Common Risk Factors in the US Treasury and Corporate Bond Markets: An Arbitrage-free Dynamic Nelson-Siegel Modeling Approach

Transcription

1 Common Risk Facors in he US Treasury and Corporae Bond Markes: An Arbirage-free Dynamic Nelson-Siegel Modeling Approach Jens H E Chrisensen and Jose A Lopez Federal Reserve Bank of San Francisco 101 Marke Sree, Mailsop 1130 San Francisco, CA Preliminary and incomplee draf - Commens are welcome Absrac The vas majoriy of he erm srucure lieraure has focused on modeling he riskfree erm srucure as implied by Treasury bond yields As fixed-income markes should be inerconneced, we combine he modeling of Treasury yields wih a modeling of he common facors presen in represenaive risky credi spread erm srucures derived from Bloomberg corporae bond daa The quesion we address is wheher we can improve our undersanding of, and our abiliy o forecas, Treasury yields by incorporaing informaion from corporae bond marke We use he arbirage-free dynamic version of he Nelson-Siegel yield-curve model derived Chrisensen, Diebold and Rudebusch (2007) o model Treasury yields and corporae bond spreads across raing and indusry caegories In addiion o he hree-facor Nelson-Siegel facors for Treasury yields, we find wo common facors a level and a slope facor are required o capure he ime series dynamics of aggregaed credi spreads We find ha he preferred specificaions of he join dynamics of all five facors have feedback effecs from he Treasury facors o he credi risk facors, bu we also find feedback effecs from he credi risk facors o he Treasury facors To deermine he significance of hese feedback effecs, we perform an ou-of-sample forecas exercise The resuls so far sugges ha he preferred Treasury yield model can easily bea he random walk and ha adding he informaion from he credi markes allows us o improve forecas performance even furher for forecas horizons up o 26-weeks The views expressed are hose of he auhors and should no be inerpreed as reflecing he views of he Federal Reserve Bank of San Francisco or he Board of Governors of he Federal Reserve Sysem We hank Rober Goldsein for many valuable commens Draf dae: Ocober 18, 2012

2 1 Inroducion Treasury bond yields are he main mechanism hrough which moneary policy affecs he real economy However, as shown by he recen urmoil in he credi markes since Augus 2007, he credi markes and heir ineracions wih he facors driving changes in Treasury yields can be equally imporan In his paper, we address he quesion of wheher informaion conained in he credi spreads of corporae bonds can be useful in forecasing Treasury yields To he bes of our knowledge, his paper is he firs o conduc a join esimaion of boh risk-free and risky facors wih feedback effecs from he Treasury marke o he credi markes and from he credi markes o he Treasury marke How migh hese feedback effecs come abou? The Treasury bond markes are impaced by he variaion of he business cycle, mainly hrough bond invesors expecaions abou he anicipaed response of he Federal Reserve o such variaion The credi markes are impaced by business cycle variaion in wo ways Firs, moneary policy changes creae marke risk in he form of ineres rae uncerainy in much he same way as observed in he Treasury bond marke Second, he business cycle variaion iself has a direc influence on credi risk in ha i impacs he abiliy of firms o pay heir debs (ie, heir probabiliy of defaul) The key quesion is, wih forward looking invesors in he credi markes, can informaion from he credi markes improve our undersanding of he dynamics of he Treasury bond marke Going back a leas o Lierman and Scheinkman (1991), i is an esablished fac ha close o 999% of he variaion in Treasury yields can be explained by hree laen variables ofen referred o as a level, slope, and curvaure facor, respecively Thus, he saring poin for our analysis is a hree-facor affine arbirage-free erm srucure model of Treasury bond yields In he choice of model, we focus on he Nelson-Siegel model, which is a robus, ye flexible model wih a very compelling inerpreaion of he laen sae variables as level, slope, and curvaure facor Chrisensen, Diebold, and Rudebusch(2007) derive he affine arbirage-free class of Nelson- Siegel erm srucure models (ie, AFDNS model) Thus, building on heir resuls, we are able o overcome he heoreical deficiency of he Nelson-Siegel model and presen a model wih a full coninuous-ime dynamic descripion of he sae variables ha is arbirage-free while mainaining he Nelson-Siegel facor loading srucure for he sae variables in he yield funcion Focusing direcly on credi risk, he variaion of he business cycle naurally causes he defaul probabiliy of all firms o exhibi some amoun of co-variaion ha, by implicaion, will show up as co-variaion in heir credi spreads In addiion, researchers have found ha credi spreads are above and beyond wha can be jusified by he acual expeced defaul loss risk This has been ermed he credi spread puzzle, which holds across raing and indusry caegories 1 Even correcing for he difference in ax reamen and liquidiy relaive o he Treasury bond marke, a significan residual componen remains in credi spreads, which is likely o be a risk premium ha reflecs overall bond invesor senimen In his paper, we ake his as evidence ha sysemaic facors are common o he credi spreads of firms across business secors and raing caegories Since he naure of any such sysemaic risk facors is so far unknown, we use laen facor models o exrac hem and examine heir ineracion wih he sysemaic componens of Treasury bond yields 2 1 See Elon e al (2001), Collin-Dufresne e al (2001), Bernd e al (2005) for discussions 2 Two recen sudies have moivaed his choice Firs, Krishnan, Richken, and Thomson (2007) use he sandard 1

3 The specifics of our model are as follows Treasury bond yields are assumed o follow a hreefacor model wih a level, slope, and curvaure facor like in he AFDNS model We proceed in wo esimaion seps We firs esimae he hree Treasury facors using Treasury yields only This gives us a preferred model ha can be used as a benchmark in erms of forecasing Treasury bond yields In a second sep, we esimae joinly he hree Treasury facors and he wo credi risk facors by combining Treasury yields wih credi spreads from each of he four business secors in our daa se By imposing he Nelson-Siegel facor srucure on he laen facors, he esimaion of even he five facor models is grealy faciliaed and allows us o perform weekly re-esimaions of our models for forecasin gpurposes over a wo-year period We hen can sudy he ou-of-sample forecasperformanceofeachmodelandlehisbeparofourmodelselecioncrieria Asshownby Chrisensen, Diebold, and Rudebusch (2007), in-sample measures of fi can be misleading and, by implicaion, do no consiue a sufficien model selecion crierion We supplemen our in-sample analysis wih he ou-of-sample forecas evaluaion in order o beer judge he appropriaeness of he models analyzed Our major empirical finding is ha here are wo common credi risk facors in all he four indusry secors we analyze: one is a level facor, he oher is a slope facor Furhermore, he resuls indicae ha he common credi risk facors across secors are very similar boh in heir esimaed pahs as well as in heir esimaed parameers for he facor dynamics One possible inerpreaion of his finding is ha hese are wo economy-wide sysemaic risk facors impacing he credi spreads of all firms independen of raing, mauriy, and business secor As far as forecasing Treasury bond yields goes, our preferred Treasury bond yield model easily beas he random walk assumpion a he 26- and 52-week forecas horizon More imporanly, adding informaion from he credi markes (ie, hese wo economy-wide sysemaic facors influencing he corporae secor) appears o improve on he forecas performance of our model for forecas horizons up o 26-weeks The remainder of he paper is srucured as follows Secion 2 describes he relaed lieraure Secion 3 describes he daa and he daa-driven moivaion for he modeling approach Secion 4 conains he model descripion Secion 5 deails he esimaion mehod Secion 6 has he in-sample esimaion resuls, while secion 7 conains he evaluaion of he forecas performance Secion 8 describes some addiional misspecificaion ess Finally, Secion 9 concludes he paper The appendix conains addiional echnical deails 2 Relaed lieraure Our sudy explicily assumes ha here exis some, as of ye, unidenified common facors ha drive he movemen of credi spreads across differen indusries, raing caegories, and mauriies The exising credi risk lieraure provides evidence of he exisence of such common facors, and hree-facor Nelson-Siegel yield curve model o fi boh he risk-free Treasury yields as well as he credi spread curves of individual firms, and hey find ha firm-level credi spread curves combined wih he risk-free yield curve conain all of he informaion necessary for predicing fuure credi spreads Thus, he Nelson-Siegel based approach appears o be jus as relevan in a credi risk seing as he NS model has proven o be for he modeling of governmen bond yield erm srucures a cenral banks around he world, see BIS (2005) for evidence on he laer Second, Diebold, Li, and Yue (2007) inroduce a global facor model for governmen bond yields based on he Nelson-Siegel approach wih a global level, slope, and curvaure facor In ha model, each currency area can load more or less heavily on he laen global level, slope, and curvaure facor Here, we apply his seup o each secor of he economy assuming ha here are laen secor-specific facors ha each raing caegory wihin he secor is more or less sensiive o 2

4 i even provides some indirec evidence of he naure of hese facors The major findings are briefly summarized in he following Duffie, Saia, and Wang (2007) find ha he probabiliy of defaul for individual firms is well deermined by wo firm-specific variables (he disance-o-defaul in a Moody s KMV EDF-syle and he railing 12-monh reurn on is sock) and wo variables common o all 2770 US indusrial firms in heir sample The wo common facors are he 3-monh US T-Bill rae and he 12-monh railing reurn on he S&P 500-index Thus, he slope-facor from he Treasury daa and he laen facor driving he S&P 500-index are key candidaes as common facors driving he acual defaul probabiliy across differen raing caegories in he US indusrial secor Equally imporan for our sudy, hey find ha neiher he 10-year Treasury yield, US personal income growh, US GDP growh, nor he AAA-BBB bond yield spread add any significan explanaory power beyond ha of he four variables menioned above Collin-Dufresne, Goldsein, and Marin (2001) sudy a sample of 688 corporae bonds and hey have wo major findings Firs, including all firm-specific defaul-relaed variables only explains abou 25% of he observed variaion in credi spreads across indusries, raings, and mauriies Second, in he residuals hey find a significan cross-correlaion across all 688 bonds 80% of which can be explained by he firs principal componen Furhermore, he facor loadings on he eigenvecor for he firs principal componen indicae ha credi spreads of all raings and mauriies have a posiive loading on his facor of abou he same size The only noable variaion in he loadings on his facor across firms is ha firms of lower credi qualiy end o load slighly more heavilyonhis facorhanhigher raedfirms This poins ohe exisenceofacommon levelfacor in credi spreads ha is no relaed o he acual defaul probabiliy of firms Their inerpreaion is ha i reflecs local supply and demand shocks (local here means confined o he corporae bond marke) An alernaive inerpreaion is ha i reflecs a risk premium facor specific o he corporae bond marke Finally, in all heir regressions, changes in he 10-year Treasury yield come across as highly significan, while changes in he 10-yr vs 2-yr slope of he Treasury yield curve rarely show up as significan The las observaion is a odds wih he resuls in Duffie e al (2007) and wih he resuls in his paper One explanaion for his difference may be ha hey are analyzing credi spread changes and yield changes whereas his paper and Duffie e al (2007) are explaining he levels of credi spreads and defaul probabiliies, respecively The differencing may eliminae some of he level effecs ha we observe Driessen (2005) includes common credi risk facors in his analysis of credi spread dynamics across 104 firms He finds ha wo common credi risk facors are needed in addiion o his wo Treasury facors in order o explain he join movemen in credi spreads across he firms in his sample Unforunaely, he provides few deails abou he characerisics of his common credi risk facors In summary, here is evidence ha Treasury facors have a role in deermining movemens in credi spreads The exac role, however, is no clear The 3-monh yield, which is a proxy for he slope facor, is observed o affec he acual defaul probabiliy which is mos likely ied o he fac ha he slope facor reflecs he curren sae of he business cycle which naurally impacs he defaul probabiliy of firms independen of indusry and raing Changes in he 10-year Treasury yield, which is a proxy for changes in he level facor of he Treasury bond yield curve, also seem o impac credi spreads Wheher his effec is relaed o he defaul probabiliy or he risk 3

5 Facor loading Firs facor Second facor Third facor Time o mauriy Figure 1: Facor loadings in he Nelson-Siegel yield funcion Illusraion of he facor loadings on he hree sae variables in he Nelson-Siegel model The value for λ is 055 and mauriy is measured in years premium on corporae bonds is no clear from he sudy by Collin-Dufresne e al (2001) Finally, credi spread dynamics appear o be driven by wo addiional common credi risk facors One facor impacs he par of he credi spread ha accouns for he defaul probabiliy, and his facor should be correlaed wih he reurns observed in he general sock marke as observed by Duffie e al (2007) The inuiion behind his is ha he developmen of he general sock marke reflecs he curren general business condiions ha would impac any firm independen of indusry and raing The oher facor is some kind of risk premium facor In heory, all bonds should be subjec o he same marginal invesor In ha case i may be reasonable ha a risk premium facor would be close o a level facor impacing he credi spread of all corporae bonds in much he same way independen of heir indusry, raing, and mauriy This would be in line wih he observaions in Collin-Dufresne e al (2001) The purpose of his paper is o propose a simple and parsimonious model ha can ie all hese pieces ogeher 3 Daa descripion and model moivaion The Nelson-Siegel model fis he yield curve a any poin in ime wih he simple funcional form 3 ( 1 e λτ ( 1 e λτ y(τ) = β 0 +β 1 )+β 2 λτ λτ e λτ), (1) where y(τ) is he zero-couponyield wih τ denoing he ime o mauriy, and β 0, β 1, β 2, and λ are model parameers The hree β parameers can be inerpreed as facors and heir corresponding facor loadings in he Nelson-Siegel yield curve funcion are illusraed in Figure 1 Due o is flexibiliy his model is able o provide a good fi of he cross secion of yields a a given poin in ime which is he primary reason for is populariy amongs financial marke 3 This is Equaion (2) in Nelson and Siegel (1987) 4

6 Mauriy Mean Sdev Skewness Kurosis Table 1: The summary saisics for he Treasury bond yields Summary saisics for he sample of weekly observed Treasury zero-coupon bond yields covering he period from January 6, 1995 o Augus 4, 2006 praciioners Alhough for some purposes such a saic represenaion appears useful, a dynamic version is required o undersand he evoluion of he bond marke over ime Diebold and Li (2006) achieve his by inroducing ime-varying parameers ( 1 e λτ ( 1 e λτ y (τ) = L +S )+C λτ λτ e λτ), (2) wherel, S, andc canbeinerpreedaslevel, slope, andcurvaurefacors(givenheirassociaed Nelson-Siegel facor loadings) Furhermore, once he model is viewed as a facor model, a dynamic srucure can be posulaed for he hree facors, which yields a fully dynamic version of he Nelson-Siegel model In he following we will argue ha he feaures of his model are relevan for modeling boh he Treasury bond yields as well as he corporae bond credi spreads considered in his paper We use Treasury yields as a proxy for he risk-free rae This choice deserves some commens given he fac ha we will be using Treasury bond yields o exrac he credi spreads ha goes ino he esimaion of he credi risk models Blanco, Brennan, and Marsh (2005) find ha for accurae pricing of credi risk he choice of proxy for he risk-free rae can be criical In heir sudy hey mach corporae bond credi spreads wih he corresponding CDS raes and find ha swap raes are he mos appropriae choice as risk-free benchmark However, for our purposes of esimaing common risk facors we hink ha his will be a second-order effec, and even if using Treasury yields should inroduce a bias, he credi spreads for all raings in each indusry will be biased in he same direcion Thus, he relaive behavior, and in paricular, he relaive facor loadings on he common credi risk facors across he differen raing caegories in each indusry should no be impaced in any significan way by his choice of risk-free benchmark Finally, given ha one purpose of his paper is o improve our undersanding of he Treasury bond marke and our abiliy o forecas hose yields, we feel ha i is naural o cener he analysis around Treasury yields and measure he credi spreads of corporae bonds relaive o he corresponding Treasury bond yields The specific Treasury yields we use are zero-coupon yields consruced by he mehod described in Gürkaynak, Sack, and Wrigh (2006) 4 and briefly deailed here For each business day from 4 The Board of Governors in Washingon DC frequenly updaes he facors and parameers of his mehod, see he relaed websie hp://wwwfederalreservegov/pubs/feds/2006/indexhml 5

7 Yield year yield 5 year yield 1 year yield 3 monh yield Time Figure 2: Time series of Treasury bond yields Illusraion of he weekly observed reasury zero-coupon bond yields covering he period from January 6, 1995 o Augus 4, 2006 The yields shown have mauriies: 3-monh, 1-year, 5-year and 10-year June 14, 1961 o he presen, 5 a zero-coupon yield curve of he Svensson (1994)-ype [ y (τ) = β e λ1τ 1 e λ 1τ β 1 + λ 1 τ λ 1 τ e λ1τ] [ 1 e λ 2τ β 2 + λ 2 τ e λ2τ] β 3 is fied o price a large pool of underlying off-he-run Treasury bonds Thus, for each business day since June 1961 we have he fied values of he four facors (β 0 (),β 1 (),β 2 (),β 3 ()) and he wo parameers (λ 1 (),λ 2 ()) From his daa se zero-coupon yields for any relevan mauriy can be calculaed as long as he mauriy is wihin he range of mauriies used in he fiing process As demonsraed by Gürkaynak, Sack, and Wrigh (2006), his model fis he underlying pool of bonds exremely well By implicaion, he zero-coupon yields derived from his approach consiue a very good approximaion o he rue underlying Treasury zero-coupon yield curve In order o mach he mauriy specrum for he corporae bond yield daa we consruc Treasury zero-coupon bond yields wih he following mauriies: 3-monh, 6-monh, 1-year, 2- year, 3-year, 5-year, 7-year, and 10-year, and we limi our sample o weekly observaions (Fridays) over he period from January 6, 1995 o Augus 4, 2006 The summary saisics are provided in Table 1, while Figure 2 illusraes he consruced ime series of he 3-monh, 1-year, 5-year, and 10-year Treasury zero-coupon yields In he lieraure on US Treasury bond yields researchers have found ha hree facors are sufficien o model he ime-variaion in cross-secions of such yields, for an early example see Lierman and Scheinkman (1991) This holds for our Treasury bond yield daa as well However, o ge some insighs abou he characerisics of hese hree facors we focus on he eigenvecors ha correspond o he firs hree principal componens They are repored in Table 2 5 The laes updae has daa unil December 28, 2007 a he ime of his wriing 6

8 The firs principal componen explains 956% of he variaion in he Treasury bond yields and is loading across mauriies is uniformly negaive Thus, when here is a shock o he firs principal componen i changes all yields in he same direcion independen of mauriy We refer o his as a level facor Likewise, i follows from he able ha he second principal componen explains abou 41% of he variaion in his daa se This facor has large negaive loadings for he shorer mauriies while i has large posiive loadings for he long mauriies Thus, a posiive shock o his facor causes shor-erm yields o move lower while long-erm yields go up, effecively creaing a seepening of he yield curve In case of a negaive shock o he second principal componen we ge he reverse movemens, shor-erm yields go up while long-erm yields move down leading o a flaening of he yield curve We refer o his as a slope facor as i deermines he slope of he yield curve Finally, he hird principal componen explains an addiional 03% of he variaion in he daa Is facor loading is a V-shaped funcion of mauriy wih large posiive loadings for he shor and long mauriies, while is loading is large negaive for he medium-erm mauriies Thus, when here is a negaive shock o he hird principal componen, he shor and long end of he yield curve moves down while he medium-erm yields move up, effecively creaing a hump shaped yield curve Similarly, a posiive shock o his facor will lead o an invered hump shaped yield curve For hese reasons his facor is naurally inerpreed as a curvaure facor In summary, for Treasury bond yields we can easily explain 999% of he oal variaion wih jus hree facors Focusing on he eigenvecors corresponding o he firs hree principal componens we see a clear paern ha is well approximaed by he facor loadings of he level, slope, and curvaure facor in he Nelson-Siegel model as illusraed in Figure 1 The corporae bond daa consiss of represenaive zero-coupon yields for a number of raing caegories across four differen US indusries The daa is downloaded from Bloomberg We focus on he following mauriies: 3-monh, 6-monh, 1-year, 2-year, 3-year, 5-year, 7-year and 10-year We neglec he 20-year and 30-year mauriies primarily because he Nelson-Siegel models used in his paper are no able o mach yields beyond he 10-year mauriy Therefore, lile is los esimaion-wise by discarding he long-erm yields 6 The foursecorswe havedaafor arehe following: US Indusrial firms, US Financial firms, US Banks, and US Uiliy firms For each secor a number of differen raing caegories are available For he US Indusrial firms we have a full sample of corporae bond yields for he BBB, A, AA, and AAA raing caegories For he US Financial firms we have a full sample for he A, AA, and AAA raing caegories while he BBB+ caegory have a 20-monh subperiod wih missing daa from April 22, 2000 o January 17, 2002 For he US Banks he BBB and A raing caegories are represened by a full sample while he AA caegory is only available as of Sep 21, 2001 Finally, for he US Uiliy firms only he A and BBB raing caegories are available However, boh are represened wih a full sample For all four secors full sample means weekly observaions (Fridays) covering he period from January 6, 1995 o Augus 4, 2006 Since he Bloomberg daa are annual discree ineres raes, we conver he corporae bond yields ino coninuously compounded yields Denoe he n-year yield a ime by r (n), hen he 6 Conrolling esimaions performed wih and wihou he long-erm yields in he daa sample confirm his 7

9 Mauriy Firs Second Third Explain Table 2: The eigenvecors of he firs hree principal componens in he Treasury bond yields The loadings of each mauriy on he hree eigenvecors ha correspond o he firs hree principal componens in he weekly Treasury zero-coupon bond yield daa covering he period from January 6, 1995 o Augus 4, 2006 corresponding zero-coupon bond price is given by P (n) = 1 (1+r (n)) n The annual yields are convered ino coninuously compounded yields hrough he equaion P (n) = 1 (1+r (n)) n = e y(n)n y (n) = 1 n ln 1 (1+r (n)) n = ln(1+r (n)) For mauriies shorer han one year we assume he sandard convenion of linear ineres raes Thus, he zero-coupon bond price corresponding o he 6-monh yield is calculaed by P (6m) = r (6m) = e 05y(6m) and he corresponding coninuously compounded yield is given by y (6m) = 2ln A similar mehod is applied o he 3-monh yields r (6m) = 2ln(1+05r (6m)) Finally, in order o conver he coninuously compounded corporae zero-coupon bond yields ino coninuously compounded credi spreads, we deduc he corresponding observed Treasury zero-coupon yield Thus, he credi spreads for each raing caegory c and indusry i are given by s i,c (τ) = y i,c (τ) y T (τ), i {Indusrials, Financials, Banks, Uiliies}, c {BBB,A,AA,AAA} Summary saisics for all he credi spreads are provided in Table 3 Across indusries he credi spreads appear o share some general characerisics, boh he average spread and he credi spread volailiy for a given mauriy increase as credi qualiy deerioraes 7 7 There is one excepion o his rule: he volailiy of he credi spreads of AA US Indusrials is marginally lower han hose of AAA US Indusrials for he mauriies from 3 o 12 monhs This may be ied o he fac ha here is a very limied number of indusrial firms wih a AAA raing in he period analyzed here Thus, his raing caegory may be subjec o a higher degree of idiosyncraic variaion in he underlying pool of bonds 8

10 US Indusrials Mauriy BBB A AA AAA Mean S dev Mean S dev Mean S dev Mean S dev No obs US Financials Mauriy BBB+ A AA AAA Mean S dev Mean S dev Mean S dev Mean S dev No obs US Banks Mauriy BBB A AA AAA Mean S dev Mean S dev Mean S dev Mean S dev na na na na na na na na na na na na na na na na No obs na US Uiliies Mauriy BBB A AA AAA Mean S dev Mean S dev Mean S dev Mean S dev na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na No obs na na Table 3: The summary saisics for he credi spreads Summary saisics for he weekly zero-coupon credi spreads for US Indusrial firms, US Financial firms, US Banks, and US Uiliy firms, respecively, for 8 fixed mauriies covering he period from January 6, 1995 o Augus 4, 2006 All numbers are measured in basis poins 9

11 To provide some preliminary, non-parameric evidence of he exisence and characerisics of common facors in he credi spread daa, we pool ogeher he 11 (indusry, raing)-combinaions for which we have a full sample, a oal of 11 8 = 88 ime series In a second sep, we derive he covariance marix for hese 88 ime series and make a principal componen analysis The resuls are provided in Table 4 The principal componen analysis reveals ha he firs hree principal componens each explain 789%, 68%, and 60% of he oal variaion in he observed credi spreads By implicaion, a hree-facor model may explain as much as 91% of he observed variaion in hese 88 ime series Beyond his poin each addiional facor conribues very lile For example, i akes 8 facors o explain 95% of he variaion, and o explain 99% requires 29 facors The approximaely 79% of he variaion explained by he firs principal componen has a somewha remarkable resemblance o he resuls in Collin-Dufresne, Goldsein, and Marin (2001) who find ha heir residuals across firms are highly correlaed and ha some 80% of ha variaion can be explained by a single sysemaic facor ha has a posiive loading for all (raing, mauriy)- combinaions, ie his facor is mos easily inerpreed as a sysemaic credi spread level facor Wheher he firs principal componen in his daa is in fac idenical o he sysemaic residual facor observed by Collin-Dufresne e al (2001) is an open quesion, bu we can confirm ha he firs principal componen is a level facor based on he observaion ha is loading is uniformly negaive for all 88 ime series Furhermore, for each (secor, raing)-combinaion he loading srucure exhibis a hump shaped paern across mauriies wih a peak beween he 5- and 7-year mauriy and wihin each secor he loading increases monoonically as credi qualiy deerioraes Overall, his paern suppors a modeling sraegy wih a common level facor for each business secor where he loadings on he common facor for he raing caegories wihin each secor are scaled up as credi qualiy deerioraes For he second principal componen we also see a sysemaic paern For all (secor, raing)- combinaions he loading on his componen is monoonically increasing wih mauriy, 8 and i always sars wih a large negaive facor loading on he 3-monh mauriy and ends wih a large posiive facor loading on he long-erm mauriies Thus, his facor describes he slope of he credi spread curves Across raing caegories wihin each of he four secors we see a endency for a larger loading a he shor and long mauriies when credi qualiy deerioraes, bu his paern is no as sysemaic as he paern observed for he firs principal componen However, for our modeling purposes he imporan hing is ha each (secor, raing)-combinaion exhibis he paern of a slope facor The variaion across raings wihin each secor we adjus for by allowing he raing caegories o load differenly on he common slope facor Finally, for he hird principal componen here is no clearly discernible paern For some (secor, raing)-combinaions i appears o be similar o a level facor For oher combinaions i has changes in sign and non-sysemaic variaion in he size of he facor loading Mos imporanly for our analysis, however, i does no exhibi he paern of a curvaure facor similar o he one observed in Table 2 for he Treasury curvaure facor Based on his evidence we do no include a hird facor in he form of a curvaure facor in he modeling of credi spreads 9 In summary, we have found ha here are wo common risk facors across he 11 differen 8 There are hree minor excepions o his rule in Table 4: he 10-year mauriy for he BBB caegory of US Banks and he 10-year mauriy for he BBB and A caegories of US Uiliy firms 9 Conrolling esimaions performed wih a hird curvaure facor included did no bring any meaningful resuls and herefore suppors his conclusion 10

12 US Indusrials Mauriy BBB A AA AAA in monhs Firs Second Third Firs Second Third Firs Second Third Firs Second Third US Financials Mauriy BBB+ A AA AAA in monhs Firs Second Third Firs Second Third Firs Second Third Firs Second Third 3 na na na na na na na na na na na na na na na na na na na na na na na na US Banks Mauriy BBB A AA AAA in monhs Firs Second Third Firs Second Third Firs Second Third Firs Second Third na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na US Uiliies Mauriy BBB A AA AAA in monhs Firs Second Third Firs Second Third Firs Second Third Firs Second Third na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na na Table 4: The eigenvecors of he firs hree principal componens in he credi spreads The loading of each mauriy on he hree eigenvecors ha correspond o he firs hree principal componens in he weekly zero-coupon credi spreads for US Indusrial firms, US Financial firms, US Banks, and US Uiliy firms covering he period from January 6, 1995 o Augus 4, 2006 The analysis is based on 88 ime series, each wih 605 weekly observaions 11

13 (secor, raing)-combinaions analyzed here As for he characerisics of he wo common credi risk facors we find ha he firs facor has he properies of a level facor wihin each (secor, raing)-combinaion, and he oher has properies normally associaed wih a slope facor Given his paern in he firs wo principal componens we feel ha he credi spread daa analyzed here is well suied for he generalized common facor decomposiion based on he Nelson-Siegel model inroduced in Diebold, Li, and Yue (2007) The idea is o replace he hree global yield facors in ha paper wih our wo secor-wide common credi risk facors and give hem an inerpreaion of level and slope Subsequenly, each raing caegory wihin he indusry can load more or less inensely on hese secor-wide common credi risk facors As a furher elaboraion, we choose o modify ha model o make i arbirage-free as deailed in Chrisensen e al (2007) This moivaes he modeling framework o be presened in he following secion 4 Model descripion In his secion he arbirage-free model of Treasury and corporae bond yields ha we are going o use in he empirical analysis is described The firs sep is o choose a model framework To ha end we choose o work wihin he reduced-form credi risk modeling framework 10 using he assumpion of Recovery of Marke Value (See Duffie and Singleon (1999) for deails) Denoe he risk-free shor rae by r, he defaul inensiy by λ Q, and he recovery rae by πq 11 Under hese assumpions he price of a represenaive zero-coupon bond is given by V(,T) = E Q [e T (r+(1 πq s )λq s )ds ] Since he loss rae 1 π Q and he defaul inensiy λ Q only appear as a produc under he RMV modeling assumpion, we replace he produc (1 π Q s )λq s by he insananeous credi spread s wihou any loss of generaliy I is his credi spread process ha we are ineresed in analyzing One imporan hing o noe here is ha for each raing caegory in a given secor we will only be making a marginal or isolaed pricing of he corporae bonds in ha caegory in he sense ha we do no ake he raing ransiions ino consideraion We recognize ha his is a heoreical inconsisency of our approach, and we cauion ha i limis he applicabiliy of he model for accurae pricing purposes However, for he purpose of exracing any common risk facors across raing caegories and indusries, which is he goal here, he model will clearly be able o cach any such effecs if presen in he daa Taking he raing ransiions ino consideraion, we believe, is a second-order effec and a fine-uning ha will no change our resuls in any significan way Therefore, we leave ha issue for fuure research 12 Wih he general modeling framework seled, he nex sep is o decide on he assumed dynamics of he risk-free rae r and he credi spread s The deails are provided in he following subsecions 10 See Lando (1998) for he echnical deails on he reduced-form modeling approach 11 If a jump risk premium exiss he defaul inensiy under he P-measure may deviae by a facor from he defaul inensiy under he Q-measure Since we only observe bond yields, he model-implied defaul inensiies and recovery raes are only meaningful when inerpreed under he Q-measure as indicaed by he noaion 12 However, he model framework presened here allows for such exensions For example he mehod used by Feldhüer and Lando (2006) can be applied immediaely in his seing under he resricion ha each raing caegory has he same facor loading for all he common credi risk facors For now we wan o avoid such resricions and herefore leave his for fuure research 12

14 41 The model for he risk-free rae For he risk-free rae we choose o build on he affine hree-facor arbirage-free approximaion of he well-known Nelson-Siegel erm srucure model presened in Chrisensen, Diebold, and Rudebusch (2007) This is a hree-facor model where he laen sae variables X T = (L T,ST,CT ) can be given he inerpreaion of level, slope, and curvaure by imposing a fixed se of resricions on he Q-dynamics of a general hree-facor affine Gaussian erm srucure model 13 dx T r = δ 0 +δ 1X T, = K Q,T (θ Q,T X T )d+σt dw Q,T The firs criical assumpion is o define he insananeous risk-free rae as he sum of he level and he slope facor r = L T +S T The second criical assumpion is ha he mean-reversion marix under he Q-marix mus have he following simple form K Q,T = λ T λ T 0 0 λ T where λ T will be idenical o λ in he sandard Nelson-Siegel model in Equaion (1) These wo assumpions also highligh he role of he curvaure facor C T This facor is absen from he insananeous risk-free rae and herefore does no impac shor-erm yields Insead, is sole role, (under he Q-measure) is o ac as a sochasic mean for he slope facor S T Finally, we follow Chrisensen e al (2007) and fix he mean vecor under he Q-measure a zero, θ Q,T = 0 They show ha his way of idenificaion is wihou loss of generaliy Imposing he above srucure on he general affine model defaul risk-free zero-coupon yields will be given by y T (τ) = LT + 1 e λt τ λ T τ [ 1 e S T λ T τ + λ T τ ] e λt τ C T + AT (τ), τ where we recognize he Nelson-Siegel facor loadings for he level, slope, and curvaure facors In addiion, he yield funcion conains he mauriy dependen erm AT (τ) τ which arises from imposing absence of arbirage on he dynamic Nelson-Siegel model 14 Given our way of idenifying he model, ie fixing θ Q,T = 0, he yield-adjusmen erm is enirely deermined by he parameerizaion of he volailiy marix Σ T Chrisensen e al (2007) find ha allowing for a maximally flexible parameerizaion of he volailiy marix Σ improves in-sample fi bu deerioraes he ou-of-sample forecas performance as he added flexibiliy is used o fi idiosyncraic variaions in-sample ha do no appear o repea hemselves when we move ou of sample For ha reason we only consider diagonal volailiy marices in his paper 13 Since he affine propery refers o bond prices, affine models only impose resricions on he facor dynamics under he pricing measure 14 The analyical formula for AT (τ) is provided in Chrisensen e al (2007) τ 13

15 42 The five-facor credi spread model Krishnan, Richken, and Thomson (2007) include he risk-free level, slope, and curvaure facor from heir Treasury bond yield analysis in heir model for firm-specific credi spreads They find ha his combinaion performs well for forecasing purposes They inerpre heir resuls as evidence ha once hey have decomposed he observed Treasury and corporae bond yields ino he risk-free level, slope, and curvaure facors and ino he credi risk-relaed level, slope, and curvaure facors, hey have exraced all he essenial informaion conained in he curren bond prices, and provided financial markes are close o being efficien, hese prices will reflec all currenly available informaion Furhermore, Krishnan, Richken, and Thomson (2007) demonsrae ha i does no improve on he abiliy of heir model o forecas o include addiional facors like for example macroeconomic ime series in he esimaion However, once hey exclude he Treasury facors from heir model, he forecasing performance is deerioraed Inspired by he resuls in Krishnan, Richken, and Thomson (2007), we include he facors of he risk-free rae direcly in he funcion for he insananeous credi spread As he Treasury curvaure facor is absen in he insananeous shor-rae process, i will also be absen from he insananeous credi spread process Thus, he insananeous credi spread is assumed o be a funcion of he level and he slope facor from he risk-free yield erm srucure in addiion o wo secor-specific common credi risk facors s i,c = α i,c 0 +αi,c L T L T +α i,c S T S T +α i,c L LS (i)+α i,c S SS (i) This srucure implies ha each raing caegory c wihin indusry i can load separaely on each of hese four facors and do i independenly of he remaining raing caegories wihin he secor We impose he Nelson-Siegel facor loading srucure on he wo common credi risk facors (L S (i),s S (i)) For ha reason he dynamics of he common credi risk facors under he Q- measure mus be assumed o be given by ( dl S (i) ds S (i) ) = ( λ S (i) )( L S (i) S S (i) ) ( )( ) σ i d+ L S 0 dw Q,L S (i) 0 σs i (i) S dw Q,SS Thus, he dynamic descripion of he five facors under he Q-measure is given by he following sysem of sochasic differenial equaions dl S (i) ds S (i) dl T ds T dc T = λ S (i) λ T λ T λ T L S (i) S S (i) L T S T C T d+ σ i L S σ i S S σ L T σ S T σ C T dw Q,LS (i) dw Q,SS (i) dw Q,LT dw Q,ST dw Q,CT Given his dynamic srucure under he Q-measure he common risk facors will preserve he Nelson-Siegel wo-facor srucure Thus, he yield of a represenaive zero-coupon bond from 14

16 indusry i wih raing c and mauriy in τ years is given by (τ) = L T + 1 e λ T τ [ 1 e λ T S T λ T τ + τ λ T τ y i,c +α i,c L L T T +α i,c 1 e λ T τ S T λ T τ S T +α i,c +α i,c 0 +αi,c L LS (i)+α i,c 1 e λs (i)τ S λ S (i)τ ] e λt τ C T + AT (τ) τ [ 1 e λ T τ S T λ T τ S S (i)+ Ai,c (τ) τ By implicaion, he corresponding zero-coupon credi spread is given by e λt τ ] C T + AT,i,c (τ;α i,c L T,α i,c S T ) τ s i,c (τ) = y i,c (τ) y T (τ) where 15 = α i,c L T 1 e λ T τ L T +αi,c S T λ T τ +α i,c 0 +αi,c L LS (i)+α i,c S S T +αi,c 1 e λs (i)τ λ S (i)τ [ 1 e λ T τ S T λ T τ S S (i)+ Ai,c (τ), τ e λt τ ] C T + AT,i,c (τ;α i,c L T,α i,c S T ) τ A T,i,c (τ;α i,c L T,αi,c S T) τ A i,c (τ) τ = σ2 L T(αi,c L T)2 6 τ 2 σ 2 S T (α i,c S T)2[ 1 2(λ T ) e λ T τ (λ T ) 3 τ σ 2 C T (α i,c S T)2[ 1 2(λ T ) (λ T ) 2e λtτ 1 4λ T τe 2λTτ σ 2 C T (α i,c S T)2[ 5 1 e 2λ T τ 8(λ T ) 3 τ = (σi L S)2 (α i,c L ) e λ (λ T ) 3 τ T τ τ 2 (σ i S S ) 2 (α i,c S )2[ 1 2(λ S (i)) 2 1 ], 1 1 e 2λ + 4(λ T ) 3 τ 3 ] τ 4(λ T ) 2e 2λT 1 e λ S (i)τ (λ S (i)) 3 τ T τ ] 1 1 e 2λ S (i)τ + 4(λ S (i)) 3 τ ] A final, equally imporan poin o ake away from he resul above is ha here are no resricions on he dynamic drif componens under he empirical P-measure Therefore, beyond he requiremen of consan volailiy, we are free o choose he dynamics under he P-measure However, o faciliae he empirical implemenaion we limi our focus o he essenially affine risk premium specificaion inroduced in Duffee (2002) In he Gaussian framework ha we are working wihin his specificaion implies ha he risk premiums are given by Γ = γ 0 +γ 1 X where γ 0 R 5 and γ 1 R 5 5 conain unresriced parameers Thus, in general, we can wrie he P-dynamics of he sae variables as dx = K P (θ P X )d+σdw P, where boh K P and θ P are allowed o vary freely relaive o heir counerpars under he Q- measure 15 See Appendix A for deails 15

17 5 Model esimaion We esimae he models inroduced in he previous subsecions using he Kalman filer Since all models considered in his paper are affine Gaussian models he Kalman filer is an efficien and consisen esimaor In addiion, he Kalman filer requires a minimum of assumpions abou he observed daa and i easily handles missing daa This moivaes our use of he sandard Kalman filer in his seing The measuremen equaion for he Treasury bond yields is given by y T = A T +B T L T S T C T +εt, while he measuremen equaion for he credi spreads can be wrien as s (i) = A S (i)+b S (i) L S (i) S S (i) L T S T C T +ε S (i) Expanding on his equaion for a given business secor we obain 16 s BBB (τ 1 ) s BBB (τ N ) s A (τ 1) s A (τ N) s AA (τ 1 ) s AA (τ N ) s AAA (τ 1 ) s AAA (τ N ) = α BBB 0 α BBB 0 α A 0 α A 0 α AA 0 α AA 0 α AAA 0 α AAA 0 + α BBB L α BBB L α A Ḷ α A L α AA Ḷ α AA L α AAA L α AAA L L S + α BBB S 1 e λs τ 1 λ S τ 1 α BBB S 1 e λs τ N λ S τ N α A S 1 e λs τ 1 λ S τ 1 α A S 1 e λs τ N λ S τ N α AA S 1 e λs τ 1 λ S τ 1 α AA S 1 e λs τ N λ S τ N α AAA S 1 e λs τ 1 λ S τ 1 α AAA S 1 e λs τ N λ S τ N S S + A BBB (τ 1 ) τ 1 A BBB (τ N ) τ N A A (τ 1 ) τ 1 A A (τ N ) τ N A AA (τ 1 ) τ 1 A AA (τ N ) τ N A AAA (τ 1 ) τ 1 A AAA (τ N ) τ N + ε BBB (τ 1 ) ε BBB (τ N ) ε A (τ 1) ε A (τ N) ε AA (τ 1 ) ε AA (τ N ) ε AAA (τ 1 ) ε AAA (τ N ) I follows from his srucure ha no all he parameers are idenified For his reason we choose he A-raing caegory o be he benchmark caegory in he sense ha we fix he consan erm a α A 0 = 0 and le he facor loadings on he wo credi risk facors be α A L = 1 and αa S = 1 This choice is moivaed by he fac ha he A-caegory is represened by a full sample of daa in all four secors, bu beyond ha his choice is wihou consequences The implicaion is ha he sensiiviies o changes in he wo credi risk facors are measured relaive o hose of he A-raing caegory in ha same indusry, while he esimaed values of he wo credi risk facors represen he absolue sensiiviy of he benchmark A-raing o hose facors 16 Here, he Treasury facors have been lef ou, and he indusry idenifier is suppressed in he noaion 16

18 For coninuous-ime Gaussian models he condiional mean vecor and he condiional covariance marix are given by E P [X T F ] = (I exp( K P ))µ P +exp( K P )X, V P [X T F ] = 0 e KPs ΣΣ e (KP ) s ds, where = T and exp( K P i ) is a marix exponenial Saionariy of he sysem under he P-measure is ensured provided he real componen of all he eigenvalues of K P is posiive This is imposed in all esimaions For his reason we can sar he Kalman filer a he uncondiional mean and covariance marix 17 X 0 = µ P and Σ 0 = The sae equaion in he Kalman filer is given by 0 e KPs ΣΣ e (KP ) s ds X i = Φ 0 i +Φ 1 i X i 1 +η i where ( i Φ 0 i = (I exp( K P i ))µ P, Φ 1 i = exp( K P i ), and η i N 0, e KPs ΣΣ e (KP ) s ) ds 0 wih i = i i 1 In he Kalman filer esimaions all measuremen errors are assumed o be iid whie noise Thus, he error srucure is in general given by ( η ε ) N [( 0 0 ), ( Q 0 0 H In he esimaion each mauriy of he Treasury bond yields has is own measuremen error sandard deviaion, σ 2 ε T (τ i ), while he measuremen errors for he credi spreads are assumed o have a uniform sandard deviaion σ 2 ε S across all raings and mauriies in each indusry Thus, given he range of mauriies in he Treasury bond yield daa considered in his paper he H-marix in general akes he following form )] H = diag(σ 2 ε T(3m),σ2 ε T(6m),σ2 ε T(1y),σ2 ε T(2y),σ2 ε T(3y),σ2 ε T(5y),σ2 ε T(7y),σ2 ε T(10y),σ2 ε S,,σ2 ε S) The linear leas-squares opimaliy of he Kalman filer requires ha he whie noise ransiion and measuremen errors be orhogonal o he iniial sae, ie E[f 0 η ] = 0, E[f 0 ε ] = 0 17 In he esimaion 0 e KPs ΣΣ e (KP ) s ds is approximaed by 10 0 e KPs ΣΣ e (KP ) s ds 17

19 Finally, he sandard deviaions of he esimaed parameers are calculaed as Σ( ψ) = 1 T [ 1 T T =1 logl ( ψ) ψ logl ( ψ) ] 1, ψ where ψ denoes he opimal parameer se 6 In-sample esimaion resuls The purpose of his secion is o find appropriae specificaions of he P-dynamics of he sae variables in our erm srucure models based on in-sample measures of fi Firs, we invesigae he dynamics of he hree Treasury facors using Treasury yields only This leads us o a preferred specificaion ha can be used as benchmark for he performance of he five-facor credi spread models sudied subsequenly Second, we analyze he appropriae specificaion of he join P- dynamics of he hree Treasury risk facors and he wo credi risk facors from each of he four business secors in our daa 61 The P-dynamics of he Treasury facors We begin he search for he mos appropriae specificaion of he dynamic ineracion of he hree Treasury facors by esimaing he model ha allows for he maximum flexibiliy in erms of dynamic ineracion beween he sae variables as hey rever back o seady sae This model is represened by he following sochasic differenial equaion for he P-dynamics dl T ds T dc T = κ P,T 11 κ P,T 12 κ P,T 13 κ P,T 21 κ P,T 22 κ P,T 23 κ P,T 31 κ P,T 23 κ P,T 33 We denoe his Case I: full K P,T θ P,T 1 θ P,T 2 θ P,T 3 L T S T C T σ + L T σ S T σ C T dw P,LT dw P,ST dw P,CT Chrisensen, Diebold, and Rudebusch (2007) find ha he parsimonious model wih independen facors under he P-measure performed well in erms of forecasing ou of sample We will refer o his as Case II: diagonal K P,T which is characerized by a mean-reversion marix given by a diagonal marix K P,T = κ P,T κ P,T κ P,T 33 However, from he esimaion resuls for hese wo specificaions i is eviden ha he Treasury facors are correlaed and have some ineresing dynamic ineracions beween, in paricular, he slope and he curvaure facor To ake his evidence ino consideraion we also analyze he cases where he mean-reversion marix under he P-measure is specified as upper- and lower-riangular marices K P,T = κ P,T 11 κ P,T 12 κ P,T 13 0 κ P,T 22 κ P,T κ P,T 33 and KP,T = κ P,T κ P,T 21 κ P,T 22 0 κ P,T 31 κ P,T 32 κ P,T 33 18

20 Treasury bond yield models Model descripion Max log likelihood Df LR p-value Case I: full K P,T na na na Case II: diagonal K P,T < Case III: upper-riangular K P,T Case IV: lower-riangular K P,T < Case V: S T ineracion only Table 5: The maximum log likelihood values for he AFDNS Treasury bond yield models The maximum log likelihood values obained for he five differen AFDNS Treasury bond yield models The daa se used consiss of weekly daa on Treasury zero-coupon bond yields covering he period from January 6, 1995 o Augus 4, 2006 We refer o hese wo specificaions as Case III: upper-riangular K P,T and Case IV: lowerriangular K P,T, respecively Given ha all models considered are nesed by he specificaion wih full K P,T -marix we can perform likelihood raio ess for he significance of he parameer resricions imposed Firs, we sar by esing he independen-facors model agains he model wih full K P -marix, ie Case I vs Case II In his case, he likelihood raio es is given by LR = 2[ ]= 401 χ 2 (6) The probabiliy of observing 401 in he χ 2 disribuion wih 6 degrees of freedom is less han and clearly rejeced Nex, we es he independen-facors model agains he model wih lower-riangular K P -marix, ie Case I vs Case IV The likelihood raio es is given by LR = 2[ ]= 784 χ 2 (3) The probabiliy of observing 784 in he χ 2 disribuion wih 3 degrees of freedom is exacly 005 Thus, he resricions imposed on he independen-facors are only weakly rejeced relaive o he lower-rinagular specificaion of K P,T Since he model in Case III wih he upper-riangular K P,T -marix has an even higher likelihood value, i immediaely follows ha he independenfacors model is also rejeced relaive o he laer model Finally, we es he model wih upper-riangular K P,T -marix agains he model wih full K P,T -marix This gives he following likelihood raio es LR = 2[ ]= 173 χ 2 (3) This es is also clearly rejeced The conclusion from his preliminary invesigaion is ha boh he lower- and he upperriangularspecificaionofk P,T addsomesignificanexplanaorypowerwherehe keylinkappears o be he slope facor For his reason we ry he following specificaion of he mean-reversion marix K P,T = κ P,T κ P,T 12 κ P,T 22 κ P,T κ P,T 33, 19