) t+1 2 t+1 2 t+1 t+1 t+1 t+1 t+1 (cid:12) (cid:12) (cid:16) (cid:0) (cid:1) (cid:17) (cid:12) (cid:12) Theparametersareestimatedbymaximizingthesumofthemonthlyloglikelihoodvalues, T−1 Q =argmax (cid:229) l (Q ), (11) t+1 Q t=1 whereT denotesthenumberofobservationsforeachseries. 8

C. DataDescription Ourestimationisbasedon13monthlyorquarterlymacroeconomicandfinancialseriesfromJanuary 1988 to June 2004. The 11 macroeconomic series are from the Federal Reserve Board. They includeseveninflation-relatedseries: theconsumerpriceindex(CPI),thecoreCPI,theproducerprice index(PPI),thecorePPI,thepersonalconsumptionexpenditure(PCE)deflator,thecorePCEdeflator, andthegrossdomesticproduction(GDP)deflator. TheGDPdeflatorisavailableatquarterlyfrequency. Allothervariablesareavailableinmonthlyfrequency. Wefirstconvertthepriceindexesintoyear-overyearpercentagechanges,andthenstandardizeeachseriesbysubtractingthesamplemeananddividing theseriesbythesamplestandarddeviation. The CPI measures the average change in the prices of a basket of goods and services bought by a typical urban household. The PPI measures the change in the selling prices received by domestic producers for all finished goods. The PCE deflator measures the average change in the prices of a basket of goods and services purchased by the typical consumer such as individuals and non-profit organizations. Their respective core measures exclude food and energy, the prices of which tend to be highly volatile. The GDP deflator measures the average change in prices of all goods and services producedbythedomesticeconomy. Among the seven inflation measures, the CPI is the most cited inflation measure, but the price changes at the wholesale level, as captured by the PPI numbers, are often passed through to the consumerpriceindexinalaterdate. Hence,trackingpricepressuresfromthePPInumbers,investorscan anticipate inflationary consequences in the coming months. On the other hand, the PCE deflator is becomingthemostwatchedpriceindexfromthestandpointofmonetarypolicyandisconsideredasa “morereliablemeasureofinflation”bytheFederalReserve1 fortwomajorreasons. First,whereasthe CPIisonlyrepresentativeofthepricepaidbyurbancustomers,thePCEdeflatorisabroadermeasure that covers both urban and rural customers. Second, the PCE deflator is a chain-weighted index that capturesshiftingspendingpatterns. Incontrast,theCPIisafixed-weightindexthatreliesonspending patterns several years ago. Each of the above three indices has a corresponding core measure that ex- 1QuotesarefromthetestimonyofAlanGreenspanbeforetheCommitteeonFinancialServices,U.S.HouseofRepresentatives,July18,2001. 9

cludes food and energy. Many economists and investors prefer the core measures because they think that shocks to energy prices are often transitory. Others disagree. Finally, since the GDP deflator includesallgoodsandservicesproducedbythedomesticeconomy,itisthemostcomprehensivemeasure ofinflation. However, theGDPdeflatorisreleasedquarterlywhileallotherinflationmeasuresarereleased monthly. In our application, we do not take a stance on which of the seven series provides the mostaccurateandtimelymeasureoftheinflationarypressure. Instead,weincludeallofthemintoour estimationandextractonecommonfactorthatcapturesthesystematicmovementsininflationpressure. Thedatasetalsoincludesfouroutputandemploymentseries: therealGDP,industrialproduction, non-farm payrolls, and the real PCE. The real GDP is available in quarterly frequency. The other threeseriesareavailableinmonthlyfrequency,butthedataonrealPCEstartatalaterdateinJanuary 1991. The real GDP growth is the broadest measure of the output growth of the domestic economy. Industrialproductionmeasures theproductionofgoods. Althoughitis lesscomprehensive, it ismore timely since the industrial production numbers are released monthly whereas the GDP numbers are releasedquarterly. Non-farmpayrollsmeasurethenumberofemployeesonfirms’payrolls. Farmsare excludedbecauseoftheirseasonalnature,whichcanskewtotalemploymentfigures. Thisnumberisa keyindicatoroftheemploymentscenariooftheeconomy,whichhasfar-reachingimplicationsforboth inflationandoutputgrowth. Onthedemandsideoftheeconomy,weincluderealpersonalconsumption expenditure,whichoftenindicateschangesinthestateoftheeconomypriortochangesinproduction. Again, we first turn the four series into year-over-year growth rates and then standardize them before weextracttherealgrowthfactor. To extract a financial market volatility factor, we include two volatility indexes: the VXO index computed from options on S&P 100 index, and the VIX volatility index computed from options on S&P 500 index. The VXO measures the one-month at-the-money Black and Scholes (1973) implied volatility on the S&P 100 index options, and the VIX is a specific portfolio of option prices that approximatetheone-monthvarianceswaprateontheS&P500indexreturns(CarrandWu(2004)). Both seriesareavailablefromBloombergindailyfrequency,buttheVIXseriesstartsatalaterdateinJanuary1990. Thetwoseriesexhibitalargeamountofshort-termvariations. Toreducenoise,wecompute the yearly moving average of the daily volatility series. Then, we sample the moving averages at the 10

end of each month and extract the volatility factor in monthly frequency. Given the documented level dependence on the volatility of volatility, we first take logs on two series and then standardize them before we extract the volatility factor. Given the positivity of volatilities, talking logs also match our Gaussianfactorspecificationbetter. Inprinciple,factorscanrotateandtheloadingscanchangeaccordinglywithoutimpactingthefinal result. However, such rotations make it difficult to interpret the economic meanings of the dynamic factors. Toimproveidentificationandenhancetheeconomicinterpretationofthefactors,weputstructuralconstraintsonthefactorloadingmatrix. Specifically,weconstrainthefirstfactortohavepositive loadingsontheseveninflationseriesandthenon-farmpayrollseriesandzeroloadingsonallotherseries. Assuch,thisfirstdynamicfactorsummarizestheinflationpressureintheeconomy. Weconstrain thesecondfactortohavenonzeroloadingsonlyonrealGDP,industrialproduction,non-farmpayroll, and the real component of the personal consumption expenditure. Thus, it summarizes the real part of the macroeconomy, which we label as a real output factor. Finally, we constrain the third factor to havenonzeroloadingsonlyonthetwofinancialmarketvolatilityindexestomakeitafinancialmarket volatilityfactor. We estimate the dynamic factors in monthly frequency. For data series that are only available in quarterly frequency or at a later date, we fill the series with missing values. Our estimation method readily accommodates missing data. At each month, we update the dynamic factors based on the availablesubsetofthedata. D. TheTime-SeriesDynamicsoftheEconomicFactors Table I reports the estimates of the factor loading matrix (H) and the measurement error variance (R y),withtheabsolutemagnitudeofthet-statisticsfortheparameterestimatesreportedinparentheses. Thelastcolumnreportsthepredictedvariation(PV),definedasoneminustheratiooftheforecasting error variance over the variance of the original series. Since each series is standardized to have unit unconditional variance, the measurement error variance reflects the relative goodness of fit for each macroeconomicseries. Thesmallerthemeasurementerrorvarianceis,thehigherpercentagevariation thethreedynamicfactorscanexplain. Similarly,PVmeasuresthepredictiveperformanceofthethree 11

dynamic factors on each of the 13 series. It also reflects the relative informativeness of 13 series in termsoftheiraffinitytotheextractedeconomicfactors. Amongtheseveninflationvariables,thesmallestmeasurementerrorvarianceandhighestpredicted variation both come from the PCE deflator, consistent with the Federal Reserve’s emphasis on this measureasamorereliablegaugeoftheinflationpressure. Ontheotherhand,thelargesterrorvariance and lowest predicted variation both come from PPI, showing that this series is the most noisy or least informative about the inflation pressure. Nevertheless, the loading estimates on all seven series are statisticallysignificantandpositive,suggestingthatallsevenmeasurescontainusefulinformationabout thestateofinflation. Hence,itisappropriatetousethemallinsteadofpickingoneagainsttheother. The non-farm payrolls number is a key indicator of the employment scenario of the economy, it has far-reaching implications for both inflation and output growth. Hence, we also allow the first factor to have a nonzero loading on the non-farm payrolls series. The loading estimates are smaller thanthoseontheseveninflationvariables,butthehight-statisticssuggestthatthisloadingestimateis strongly significant and that non-farm payrolls are indeed informative about the inflation pressure of theeconomy. Among the four output and employment series, non-farm payrolls also have the highest loading andhighestt-statisticsonthesecondfactor. Furthermore,themeasurementerrorvarianceestimateon non-farm payrolls is neither visually nor statistically different from zero and the predicted variation is highest among all 11 macroeconomic series, showing that the non-farm payrolls series is the most informativeabouttheeconomy. Again,however,allfourserieshavesignificantlypositiveloadingson thesecondfactor. Hence,theyareallinformativeabouttherealsideoftheeconomy. Finally,forthetwovolatilityindexseriesVXOandVIX,themeasurementerrorvarianceestimates are both small and the predicted variations are high. The loading estimates on the two series are also similar,suggestingthatthetwoindicesmovecloselytogether. Table II reports the parameter estimates for k , which control the dynamics of the three macroeconomicandfinancialfactors. Foridentification,weassumealowertriangularstructureforthek matrix. Thus, the ranking of the three factors determines their dependence structure. We let inflation be the 12

first factor. Thus, the prediction of this factor only depends on its own past value. The estimate of 0.1139 corresponds to a monthly autocorrelation of 0.9906, showing the high persistence of inflation. The second factor is real output, the conditional mean of which depends on lagged values of both the inflation factor and the output factor itself. The estimate of 0.2007 corresponds to a monthly autocorrelationof0.9834,lowerthanthatfortheinflationfactors. Theoff-diagonalterm0.4891indicatesthat past values of the inflation factor predict negatively on the changes in the real output. Thus, the two macroeconomicfactorsshownegativecross-correlation. Thethirdfactoristhevolatilityfactor,which responds negatively to inflation, but positively to output. The small diagonal value suggests that the volatilityfactorishighlypersistent,inpartreflectingtheeffectofourmovingaveragesmoothing. Given the parameter estimates, the Kalman filter generates the ex post updated values of the three dynamic factors from the 13 observed series. Figure 1 plots the time series dynamics of the three extracted economic factors. The solid line depicts the inflation factor, the dashed line depicts the real outputgrowthfactor,andthedash-dottedlinedepictsthefinancialmarketvolatilityfactor. Theinflation factorhadahikeinearly1991,coincidingwiththespikeininflationpressurecausedbyenergyshocks during the first Gulf War. The inflationary pressure quickly receded and stayed low for the rest of the sampleperiod. [Figure1abouthere.] The dashed line for the real output growth shows two periods of sharp slowdown and one period of prolonged high output growth. The first slowdown coincided with the 1991–1992 recession. From mid 1994 to late 2000, the output growth factor remained at high values with some fluctuation. The factorstartedanotherverysteepfallinearly2001,reflectingthesharpslowdownoftheoutputgrowth, and reached the bottom in the second quarter of 2002. The output growth has picked up since then. Thisupswingisstillcontinuingasofnow,andthecurrentlevelofthisfactorisstillwaybelowitslevel reachedin2000. The volatility factor extracted from stock index options, as shown by the dash-dotted line, started high in late 1980s, showing the lingering effects of the 1987 stock market crash. The first Gulf War causedaspikeonthestockmarketvolatility,butotherwisethevolatilitystayedlowbetween1992and 13

1997. Thestockmarketvolatilityincreasedinlate1997followingtheAsiancrisesandthentheRussian default and the ensuing hedge fund crisis. Stock market volatility peaked around late 2002 and early 2003afteraseriesofcorporatescandalsincludingEnronandWorldComdefaultsandthewarinIraq. Sincethespringof2003,thestockmarketvolatilitystartedtocomedown. II. A No-Arbitrage Dynamic Term Structure Model of Interest Rates and Credit Spreads with Observable Economic Factors Weproposeadynamictermstructuremodelthatappliesno-arbitrageargumentstolinkthedynamic economic factors extracted in the previous section to the whole term structure of default-free interest rates on Treasury bonds and credit spreads on corporate bonds at different credit rating classes and industrysectors. A. MarketPricesofFactorRisksandRisk-NeutralFactorDynamics The previous section has specified the factor dynamics in equation (1), or its discrete version in equation (3). To price Treasury and corporate bonds based on the extracted dynamic factors, we need tospecifyhowthemarketpricesrisksthatareinherentinthedynamicfactors. Weconsideraflexible affinespecificationforthemarketpriceofrisksinthedynamicfactors, g (X)=g +g X, (12) t 0 1 t where g is an (n×1) vector and g is an (n×n) matrix, which we constrain to be lower triangular. 0 1 Thus, the market price has both a constant component and a component that varies with the factor level. Via model estimation, we determine the magnitude of the market prices and whether and how theyvarywiththestateoftheeconomy. The market price specification in (12) and no-arbitrage arguments dictate that there exists a riskneutralmeasureQthatisabsolutelycontinuouswithrespecttothestatisticalmeasureP,suchthatthe fair value of a financial asset,V, is equal to the expected value, with the expectation taken under this t 14

new measure Q, of its future payoff streams, p ,s>t, discounted by the corresponding instantaneous s risk-freeinterestrater : s V =EQ t (cid:20) Z ¥ exp − t (cid:18) Z s r du p ds F , (13) u s t t (cid:19) (cid:12) (cid:21) (cid:12) (cid:12) whereEQ[·|F ]denotestheexpectationoperatorundermeasureQ (cid:12) conditionalonthefiltrationF . The t t measurechangefromthestatisticalmeasurePtotherisk-neutralmeasureQisdefinedbythefollowing exponentialmartingale, dQ ≡E − dP(cid:12) (cid:18) (cid:12)t (cid:12) (cid:12) Z t g (X )⊤dW =E − s s 0 (cid:19) (cid:18) Z t (g +g X )⊤dW , (14) 0 1 s s 0 (cid:19) where E(·) denotes the Dole´ans-Dade exponential operator. According to the Girsanov theorem, the factordynamicsremainOrnstein-Uhlenbeckundertherisk-neutralmeasureQ, dX =k Q q Q−X dt+dW Q , (15) t t t (cid:16) (cid:17) withk Qq Q=−g beingan(n×1)vectorandk Q=k +g beingan(n×n)lowertriangularmatrix. 0 1 B. TheTermStructureofTreasuryYields TomodelthetermstructureofTreasuryyields,weassumethattheinstantaneousdefault-freeinterestrateisaffineinthethreedynamicfactors, r =r(X)+e r, r(X)=a +b⊤X, (16) t t t t r r t where e r denotes the instantaneous interest rate moves that are not explained by the three dynamic t factors. Bydesign,X orr(X)isorthogonaltoe r. t t t According to the fundamental valuation equation in (13), the time-t fair value of a default-free zero-couponbondwithtime-to-maturityt is B(t,t ) = EQ exp − (cid:20) (cid:18) Z t+t r(X )ds F EQ exp − s t t (cid:19)(cid:12) (cid:21) (cid:20) (cid:18) (cid:12) (cid:12) (cid:12) Z t+t e rds F , t s (cid:19)(cid:12) t (cid:21) (cid:12) = B(X t ,t )E(t,t ), (cid:12) (cid:12) (17) 15

wherethemultiplicativedecompositionfollowsfromtheorthogonalityassumptionbetweenX ande r. We leave the dynamics of e r unspecified and regard E(t,t ) as an error term on the zero-coupon bond t pricethatisnotexplainedbythethreedynamiceconomicfactors. The specification of the Q-dynamics for the factors X in (15) and the instantaneous default-free t interest-ratefunctionr(X)in(16)satisfytheaffineconditionofDuffieandKan(1996)andDuffie,Pan, t and Singleton (2000). Thus, we can solve B(X,t ) as an exponential-affine function of the economic t factors, B(X,t )=exp −a(t )−b(t )⊤X , (18) t t (cid:16) (cid:17) wherethecoefficients[a(t ),b(t )]aresolutionstothefollowingordinarydifferentialequations: a′(t ) = a −b(t )⊤g −b(t )⊤b(t )/2, r 0 ⊤ b′(t ) = b − k Q b(t ), (19) r (cid:16) (cid:17) subject to the boundary conditions a(0)=0 and b(0)=0. The ordinary differential equations can be readilysolvedusingstandardnumericalprocedures. Given the exponential-affine solution in (18), the continuously compounded spot rates are affine functionsofthethreeeconomicfactors, lnB(X,t ) a(t ) b(t ) ⊤ R(X,t )≡− t = + X. (20) t t (cid:20) t (cid:21) (cid:20) t (cid:21) t Theobservedspotrate,R(t,t ),canbewrittenas R(t,t )=R(X,t )+e(t,t ), (21) t where e(t,t ) ≡ −lnE(t,t )/t denotes the portion of the spot rate that is not explained by the three economicfactors. Inourestimation,wetreate(t,t )asthemeasurementerrorterm. 16

C. TheTermStructureofCorporateYieldsandCreditSpreads For defaultable bonds, Duffie and Singleton (1999) and Duffie, Pedersen, and Singleton (2003) show that the valuation can be written in analogous forms by adjusting the risk-free discounting with an instantaneous credit spread. Specifically, the time-t value of a zero-coupon defaultable bond with time-to-maturityt ,D(t,t ),canbewrittenas, D(t,t )=EQ exp − (cid:20) (cid:18) Z t+t (r +s )du F , (22) u u t t (cid:19)(cid:12) (cid:21) (cid:12) (cid:12) (cid:12) wheres denotestheinstantaneousdefaultspread,whichcanbethoughtofasareduced-formproduct t ofdefaultprobabilitiesandlossgivendefault. Inaddition,itcanalsobeusedtocapturespreadsinduced byliquidityandotherfactors. Topricecorporatebondsatacertaincreditratingclass(and/orindustrysector)i,weassumethatthe instantaneouscreditspreadforthatratingclass,si,isanaffinefunctionofthethreeeconomicfactors, t si =si(X)+e i, si(X)=a +b⊤X, (23) t t t t i i t wheree i denotestheportionofthespreadthatisnotexplainedbythethreeeconomicfactors. t Then, wecanshowanalogouslythatthefairvalueofthezero-couponbondintheithcreditrating groupisalsoexponentialaffineinthethreedynamiceconomicfactors, D (t,t )=D (X,t )E (t,t ), with D (X,t )=exp −a (t )−b (t )⊤X , (24) i i t ir i t i i t (cid:16) (cid:17) where E (t,t ) is the error term induced by the unexplained movements in both the default-free rate ir and the credit spread: e r+e i, and the coefficients [a(t ),b(t )] are solutions to the following ordinary i i differentialequations: a′(t ) = (a +a)−b(t )⊤g −b(t )⊤b(t )/2, i r i i 0 i i ⊤ b′(t ) = (b +b)− k Q b(t ), (25) i r i i (cid:16) (cid:17) 17

subjecttotheboundaryconditionsa(0)=0andb(0)=0. i i Thecontinuouslycompoundedspotrateonthedefaultablebondisaffineintheeconomicfactors, lnD(X,t ) a(t ) b(t ) ⊤ R(X,t )≡− i t = i + i X. (26) i t t (cid:20) t (cid:21) (cid:20) t (cid:21) t Theobservedspotrateonthedefaultablebondcanbewrittenas lnD(t,t ) R(t,t )≡− i =R(X,t )+e (t,t ), (27) i t i t ir withe (t,t )≡−lnE (t,t )/t . ir ir We define the credit spread on the corporate bond as the difference between the spot rate on the defaultablebondandthecorrespondingspotrateontheTreasury: a(t )−a(t ) b(t )−b(t ) ⊤ S (t,t )≡R (t,t )−R(t,t )= i + i X +e(t,t ), (28) i i (cid:20) t (cid:21) (cid:20) t (cid:21) t i withe(t,t )=e (t,t )−e(t,t ). Thus,viano-arbitragearguments,welinkthecreditspreadsacrossall i ir maturitiesatacertaincreditratingclasstotheobservabledynamiceconomicfactors. Theno-arbitrage linksaredeterminedbythefactordynamics,themarketpricesoffactorrisks,andbytheinstantaneous default-freeinterestrateandcreditspreadasfunctionsofthesefactors. Themodelprovidesthetheoretical basis and economic insights on the determinants of the Treasury yields and corporate bond credit spreads. III. Data and Estimation A. ConstructingtheTreasuryYields TheTreasuryyieldsdataaremonthlycontinuouslycompoundedspotratesobtainedfromtheFederal Reserve Board, which extracts the rates from the Treasury notes and bond prices following the procedure proposed by Svensson (1995). The spot rates are available at 12 maturities: three months, 18

sixmonths,andeveryyearfromonetotenyears. Weusethesamesampleperiodasfortheeconomic factorsfromJanuary1988toOctober2004. TableIIIreportsthesummarystatisticsoftheTreasuryyields. Overthe15yearsofsampleperiod, the mean Treasury yield show an upward sloping term structure. The standard deviation are larger forshort-termyieldsthanforlong-termyields. BothskewnessandkurtosisestimatesfortheTreasury yields are small. The Treasury yields show strong persistence, with monthly autocorrelation ranging from0.973to0.986. The left panel of Figure 2 plots the time series of Treasury yields over the 15-year sample period. The plot shows two periods of low interest rates with steep yield curves at around 1993 and 2003, respectively, each incidence following the trough of the real output factor in Figure 1. The Treasury interestratesarehighandthetermstructureflatinthelate1980sandalsointheextendedhigh-growth periodofthelater1990s. [Figure2abouthere.] TherightpanelinFigure2plotsthetermstructureoftheTreasuryspotratesateachmonthofour sampleperiod. Theboldsolidlinedenotesthemeanupward-slopingtermstructure. Oursampleperiod has witnessed a variety of term structure patterns, including upward sloping, flat, and hump-shaped termstructures. B. ConstructingtheCorporateBondYields We construct continuously compounded spot rate for each credit rating class using month-end prices on corporate bonds that are either in the Merrill Lynch U.S. Corporate Master Index or the MerrillLynchU.S.HighYieldIndex. TheseindicestrackthepricesofU.S.dollar-denominatedinvestmentgradeandhighyieldcorporatepublicdebtissuedintheU.S.domesticbondmarket. TheMerrill Lynch data set covers the period from January 1997 to June 2004. To construct a long time-series of corporate bond yield sample, we augment the Merrill Lynch data by the Lehman Brothers Fixed IncomedatabasefromJanuary1988toDecember1996. TheLehmandatacoverstheperiodfromJanuary 19

1973toMarch1998,butthereareveryfewnoncallablebondsthatwereissuedbeforemid1980s. We estimateourmodelsbasedondatafromJanuary1988toOctober2004. We enforce the following bond selection criteria. First, we consider only straight bonds without option features. Callable, putable, convertible and bonds with sinking fund clause are dropped from our sample. Second, bonds with remaining maturities less than one year or greater than 35 years are eliminated. Third, only those bonds that have fixed coupon schedule and pay fixed rate semiannual coupons are included. Fourth, we include only senior unsecured bonds, where bond seniority information are obtained from Moody’s Investors Services. Finally, for the Lehman data, bond prices thatarecalculatedusingamatrixmethodareexcluded. Theresultingbondsamplehas337,990bondmonthobservations. Continuouslycompoundedcorporatespotratesforeachletter-graderatingclassareestimatedusing the Nelson and Siegel (1987) model and a procedure detailed in Bolder and Streliski (1999). For example, for the AA credit rating class, there are a total of 47031 bond-month observations. The Nelson-Siegel model is estimated for each month on this sub-sample of AA bonds to extract the spot yield curves for the AA credit rating class. We repeat the same procedure for each of the following rating classes: AAA, AA, A, BBB, BB, and B. Yield spread for each rating class is calculated as the difference between the spot yield of the rating class and the maturity-matched Treasury yield. The maturityforthecreditspreadgoesfromonetotenyearseveryyear. Table IV reports the summary statistics of credit spreads at different maturities and rating classes. Themeancreditspreadincreaseswithdecliningcreditratings. ThemeanspreadsforthethreeArating classesareclosetooneanotherbetween71and89basispoints,butthespreadincreaseacceleratesas theratingfurtherdeclines,especiallyaftertheratinggoesbelowinvestmentgrade. FromAtoBB,the meanspreadalmostdoublesforeachletterdowngrade. Acrossmaturitieswithineachratingclass,the mean term structure is relatively flat. The standard deviations of credit spreads on AAA, AA, and A bondsareinthesamerange,butthestandarddeviationestimatesonBBB,BB,andBbondsaremuch largerandincreasewithdecliningratings,moresoatshortthanatlongmaturities. Creditspreadsalso showhighpersistence,moresoforhighcreditratingclassesandmoderatematurities. 20

Figure3plotsthetimeseriesofthecreditspreads,witheachpaneldenotingonecreditratingclass andeachlinedenotingonematurity. Weusethesamescalingforthefourpanelsoninvestment-grade spreads (AAA, AA, A, and BBB), but use increasingly larger scales for high yield spreads (BB and B)toaccommodatethemuchwiderspreads. Asidefromthescaledifferences, thetimeseriesplotsin the six panels show common movements that are in line with the state of the economy. For all rating classes, we observe two common periods of high spreads, one in early 1990s and the other in early 2000,correspondingtothetworecessionsinoursampleperiod. [Figure3abouthere.] Figure 4 plots the term structure of the credit spreads at each month, with the bold solid lines denoting the mean term structure. Although the mean term structures are relatively flat for all six ratingclasses,thetermstructureateachmonthhasshowndifferentpatterns,includingupwardsloping, downwardsloping,flat,andhump-shapedtermstructures. [Figure4abouthere.] C. EstimatingtheNo-ArbitrageLinks Forestimation,wecastthedynamictermstructuremodelintoastate-spaceform,extractthedistributionsofthestatesateachdatebyusinganefficientfilteringtechnique,andestimatethemodelusing quasimaximumlikelihoodmethod,assumingnormalforecastingerrorsontheobserveddataseries. Thestatisticaldynamicsofthethreeeconomicfactorshavealreadybeenestimatedintheprevious sectionsusingthe13macroeconomicandfinancialtimeseries,withtheestimatesreportedinTableII. We take these parameter estimates as given and estimate the remaining parameters that determine the no-arbitragelinksrepeatedlyusingdatafromeachmarket. First, we estimate the market prices of factor risks, and the default-free instantaneous interest rate function using the 12 Treasury spot rate series. Based on this estimation, we determine the impacts of the economic factors on the Treasury yield curve. Then, we re-estimate the market prices of factor 21

risks,andalsoestimatetheinstantaneouscreditspreadfunctionateachcreditratinggroupusingtheten creditspreadseriesinthatratinggroup. Werepeatthisestimationprocedureforeachofthesixcredit ratingclasses. Theproceduregeneratessevensetsofestimatesonthemarketpricesofrisks,onesetfor eachmarket. Marketefficiencydictatesthatdifferentmarketsshouldpricethesameriskthesameway. Hence,thesevensetsofparameterestimatesshouldbeclosetooneanother. Largedeviationssuggest eithermarketsegmentationormodelmisspecification. Foreachestimation,thestatepropagationequationremainsthesameasin(3). Thedifferenceliesin the different measurement equations. For the Treasury market estimation, the measurement equations areintermsofthecontinuouslycompoundedTreasuryspotrates, R(t,t )=R(X t ,t )+e(t,t ), t =0.25,0.5,1,2,3,4,5,6,7,8,9,10years. (29) wheree(t,t )istreatedasthemeasurementerror. Forestimationonthecorporatebondmarketatafixed creditratingclassi,themeasurementequationsaredefinedonthecreditspreads, S i (t,t )=S i (X t ,t )+e i (t,t ), t =1,2,3,4,5,6,7,8,9,10years. (30) whereS(X,t )denotesthespreadexplainedbythethreeeconomicfactorsande(t,t )denotestheunexi t i plainedcomponent,whichistreatedasthemeasurementerror. ToapplyKalmanfilter,weassumethat themeasurementerrorsarenormallydistributed. Wefurtherassumethattheyaremutuallyindependent butwithdifferentvariance. Since the Treasury yields and credit spreads are both affine in the economic factors as shown in equations (20) and (28), respectively, we can use the Kalman filter to obtain the ex ante forecasts and ex post updates on the conditional mean and covariance of the three dynamic factors via the iterative procedure defined by equations (4) to (9). Furthermore, since we have already extracted the dynamic economic factors X in an earlier section fromthe macroeconomic and financial series, we now regard themasobservableseries. Hence,theexpostupdatesare, X =Xm ; V =0, (31) t+1 t+1 t+1 b b b 22

where Xm denotes the dynamic factors extracted in the earlier section. The ex post variance is zero t+1 becausebwetreatXm asobservable. t+1 b The model parameters are estimated by maximizing the log likelihood function defined on the forecasting errors of the Treasury yields and credit spreads, respectively. The procedures estimate the followingparameters: Q ≡[a ,b ,g ,k Q]andthemeasurementerrorvariancefromtheTreasuryyields, r r 0 andQ ≡[a,b,g ,k Q]andthemeasurementerrorvariancefromthecreditspreadsofeachratinggroup. i i 0 Since the bond pricing equations ask for direct input for the risk-neutral mean-reversion coefficient matrixk Q =k +g . Weestimatek Q directly,insteadofestimatingthemarketpricecoefficientmatrix 1 g andthencombiningitwiththepreviousestimateonk . 1 IV. Economic Determinants of Treasury and Default Term Structures Byestimatingthedynamictermstructuremodels,wequantifytheimpactsofeacheconomicfactor on the term structure of Treasury yields and credit spreads at different credit rating classes. Furthermore,welinktheseimpactstotheunderlyingfactordynamicsandmarketpricesoffactorrisks. A. PredictivePerformanceoftheEconomicFactors Togaugetheperformanceofthethreedynamiceconomicfactorsinexplainingthevariationofthe term structure of Treasury and corporate bond yields, Table V reports the predicted variation (PV) of theTreasuryyieldsandcorporatebondcreditspreads,definedasoneminustheratiooftheforecasting errorvarianceoverthevarianceoftheTreasuryspotratesandcorporatecreditspreads,respectively. Themodelperformanceisrelativelyuniformacrossmaturities. Thethreeeconomicfactorspredict over 70 percent of the variation in the Treasury yields, and about 30–60 percent of the variation in the corporate credit spreads at the four rating classes. The performance is better than most regression analysisresults,showingtheenhancedpowerofpredictioninusingthedynamicfactorsextractedfrom manymacroeconomicandfinancialseries, insteadofusingafewrawseriesthemselves. Ontheother hand, the results also show that even with the dynamic factor approach, about half of the variation in creditspreadsarestillnotexplainedbythethreemajordimensionsoftheaggregateeconomy. 23

B. MarketPricesofEconomicRisk Table VI reports the estimates and t-statistics of the model parameters that determine the market prices of economic factor risks. These market prices, joint with the time series dynamics in Table II, determine the risk-neutral factor dynamics, which play important roles in the Treasury and corporate bondpricingacrossdifferentmaturities. Weestimatethemarketpricesandhencerisk-neutraldynamicsrepeatedlyusingthetermstructure of Treasury yields and also the term structure of credit spreads at each of the six credit rating classes. Thus, we obtain seven sets of estimates. Similar estimates across Treasury and the different credit ratingclassesprovideevidenceonmarketintegrationandtherobustnessofthedynamicspecification. On the other hand, different estimates would suggest either measurement noise or evidence of market segmentationthatdifferentmarketspricetheeconomicriskdifferently. Theestimatesong measuretheconstantportionofthemarketprice. Theyalsodefinetheconstant 0 component of the risk-neutral drift of the dynamic factors. Across the five sets of the estimates, the moststatisticallysignificantarethenegativeestimatesontheoutputfactor(thesecondelementofg ), 0 suggestingthattheoutputfactorhasanegativemarketprice. Theestimatesonthevolatilityfactorare strongly positive across all five markets, but only one of the five estimates is statistically significant under 95 percent confidence level. The most inconsistent across markets are the estimates on market price of the inflation factor. The market price estimates on the inflation factor are positive for the Treasury and AAA rating class, but negative for the other three rating classes. Nevertheless, four of the five estimates are not significantly different from zero, suggesting large estimation errors on this constantcomponentofmarketpriceforinflationrisk. Instead of estimating the proportional component of the market price of risk g and then deriving 1 the risk-neutral dynamics k Q, we directly set k Q as the free parameter because it is k Q that directly enters thebondpricingequation. Overall, the smallerthe estimates, themorepersistenttheeconomic factorsareundertherisk-neutralmeasure,andhencethemorelong-lastingthefactorimpactsbecome. Table VI shows that the first diagonal element of k Q is very small, suggesting that the inflation factorishighlypersistentundertherisk-neutralmeasure. Hence,shocksonthisfactorarelikelyimpact 24

thetermstructureofinterestratesandcreditspreadsatbothlongandshortmaturities. Incontrast,the estimatesfortheseconddiagonalelementofk Q aremuchlarger,andallarelargerthanthetime-series counterpart, suggestingthat shocks on the real output factor dissipates fasteracross the term structure thanshocksontheinflationrisk. Forthethirddiagonalelementofk Qthatcontrolstherisk-neutralpersistenceandthetermstructure impact of the volatility factor, the estimate is the largest at 0.7241 from the Treasury market, but the estimatesobtainedfromthecorporatebondsmarketsaresmaller,andincreasinglysoasthecreditrating declines. Theincreasingrisk-neutralpersistencewithdecliningcreditratingdictatesthattheimpactof thevolatilityfactorbecomesmorelong-lastingforcorporatebondsatlowercreditratingclasses. C. EconomicDeterminantsofTreasuryYieldTermStructure Table VII reports the coefficient estimates on the instantaneous default-free interest rate function in the column under “Treasury.” The intercept estimate a measures the long-run mean of the inr stantaneous default-free interest rate, which has an estimate of 4.86 percent. The loading coefficient estimates, b , measure the contemporaneous response of the short rate to unit shocks on the three r macroeconomicandfinancialfactors. Theestimatesforthethreeelementsareallpositiveandstrongly significant,suggestingthatinflation,output,andfinancialmarketvolatilityallhavepositiveimpactson theshortrate. Equation (19) illustrates how the short rate function (a ,b ) interacts with the risk-neutral factor r r dynamics(g ,k Q)todetermineboththemeantermstructureofTreasuryyields,asmeasuredbya(t )/t , 0 andthefactorloadingsacrossthetermstructure,asmeasuredbyb(t )/t . Figure5plotsthemeanterm structureintheleftpanelandloadingsofthethreefactorsintherightpanel. Theupwardslopingmean term structure in the left panel is consistent with data observation. The three lines in the right panel showthecontemporaneousresponseoftheTreasuryyieldcurvetounitshocksonthethreeeconomic factors, respectively. The solid line denotes the response to the inflation factor, which is positive and thestrongestamongthethreelines. Furthermore,theresponseislargenotonlyatshortmaturities,but alsolaststhroughlongmaturitiesduetothehighrisk-neutralpersistenceoftheinflationfactor. 25

[Figure5abouthere.] Thedashedlinedenotestheimpactsoftherealoutputfactor,whichisalsopositive,butsmallerin magnitude. Furthermore,duetothelowerrisk-neutralpersistenceofthisrealoutputfactor,theimpacts dissipatefasterasmaturityincreases. Foraunitshocktotherealoutputfactor,theresponseofone-year Treasuryrateisabouttwiceasmuchastheresponseoftheten-yearTreasuryrate. Finally, the dash-dotted line captures the impacts of the financial market volatility factor, which are positive but small in magnitudes. Furthermore, its impacts also decline quickly with increasing maturities. Theimpactsareclosetozeroatlongmaturities. Factoranalysisonthetermstructureofinterestrates,e.g.,LittermanandScheinkman(1991),Knez, Litterman,andScheinkman(1994),andHeidariandWu(2003),identifiesthreekeyfactors,whichare oftenreferredtoasthelevelfactor,theslopefactor,andthecurvaturefactor,respectively. LuandWu (2004)showthattheinflationfactorandtheoutputfactorgeneratealevelandslopeeffectontheterm structure, respectively, whichmatchtheroleofthefirsttwostatisticalfactors. HeidariandWu(2003) show that the third statistical factor in interest rates is highly correlated with the interest rate option implied volatilities. Our proposed three-dimensional decomposition of the aggregate economy is in linewithsuchevidence. ThefactorloadingplotsinFigure5alsorevealsimilarinterpretationsforthe threeeconomicfactors. D. EconomicDeterminantsofCreditSpreadTermStructure Table VII also reports the parameter estimates for the instantaneous credit spread function under eachofthesixcreditratingclasses. Theestimatesfora measurethefixedcomponentoftheinstantai neousspreadinducedbycreditrisk. TheestimateforthisinterceptisclosetozeroforAAAbonds,but becomesincreasinglypositiveastheratingdeclines,indicatinganincreasedaveragecompensationfor theincreasedriskatlowercreditratingclasses. Theloadingcoefficientestimatesonthethreeeconomicfactorsvaryacrossdifferentratingclasses. These loading coefficients interact with the risk-neutral factor dynamics k Q to determine how shocks 26

in the three factors impact the term structure of corporate bond yields and hence the credit spreads. Figure 6 plots the three elements of (b(t )−b(t ))/t for each credit rating group, which measure the i contemporaneous responses of the credit spread term structure to unit shocks on each of the three dynamiceconomicfactors. Theresponsesarecomputedaccordingto(28)andtheordinarydifferential equations in (19) and (25). Each panel in Figure 6 corresponds to one rating class and each line corresponds to the response of the credit spread term structure to one of the three economic factors. Foreaseofcomparison,weusethesamescaleforthefourinvestmentgradepanels(AAA,AA,A,and BBB).Weusealargerscaleforthehigh-yieldpanels(BBandB)toaccommodatethelargerresponses. [Figure6abouthere.] Thesolidlineineachpaneldenotestheresponseofthecreditspreadtermstructurestotheinflation factor. Theresponsesarepositiveacrossallratinggroups. Withineachratinggroup,theresponsesare persistentacrossmaturities,consistentwithitspersistentrisk-neutraldynamics. Thepositiveresponses suggest that increasing inflation not only increases the Treasury rate across all maturities, but also widensthecreditspreadsoncorporatebonds. The dashed line in each panel shows the response of the credit spread term structure to the real outputfactor. TheresponsesareslightlypositiveforAAAandAAcreditspreads,butbecomenegative for A credit spreads, and very much so for the BBB, BB, and B credit rating classes. Furthermore, the negative responses are larger at short maturities than at long maturities. Thus, although a positive output shock increases the interest rate level and flattens an otherwise upward sloping yield curve, it reduces the pricing and/or risk of corporate default and narrows the credit spread at low credit rating classes,particularlysoatshortmaturities. The dash-dotted line in each panel plots the response of the credit spreads to the financial market volatility factor. The responses are positive for all rating groups, and increasingly so with declining creditratings. Furthermore,inallcases,theimpactsincreasewithmaturities,consistentwiththehigh risk-neutralpersistenceforthisvolatilityfactorestimatedfromthecorporatebondmarket. Therefore, whereas the Treasury yields are dominated by macroeconomic forces, the financial market volatility 27

plays an increasingly important role on the credit spreads, particularly at low rating classes and long maturities. V. Credit Spreads Term Structure For Financial and Corporate Sectors To investigate whether corporate bond spreads at different industry sectors react differently to the economicshocks,wefurtherdividethecorporatebondssamplewithineachcreditratingclassintotwo broadindustrysectors: financial(F)andcorporate(C).Weconstructthetermstructureofcreditspreads foreachindustrysectorandratingclass,withtheexceptionofBBandBratingclasses,wherelackof datapreventusfromobtainingreliabletermstructureestimatesforthetwoindustrysectors. Figure 7 compares the time series of the credit spreads for these two industry sectors under each of the four rating classes. The common movements are similar to what we have observed from the aggregate plots in Figure 3. Comparing the time series between the two industry sectors within each creditratingclass,wefindthatthecreditspreadsareslightlyhigherforthefinancialfirmsthanforthe corporatefirms. [Figure7abouthere.] We repeat the estimation on each of eight new classifications of credit spreads. The parameter estimates are roughly in the same range as those obtained before. o save space, we do not report the parameterestimates. Theyareavailableuponrequest. Figure8plotsthecontemporaneousresponseof thecreditspreadstounitshocksonthethreeeconomicfactors. Weapplythesamescaletoallpanelsfor easeofcomparison. Asbefore,wefindthatthetheinflationfactorhaspositiveandpersistentimpacts on the term structure of credit spreads, but the impacts become smaller at lower credit ratings. The real output factor has slightly positive impacts on credit spreads at high credit rating classes, but the impacts become negative at lower rating classes. The impact of the volatility factor is small at high ratingclasses,butbecomesstronglypositiveatlongratingclasses. [Figure8abouthere.] 28

Comparing the responses between the two industry sectors within each rating class, we find that the responses of the credit spreads in the financial sector are stronger in absolute magnitudes than the response of the spreads in the corporate sector, especially at lower rating classes. For the financial sector,theimpactsoftheinflationfactoraremorepositiveacrossallratingclasses; theimpactsofthe real output factor are more negative for the BBB class; and the impacts of the volatility factor more positiveforAA,A,andBBBclasses. Therefore,overallthefinancialsectorismoresensitivetochanges intheeconomicenvironments. VI. Conclusion We use a dynamic factor model to summarize the information in many observed macroeconomic and financial data series and to provide a no-arbitrage link between the dynamic economic factors andthetermstructureofTreasuryyieldsandcreditspreadsofcorporatebondsatdifferentcreditrating classesandindustrysectors. Byestimatingthemodel,wequantifytheimpactsofmanymacroeconomic andfinancialseriesonthewholetermstructureofTreasuryyieldsandcreditspreads. Wealsolearnthe fundamentalsourcesbehindtheimpacts: theeconomicfactordynamicsandmarketpricesofeconomic risks. We find that positive inflation shocks increase Treasury yields and credit spreads across all maturities and credit rating classes. The impacts on the Treasury yields are the strongest. The impacts on thecreditspreadsaresmaller,anddeclinewithloweringcreditratings. Thus,apositiveinflationshock moves up both the benchmark yield curve and the credit spread, more so for the Treasury rates and spreadsonhighcreditratingclassesthanforthespreadsonlowratingclasses. PositiveshocksontherealoutputgrowthalsoincreasetheTreasuryyields,moresoatshortmaturitiesthanatlongmaturities. Theimpactsonthecreditspreadsarepositiveforhighcreditratingclasses, but become negative and increasingly so as we move to lower credit rating classes. Furthermore, the impacts are more negative at shorter maturities than at longer maturities. Thus, a positive shock to the real side of the economy increases the benchmark interest-rate level, flattens an otherwise upward slopingyieldcurve,butnarrowsthecreditspread,particularlyatshortmaturitiesandlowcreditrating classes. 29

The financial market volatility factor has small and transient impacts on the Treasury yield curve, but strongly positive and persistent impacts on the credit spreads across the whole term structure. Therefore, although increasing financial market volatility has only small impact on the benchmark yieldcurve,itwidensthecreditspread,moresoatlongmaturitiesandlowratingclasses. Whenwefurtherdecomposeeachcreditratingclassintotwoindustrysectors: financialandcorporate,wefindthatthecreditspreadsinthefinancialsectorsareonaveragewiderandmorevolatilethan the spreads in the corporate sector, especially at lower rating groups. Furthermore, the credit spreads inthefinancialsectoralsorespondmorestronglytochangesineconomicclimates. This paper integrates the strength of two strands of extant literature: the economic intuition of regressionanalysisonobservedvariablesandtheinternalconsistencyofthedynamictermstructuremodeling. Ourresultsshowgreatsynergyfromtheintegration,andabundantpotentialforfutureresearch. First, a dynamic factor model can be used in many research areas as an efficient way to reduce noise and strengthen the signal content of many explanatory variable choices. Instead of choosing one over another, a dynamic factor model allows us to include them all, with the information filtered out from the factors. This approach becomes more important with the increasing availability of large amounts of data. Second, no-arbitrage arguments can be used to add discipline, economic rigor, and internal consistency in many applications of factor analyses. The no-arbitrage modeling literature is long and wellestablished,butlinkingittoobservableeconomicfundamentalsisonlyatitsnascentstage,leaving wideopenopportunitiesforfutureexplorations, notonlyforlinkagesbetweeninterestrates/exchange rates and aggregate economic factors, but also for linkages between corporate bonds/stocks and firm levelfundamentals. 30

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TableI ExtractingSystematicDynamicFactorsFromMacroeconomicandFinancialData Entries report the estimates and the absolute values of the t-statistics (in parentheses) of parameters (H)thatlinkeachobserveddataseriestothethreesystematicdynamicfactors. EntriesunderH denote i the loadings of each series on the ith factor. Entries under Ry report the measurement error variance estimates and t-statistics for each series. The last column (PV) reports the predicted variation of the factorsoneachseries,definedasoneminustheratiooftheforecastingerrorvarianceoverthevariance of the original series. The parameters are estimated with maximum likelihood method and Kalman filtering using macroeconomic and financial data series listed below. The macroeconomic data are from the Federal Reserve Board, the volatility series are downloaded from Bloomberg. The sample periodisfromJanuary1988toJune2004. Series H H H R y PV 1 2 3 CPI 0.439 (6.45) — — — — 0.081 (4.20) 0.891 CoreCPI 0.415 (4.06) — — — — 0.181 (4.80) 0.841 PPI 0.316 (2.92) — — — — 0.523 (3.58) 0.424 CorePPI 0.403 (7.45) — — — — 0.224 (8.39) 0.767 PCEDeflator 0.454 (8.47) — — — — 0.020 (4.47) 0.949 CorePCEDeflator 0.424 (4.44) — — — — 0.143 (3.67) 0.857 GDPdeflator 0.437 (7.81) — — — — 0.085 (4.40) 0.929 RealGDP — — 0.277 (5.67) — — 0.399 (4.48) 0.571 IndustrialProduction — — 0.299 (8.43) — — 0.314 (7.89) 0.641 Non-farmPayrolls 0.169 (4.50) 0.379 (10.7) — — 0.000 (0.00) 0.988 RealPCE — — 0.228 (6.89) — — 0.548 (7.32) 0.449 VXO — — — — 0.391 (14.0) 0.000 (0.00) 0.987 VIX — — — — 0.379 (13.6) 0.020 (4.60) 0.974 34

TableII TimeSeriesDynamicsoftheEconomicFactors Entriesreporttheparameterestimatesandtheabsolutevaluesofthet-statistics(inparentheses)ontimeseriesdynamicsofthethreeeconomicfactors. Thedynamicsareestimatedwithmaximumlikelihood method and Kalman filtering with 13 macroeconomic and financial data series. The macroeconomic data are from the Federal Reserve Board. The volatility series are downloaded from Bloomberg. The dataaremonthlyfromJanuary1988toJune2004. DynamicFactors(X) k in: dX =−k Xdt+dW Inflation 0.1139 0 0 (0.62) — — RealOutput 0.4891 0.2007 0 (2.71) (1.58) — Volatility 0.1484 -0.1790 0.0625 (1.00) (1.42) (0.49) 35

TableIII SummaryStatisticsofTreasuryYields Entries report mean (‘Mean’), standard deviation (‘Std’), skewness (‘Skew’), kurtosis (’Kurt’), and monthlyautocorrelation(‘Auto’)ofthecontinuouslycompoundedTreasuryyieldsatdifferentmaturities. DataaremonthlyfromJanuary1988toJune2004,obtainedfromtheFederalReserveBoard. Maturity Mean Std Skew Kurt Auto 3m 4.777 2.081 -0.120 -0.580 0.986 6m 4.887 2.114 -0.183 -0.571 0.986 1y 5.083 2.090 -0.244 -0.525 0.985 2y 5.395 1.953 -0.273 -0.438 0.981 3y 5.636 1.821 -0.246 -0.417 0.978 4y 5.833 1.715 -0.195 -0.457 0.976 5y 6.002 1.632 -0.135 -0.531 0.975 6y 6.149 1.565 -0.074 -0.618 0.974 7y 6.278 1.510 -0.016 -0.702 0.973 8y 6.391 1.464 0.037 -0.775 0.973 9y 6.491 1.424 0.085 -0.835 0.973 10y 6.578 1.389 0.126 -0.883 0.973 36

TableIV SummaryStatisticsofCreditSpreadsonCorporateBonds Entriesreportmean,standarddeviation,andmonthlyautocorrelationofthecreditspreadsoncorporate bondsateachmaturityandcreditratingclass. Thespreadsaredefinedasthedifferenceinpercentage pointsbetweencontinuouslycompoundedspotratesatacertaincreditratinggroupandthecorrespondingTreasuryspotrates. CorporatebondspotratesareextractedusingNelson-Siegelmethodfromthe corporate bond data. Data are monthly from January 1988 to June 2004, obtained from the Federal ReserveBoardandMerrillLynch. Maturity 1 2 3 4 5 6 7 8 9 10 SampleMean AAA 0.714 0.713 0.735 0.754 0.762 0.760 0.751 0.737 0.720 0.703 AA 0.749 0.747 0.772 0.800 0.821 0.833 0.837 0.836 0.830 0.823 A 0.890 0.937 0.993 1.039 1.070 1.088 1.096 1.095 1.090 1.081 BBB 1.519 1.472 1.480 1.509 1.539 1.565 1.584 1.598 1.606 1.611 BB 3.828 3.320 3.079 2.985 2.969 2.990 3.027 3.067 3.104 3.136 B 6.068 6.389 6.456 6.362 6.191 5.984 5.762 5.531 5.297 5.061 SampleStandardDeviation AAA 0.392 0.298 0.290 0.302 0.311 0.314 0.315 0.315 0.315 0.315 AA 0.263 0.255 0.270 0.280 0.288 0.295 0.303 0.311 0.320 0.330 A 0.305 0.327 0.346 0.350 0.350 0.348 0.348 0.351 0.356 0.362 BBB 0.555 0.606 0.621 0.614 0.596 0.575 0.555 0.538 0.526 0.517 BB 2.167 1.508 1.241 1.143 1.109 1.102 1.114 1.138 1.170 1.207 B 4.789 4.232 4.184 4.030 3.734 3.355 2.942 2.531 2.157 1.863 MonthlyAutocorrelation AAA 0.920 0.949 0.966 0.971 0.972 0.972 0.972 0.970 0.967 0.963 AA 0.908 0.940 0.951 0.956 0.960 0.963 0.965 0.966 0.965 0.964 A 0.925 0.954 0.960 0.962 0.963 0.963 0.964 0.964 0.963 0.962 BBB 0.916 0.955 0.965 0.968 0.968 0.968 0.967 0.966 0.964 0.962 BB 0.813 0.891 0.926 0.927 0.924 0.924 0.926 0.926 0.924 0.920 B 0.660 0.833 0.911 0.932 0.936 0.935 0.931 0.925 0.913 0.891 37

TableV PredictivePoweroftheDynamicEconomicFactorsontheTermStructureofTreasuryYields andCreditSpreads Entries report the predicted variation (PV) of the dynamic factors on the Treasury yields and and corporatecreditspreads,definedasoneminustheratiooftheforecastingerrorvariancetothevarianceof the Treasury spot rate and corporate credit spread, respectively. Treasury yields and credit spreads at eachratingclassesareforecastedbythreeeconomicfactorsaccordingtoano-arbitragedynamicterm structuremodel. Theforecastsaremadeaccordingtotheestimatedfactordynamics. Maturity 1 2 3 5 5 6 7 8 9 10 Treasury 0.754 0.739 0.725 0.719 0.718 0.720 0.721 0.721 0.720 0.718 AAA 0.224 0.564 0.672 0.644 0.594 0.548 0.511 0.484 0.467 0.457 AA 0.436 0.544 0.525 0.477 0.426 0.387 0.365 0.356 0.352 0.346 A 0.406 0.535 0.511 0.477 0.442 0.411 0.385 0.367 0.354 0.343 BBB 0.389 0.585 0.643 0.658 0.649 0.626 0.594 0.559 0.522 0.486 BB 0.329 0.470 0.492 0.464 0.457 0.479 0.516 0.553 0.582 0.600 B 0.338 0.581 0.591 0.561 0.539 0.529 0.525 0.521 0.506 0.460 38

TableVI MarketPricesofEconomicRisk Entries report the parameter estimates and absolute magnitudes of the t-statistics (in parentheses) on market prices of economic factor risks that, joint with the time series dynamics in Table II, determine the risk-neutral factor dynamics. The parameters are estimated from Treasury yields and corporate creditspreadsateachofthesixcreditratingclasses. g denotestheconstantcomponentofthemarket 0 priceofthefactorrisks,whichdeterminestheconstantcomponentoftherisk-neutraldriftofthefactors. k Q=k +g definesthemean-revertingpropertyofthedynamicfactorsundertherisk-neutralmeasure, 1 with g denoting the proportional component of the market prices. The parameters are estimated with 1 maximum likelihood methods, using Treasury yields and corporate bond credit spreads, respectively. DataarefromtheFederalReserveBoardandMerrillLynch,monthlyfromJanuary1988toJune2004. g k Q 0 TreasuryYield 0.0115 (0.08) 0.0117 (0.13) 0 - 0 - -1.7951 (2.32) 0.3328 (0.66) 0.3481 (9.22) 0 - 3.5003 (1.42) -0.9021 (0.54) -0.1689 (0.63) 0.7241 (12.5) Creditratinggroup: AAA 0.1009 (0.15) 0.0022 (0.17) — — — — -3.5325 (6.52) 0.2862 (1.41) 1.1672 (4.46) — — 4.9724 (0.82) -0.1839 (0.36) -2.2665 (3.71) 0.1022 (10.29) Creditratinggroup: AA -1.2959 (10.56) 0.1038 (9.16) — — — — -1.5492 (4.55) -0.0464 (0.49) 0.4035 (5.20) — — 7.9824 (20.38) -0.0720 (0.51) -0.3207 (1.89) 0.0959 (10.11) Creditratinggroup: A -0.2805 (0.19) 0.0877 (0.09) — — — — -1.9276 (0.57) 0.0683 (0.03) 0.3890 (1.89) — — 2.9054 (0.17) -0.2062 (0.02) -0.1850 (0.42) 0.1636 (7.55) Creditratinggroup: BBB -0.4496 (0.42) 0.0495 (0.11) — — — — -2.5961 (1.08) 0.1104 (0.06) 1.5610 (0.89) — — 3.0067 (0.41) -0.0113 (0.00) -1.3348 (0.69) 0.1651 (5.50) Creditratinggroup: BB -0.9907 (0.69) 0.0054 (0.32) — — — — -0.8301 (0.13) 1.0995 (0.56) 0.2557 (0.33) — — 3.6565 (0.63) -0.6077 (0.71) 0.0180 (0.05) 0.0178 (0.42) Creditratinggroup: B -0.8123 (0.17) 0.0042 (2.99) — — — — -1.9412 (0.16) 0.9220 (1.16) 0.4502 (0.63) — — 3.2056 (0.18) 0.7934 (4.91) -0.1414 (0.98) 0.0054 (1.08) 39

TableVII TheInstantaneousDefault-FreeInterestRateandCreditSpreadFunctions Entries report the parameter estimates and absolute magnitudes of the t-statistics (in parentheses) on theinstantaneousdefault-freeinterestratefunctionandtheinstantaneouscreditspreadfunctionunder different rating classes. The parameter a and a re the corresponding intercepts and b and b are the r i r i corresponding loading vector on each of the three factors. The parameters are estimated with maximumlikelihoodmethodsandKalmanfilter,usingcorporatebondyieldspreadsoverthecorresponding Treasuryyieldatmaturitiesfromonetotenyears. DataaremonthlyfromJanuary1988toJune2004, obtainedfromtheFederalReserveBoardandMerrillLynch. Ratings Treasury AAA AA A BBB BB B Intercepts(a /a) 0.0486 0.0000 0.0053 0.0063 0.0100 0.0345 0.0724 r i (56.3) (0.88) (7.42) (4.23) (1.91) (2.98) (0.29) FactorLoadings(b /b): r i Inflation 0.0107 0.0020 0.0011 0.0011 0.0007 -0.0008 0.0070 (18.1) (4.11) (4.88) (1.25) (0.23) (0.41) (2.24) RealOutput 0.0073 0.0014 0.0002 -0.0001 0.0012 -0.0052 -0.0126 (23.8) (2.01) (1.20) (0.50) (0.29) (5.86) (4.54) Volatility 0.0025 -0.0006 -0.0006 -0.0002 0.0002 -0.0005 0.0006 (31.1) (12.69) (10.93) (2.29) (1.34) (1.77) (4.57) 40

6 4 2 0 −2 −4 −6 90 93 95 98 01 04 srotcaF cimonocE Inflation Output volatility Figure 1. Time series dynamics of economic factors. The solid line denotes the time series of the extracted inflation factor, the dashed line denotes the time series of the extracted real output growth factor,andthedash-dottedlinedenotesthefinancialmarketvolatilityfactor. 41

10 9 8 7 6 5 4 3 2 1 0 90 93 95 98 01 04 % ,dleiY yrusaerT 10 9 8 7 6 5 4 3 2 1 0 0 2 4 6 8 10 Maturity, Years % ,dleiY yrusaerT Figure 2. Time series and term structure of Treasury yields. Lines in the left panel plot the time seriesofTreasuryyieldsatdifferentmaturitiesfromthreemonthstotenyears. Linesintherightpanel plot the term structure in each month, with the bold solid line denoting the mean term structure. Data arefromtheFederalReserveBoard. 42

400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: AAA 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: AA 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: A 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: BBB 1200 1000 800 600 400 200 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: BB 3500 3000 2500 2000 1500 1000 500 0 90 93 95 98 01 04 spB ,daerpS tiderC Credit Rating: B Figure 3. Time series of credit spreads on corporate bonds. Lines denote the time series of credit spreads. Each panel is for one credit rating group. The ten lines in each panel denote ten maturities from one to ten years. Data are monthly from January 1988 to June 2004, obtained from the Federal ReserveBoardandMerrillLynch. 43

200 150 100 50 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: AAA 160 140 120 100 80 60 40 20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: AA 200 180 160 140 120 100 80 60 40 20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: A 350 300 250 200 150 100 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: BBB 1100 1000 900 800 700 600 500 400 300 200 100 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: BB 3000 2500 2000 1500 1000 500 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,daerpS tiderC Credit Rating: B Figure 4. Term structure of credit spreads on corporate bonds. Lines denote the term structure of corporate credit spreads at different times. Each panel is for one credit rating group. Each line is foronemonthfromJanuary1988toJune2004,obtainedfromtheFederalReserveBoardandMerrill Lynch. Theboldlineineachpanelrepresentsthemeantermstructure. 44

6.6 6.4 6.2 6 5.8 5.6 5.4 5.2 5 1 2 3 4 5 6 7 8 9 10 Maturity, Years % ,evruC dleiY eerF−tluafeD naeM Treasury Spot Rates 1.4 1.2 1 0.8 0.6 0.4 0.2 0 1 2 3 4 5 6 7 8 9 10 Maturity, Years % ,sgnidaoL rotcaF Treasury Spot Rates Inflation Output Volatility Figure5. Meantreasuryyieldcurveandfactorloadings. Theleftpanelplotsa(t )/t ,whichdeterminesthemeanspotratecurvefortheTreasurybond. Linesintherightpanelplotthethreeelements of b(t )/t , which measure the contemporaneous response of Treasury spot rates to unit shocks on the threemacroeconomicandfinancialfactors. 45

15 10 5 0 −5 −10 −15 −20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: AAA 15 10 5 0 −5 −10 −15 Inflation Output Volatility −20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: AA Inflation Output Volatility 15 10 5 0 −5 −10 −15 −20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: A 15 10 5 0 −5 −10 Inflation −15 Output Volatility −20 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: BBB Inflation Output Volatility 100 80 60 40 20 0 −20 −40 −60 −80 −100 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: BB 100 80 60 40 20 0 −20 −40 −60 Inflation Output −80 Volatility −100 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Credit Spreads under Rating: B Inflation Output Volatility Figure 6. Contemporaneous response of the term structure of credit spreads to unit shocks on the dynamic economic factors. Lines denote the contemporaneous response of the term structure of creditspreadsatdifferentcreditratingclassestounitshocksonthethreeeconomicfactors. Eachpanel denotesonecreditratinggroup. Ineachpanel,thesolidlinedenotestheimpactoftheinflationfactor, thedashedlinedenotestheimpactoftherealoutputgrowthfactor,andthedash-dottedlinedenotesthe impactofthefinancialmarketvolatilityfactor. 46

400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: AAA; Sector: F 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: AAA; Sector: C 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: AA; Sector: F 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: AA; Sector: C 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: A; Sector: F 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: A; Sector: C 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: BBB; Sector: F 400 350 300 250 200 150 100 50 0 90 93 95 98 01 04 spB ,daerpS tiderC Rating: BBB; Sector: C Figure 7. Time series of credit spreads across industry sectors and credit rating classes. Lines denote the time series of credit spreads. Each panel is for one credit rating class and industry sector. Thetenlinesineachpaneldenotetenmaturitiesfromonetotenyears. DataaremonthlyfromJanuary 1988toJune2004,obtainedfromtheFederalReserveBoardandMerrillLynch. 47

20 15 10 5 0 −5 −10 −15 −20 −25 −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: AAA; Sector: F 20 15 10 5 0 −5 −10 −15 −20 Inflation Output −25 Volatility −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: AAA; Sector: C Inflation Output Volatility 20 15 10 5 0 −5 −10 −15 −20 −25 −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: AA; Sector: F 20 15 10 5 0 −5 −10 −15 −20 Inflation Output −25 Volatility −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: AA; Sector: C Inflation Output Volatility 20 15 10 5 0 −5 −10 −15 −20 −25 −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: A; Sector: F 20 15 10 5 0 −5 −10 −15 −20 Inflation Output −25 Volatility −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: A; Sector: C Inflation Output Volatility 20 15 10 5 0 −5 −10 −15 −20 −25 −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: BBB; Sector: F 20 15 10 5 0 −5 −10 −15 −20 Inflation Output −25 Volatility −30 1 2 3 4 5 6 7 8 9 10 Maturity, Years spB ,skcohS rotcaF ot sesnopseR Rating: BBB; Sector: C Inflation Output Volatility Figure8. Contemporaneousresponseofthetermstructureofcreditspreadstounitshocksonthe economicfactors. Linesdenotethecontemporaneousresponseofthetermstructureofcreditspreads atdifferentcreditratingclassestounitshocksonthethreemacroeconomicandfinancialfactors. Each panel denotes one credit rating group and industry sector. In each panel, the solid line denotes the 48 impact of the inflation factor, the dashed line denotes the impact of the real output growth factor, and thedash-dottedlinedenotestheimpactofthefinancialmarketvolatilityfactor.