Modeling Fixed Income Excess Returns

Transcription

1 Modeling Fixed Income Excess Reurns Basma Bedache Deparmen of Economics Wayne Sae Universiy Deroi, MI, 480 Phone:(313) Fax:(313) Chrisopher F. Baum Deparmen of Economics Boson College Chesnu Hill, MA, 0467 phone:(617) Fax:(617) April 14, 000 We acnowledge suggesions o ered by paricipans a he Fourh Annual Meeings of he Sociey for Compuaional Economics and seminar paricipans a Wayne Sae Universiy. A previous version of his paper was circulaed as Condiional heerosedasiciy models of excess reurns: How robus are he resuls? Please address correspondence o Basma Bedache.

2 Modeling Fixed Income Excess Reurns Absrac 1 Inroducion The excess reurn on a period bond is de ned as he di erence beween he holding period (or ex pos) premium from holding he long mauriy bond as compared o he one-period securiy 1 Excess reurns earned in xed-income mares have been modeled using he ARCH-M model of Engle e al. and is varians. We invesigae wheher he empirical evidence obained from an ARCH-M ype model is sensiive o he de niion of he holding period (ranging from 5 days o 90 days) or o he choice of daa used o compue excess reurns (coupon or zero-coupon bonds). There is robus suppor for he inclusion of a erm spread in a model of excess reurns, while he signi cance of he in-mean erm depends on characerisics of he underlying daa. Keywords: erm premium, excess reurns, GARCH modeling JEL: E43, C, C5 Excess reurns or bond erm premia have been sudied exensively in he empirical nance lieraure. reurn on he bond and he one-period ineres rae. and is commonly referred o as he ex pos For example, if a bond mauring in 5 years is held for one monh, he (one-monh) excess reurn on his bond is calculaed as he di erence beween he reurn realized from holding he 5-year bond for one monh and he ineres rae on a one-monh securiy. Oher recen sudies have analyzed he role of ime-variaion in erm premia in a di eren seing. Tzavalis and Wicens (1997) nd ha when hey accoun for a ime-varying erm premium by including excess reurns in a regression of he change in he long erm ineres rae on he yield spread, heir esimaed coe ciens are consisen wih he predicions of he expecaions heory. Using a discree ime approximaion for he erm premium which is derived in he coninuous ime CIR model, Klemosy and Piloe (199) esimae he ex ane erm premium as a 1 The excess reurn represens he realized erm or liquidiy premium. One of he rs sudies o empirically model ime variaion in erm premia is Engle, Lilien, and Robins (1987) where he auhors inroduced he auoregressive condiional heerosedasiciy-in-mean model (ARCH-M). In he ARCH-M model, he erm premium depends on he condiional variance of he underlying ineres rae process. A number of sudies applying and exending his mehodology followed (see Bollerslev, Chou and Kroner (199) for a survey of he lieraure). More recenly, Brunner and Simon (1996) nd ha he exponenial generalized auoregressive condiional heerosedasiciy-inmean model (EGARCH-M) is a good represenaion of weely excess reurns on en-year Treasury noes and long-erm Treasury bonds.

3 Alhough condiional heerosedasiciy models have been widely applied o excess reurns, here have been no sudies on he robusness of empirical ndings ha are based on hese models wih respec o varying characerisics of he daa ha are used in calculaing bond erm premia. This paper aemps o ll his gap in he lieraure by providing evidence on he sensiiviy of empirical resuls o he choice of holding period over which excess reurns are measured, he daa se which is used o calculae excess reurns, and he bonds erm o mauriy. This is a paricularly ineresing issue since he empirical lieraure has largely focused on modeling excess reurns ha are calculaed from Treasury bills. We documen and compare he ime series characerisics of excess reurns on medium o long-erm Treasury securiies for various sampling inervals which we se equal o he holding period. Using he class of condiional heerosedasiciy models which have been previously used in he lieraure, we analyze excess reurns series for hree holding periods: ve days (one wee), one monh, and hree monhs, and for a number of mauriies. Our sudy conribues o he lieraure in wo oher respecs. We employ a general condiional heerosedasiciy model which allows for he inclusion of in-mean e ecs, lagged reurns, and a erm spread variable which has been found o be signi can in previous sudies. In addiion, he model allows for a general class of disribuions for he error erm raher han assuming ha he errors are normal. Finally, we provide new evidence on modeling excess reurns of medium-erm zero coupon bonds and medium- o long-erm coupon bonds which have no received aenion in he lieraure. Our ndings can be summarized as follows. The model of excess reurns presened here, incorporaing he erm spread, condiional heerosedasiciy, and a more exible error disribuion, appears o be quie robus o variaions in enor, holding period, and he form of he underlying daa. This robus behavior holds up over a lenghy sample period encompassing srucural changes in he behavior of he Federal Reserve as a dominan force in he Treasury mares, wih concomian ucuaions in he volailiy of ineres raes and reurns from holding xed-income securiies. Alernaive sampling inervals and holding periods, ranging from ve days o 90 days, do no appear o have a qualiaive impac on hese ndings. Explici consideraion of he condiional heerosedasiciy of excess reurns series appears o be an essenial componen of a characerizaion of heir sochasic properies. However, inclusion of an in-mean erm in he condiional mean equaion is no consisenly suppored by our ndings. The res of he paper is consruced as follows. Secion presens he heoreical bacground funcion of he level and he variabiliy of he risless ineres rae process. They nd ha he erm premium varies posiively wih he variabiliy of he ineres rae and negaively wih is level.

4 and mehodology underlying our invesigaion, while he daa used in he analysis are described in Secion 3. The following secion presens our empirical resuls. Secion 5 concludes wih an ouline of furher research. Theoreical bacground and mehodology Le R be he yield per period on a zero-coupon bond wih periods o mauriy. The one period holding reurn on his -period bond is given by: (, 1) 1 +1 H = R ( 1) R (1) According o he expecaions heory, invesors see o equalize expeced holding reurns from all available invesmen sraegies. Therefore, he expeced reurn from holding any -period bond for one period (say one monh) is relaed o he yield on a one-period bond as shown in () below. (, 1) 1 EH = R + TP () E is he expecaions operaor condiional on informaion available a ime and TP is a consan erm premium. The one-period excess holding reurn on a -period bond ( XR ) is de ned as he di erence beween he ex pos holding reurn and he one-period yield which, assuming ha expecaions are raional, is equal o he unobservable erm premium plus noise unforecasable a ime : (, 1) 1 XR = H R = TP + (3) Given he assumpion of a consan erm premium, he expecaions hypohesis implies ha excess reurns are unforecasable using informaion available a ime. Equaion (3) can be used o es his implicaion by regressing XR signi cance. Previous sudies have found ha excess reurns are forecasable using variables such as he spread beween he -period and he one-period raes (see for example Campbell (1995)). Evidence of he predicabiliy of excess reurns has been inerpreed o indicae he exisence of a ime-varying erm premium. To explicily model he behavior of he (unobserved) erm premium using he ime series of (observable) excess reurns as in equaion (3), Engle e al. (1987) assume ha erm premia are given by: on observable and relevan variables and esing for heir TP = + h (4) 3

5 where h is he sandard deviaion of he holding reurn forecas error,, which represens he ris of holding he period bond. This formulaion assumes ha he erm premium has a less han proporional relaion o he variance of he holding reurn, h, consisen wih a wo-asse model in which he supply of he risy asse is xed. Subsiuing (4) ino (3) and specifying an auoregressive process for h, he model, ermed he auoregressive condiional heerosedasiciy in mean (ARCH-M) model, is formulaed as: XR = + h +, N (0,h) (5) q h = c+ qi i (6) i=1 Engle e al. (1987) nd ha he ARCH-M model wih q = 4 is a good represenaion of quarerly excess reurns on six-monh Treasury bills. 3 Since our sudy employs a number of excess reurn series which vary by mauriy and sampling frequency, we consider a more general version of he model in (5)-(6), a GARCH-M, which allows for auoregressive erms and exogenous variables in he mean and variance equaions. In addiion, he model is esimaed using a generalized error disribuion (GED) when we nd evidence ha he errors are non-normal. The general model is formulaed as: 0 Arlag exog i i i i i=1 i=1 q p vexog i i i i i i i=1 i=1 i=1 XR = a + a XR + b X + mh +, (0,h ) (7) h = c+ q + p h + d Z (8) Where Arlag is he number of auoregressive lags in he mean equaion and exog and vexog are he number of exogenous variables, X and Z, enering he mean and he variance equaions, i respecively. An exogenous variable which is liely o be signi can in he mean of i spread beween he long erm yield and he one-period yield, S = R R. To illusrae his i is helpful o rewrie equaion () as: 1 XR is he 3 (, 1) TP = E H R (9) h (, 1) 1 Afer subsiuing for H from (1), adding and subracing R, and rearranging we ge: Engle e al. s (1987) nal model uses log variance equaion. in he mean equaion and applies declining weighs on he lags in he 4

6 TP = ( R R ) + ( 1)( R E R ) (10) Equaion (10) shows ha he erm premium can be decomposed ino wo componens: he spread beween he long and he shor erm yields and a muliple of he expeced change in he long erm yield. Therefore, he spread is liely o be an imporan predicor of he erm premium and may conain informaion abou he erm premium which is no capured in he condiional variance erm (he in-mean erm), h. Since h is he sandard deviaion of he forecas error of he holding period reurn which is he same as he forecas error of he fuure long-erm yield, ER in-mean erm in equaion (7) re ecs he second erm in (10), while he spread re ecs he rs erm. Engle e al. (1987) sugges ha he spread can predic excess reurns since i represens informaion abou he risiness of he long erm bond, which is modeled using he in-mean erm h. To es his proposiion, hey compare he coe ciens on he spread from an equaion regressing he excess reurn on a consan and he spread o ha from he ARCH-M model. They nd ha he spread remains signi can bu a a lower level of signi cance and ha is coe cien is drasically reduced. Thus, hey conclude ha he spread s explanaory power is merely a re ecion of he informaion capured by h. 1 +1, he In he empirical resuls below, we include he spread as a regressor in equaion (7) and es wheher he resuls are consisen wih he relaionshipdescribed in (10). We also consider he inclusion of he spread as an explanaory variable in he condiional variance equaion. The spread could a ec he erm premium indirecly hrough he variance equaion if he spread is correlaed wih uncerainy abou he long erm yield. For example, he spread may increase when uncerainy (or ris) abou he long-erm yield increases if invesors require a higher yield on he long-erm bond o compensae for he increased uncerainy. Our approach o selecing a nal speci caion of he model in (7)-(8) for each of he excess reurns series is as follows. We begin by esimaing he ARCH(4)-M model which has been previously used in he lieraure for each series and conducing a number of diagnosic ess on he model s residuals. We es for serial correlaion in he sandardized residuals and squared residuals for up o 4 lags, and conduc F-ess for he presence of remaining ARCH e ecs up o 4 lags. Ensuring ha he squared residuals are properly modeled is paricularly imporan since he in-mean erm canno be consisenly esimaed if he condiional variance is misspeci ed. We also es for normaliy and for he presence of sign bias in he sandardized residuals. We use he GARCH(1,1)-M model in cases where we deec he presence of high order ARCH in he residuals and we include lagged excess reurns in he mean equaion in cases where we found serial correlaion in he residuals. Where we rejec normaliy, we employ he generalized error disribuion, deailed in Appendix 5

7 3 Descripion ofhe daa 5 5 C, o obain esimaes of he model parameers. Therefore, we choose he nal speci caion for which he residuals and squared residuals are free from any serial correlaion, remaining ARCH e ecs and sign bias. Afer selecing he bes univariae model for each excess reurns series, we reesimae he models by adding he spread as an explanaory variable in he mean equaion and checing he esimaed residuals as previously described. In he empirical resuls presened in Secion 4, he nal speci caion for each series includes he spread as an explanaory variable. We use wo di eren daa ses derived from U.S. Treasury securiies quoaions in our analysis. The rs daase, used o consruc excess reurns for 1-, -, 3-, 4-, and 5-year securiies, maes use of he Fama-Bliss Discoun Bonds le and he CRSP Risfree Raes le from CRSP s monhly daase. The Fama-Bliss le conains monhly price and yield quoaions on zero-coupon securiies for one- o ve-year enors generaed from fully axable, non-callable, non- ower Treasury securiies. Deails of is consrucion are given in he 1997 CRSP Monhly US Governmen Bond File Guide (p.). The Risfree Raes le conains nominal one- and hree-monh ris free raes, consruced from Treasury bill quoaions. Yields are expressed as coninuously compounded 365-day raes. Approximae holding period reurns are calculaed from hese quoaions, as described in Appendix A. The second daase, used o consruc excess reurns for several enors, maes use of he CRSP daily daase s Fixed Term Indices les, combined wih individual securiies quoaions from he daily CRSP le and a measure of he daily Fed Funds rae from he Federal Reserve Board. The Fixed Term Indices les conain price, yield and duraion quoaions for 1-, -, 5-, 7-, 10-, 0- and 30-year erms, consruced by CRSP from individual Treasury securiies quoaions as described in he 1997 CRSP US Governmen Bond File Guide (pp.3-5). The ideni caion of he underlying individual securiy allows us o calculae an approximae holding period reurn for ve-day (onewee), one-monh and hree-monh holding periods, as described in Appendix B. These measures, available a a daily frequency, provide us wih a much richer se of holding period reurns in wo senses. Firs, hey gauge he reurn from holding medium and long-erm Treasuries, whereas mos research has been based on reurns from Treasury bills, of no more han one year enor. Second, hey are available a a daily frequency, faciliaing he sudy of a sizable sample of weely excess reurns, An alernaive o using a non-normal disribuion would be o employ he quasi-maximum lielihood esimaion (QMLE) mehod developed by Bollerslev and Wooldridge (199). The auhors provide compuable formulas for asympoic sandard errors ha are valid when a normal log-lielihood funcion is maximized, even when he assumpion of normaliy is violaed. 6

8 whereas much of he lieraure uilizes monhly series. 7 6 Alhough compuaions wih hese daa mus ae ino accoun he naure of he quoaions (measures of yield on coupon securiies, raher han he more analyically racable zero-coupon insrumens) hese wo advanages ouweigh he di culies. Descripive saisics for he excess reurns series calculaed for his sudy are given in Table 1. Panels A, B, and C presen he descripive saisics for quarerly, monhly, and ve-day excess reurns, respecively. Series denoed Z are derived from he zero-coupon quoaions, while series labeled C are consruced from he Fixed Term Indices quoaions on coupon securiies. Di erences in mehodology of excess reurns compuaion seem o be mos apparen for he quarerly holding period reurns a he one-year enor, in which he zero-based series have a considerably higher mean and lower variance han he coupon-based series. For he oher wo overlapping enors (wo-year and ve-year) here is much closer agreemen beween he series. The discrepancy beween means persiss for he monhly excess reurns, where he zero-based measures have a much higher mean for each of he hree overlapping enors, bu similar variances. The variances of excess reurns increase wih enor in almos every insance for boh zero-based and coupon-based excess reurns. Reurns are generally posiively sewed, wih greaer sewness a shorer enors. Excess urosis is presen in almos all series, and is mos pronounced for shorer enors for mos of he series. A signi can degree of auocorrelaion is apparen in all bu he quarerly series calculaed from zero-coupon securiies. Since esimaion of models for excess reurns depends on he saionariy of he series ( nie uncondiional second momens), we perform Augmened Dicey-Fuller and Phillips-Perron uni roo ess on each of he excess reurns series. The null hypohesis of nonsaionary behavior is decisively rejeced for all series. 4 Empirical Resuls We have esimaed ARCH and GARCH models of excess reurns, as described in he previous secion, for several mauriies and daa frequencies, maing use of wo disinc daases. In his secion, we presen and inerpre our ndings from his se of esimaes. 6 Several common feaures emerge. Firs, i appears ha low-order ARCH and GARCH models adequaely describe he This daa se also allows us o consruc excess reurns for a one-day holding period. Our preliminary analysis indicaes ha he ime series characerisics of hese daily daa are considerably di eren from heir weely, monhly and quarerly counerpars. Esimaion of hese models requires he use of highly nonlinear and compuaionally inensive mehods. Given ha such analysis shifs he focus of he paper away from is purpose, we chose o defer analysis of he daily daa o fuure research. 7 The uni roo es resuls are available from he auhors upon reques. 7

9 sochasic properies of excess reurns series. Second, he erm spread is signi can in he condiional mean equaion for excess reurns, for all mauriies and daa frequencies, in boh daases. Third, an ARCH-in-mean erm is an imporan componen of some models of excess reurns, over and above he role of he erm spread, bu fails o be universally signi can. Fourh, he error processes in hese models are generally non-normal; a generalized error disribuion wih fa ails is a more appropriae represenaion of heir urosis. hese ndings. 8 We urn now o a deailed invesigaion of 4.1 Resuls for he Zero-Coupon Daase The zero-coupon daase, described in Secion 3, conains monhly quoaions on one- o ve-year zero coupon securiies consruced from Treasury quoaions. We have seleced he one-, wo-, and ve-year enors for our modeling, all of which overlapwih he available enors from he Fixed Term Indices daase. We models o hese hree enors for wo holding periods: monhly and quarerly. A summary of he speci caions used appears as Table. include an addiional esimaed parameer (he ail-hicness parameer). The quarerly series 8 Table. Excess Reurns Model Speci caions for he Zero-Coupon Daase Holding Period Tenor Quarerly Monhly 1year year 5year ARCH()-M ARCH()-M ARCH()-M, GED AR1-GARCH(1,1)-M, GED AR1-ARCH(4)-M, GED AR1-ARCH(4)-M, GED In his summary, all models include an in-mean erm (-M) which is he condiional sandard error of excess reurns. Models including GED are wih a generalized error disribuion, and condiional mean equaions conain only a consan, erm spread, and in-mean erm; he monhly series also include a lagged value of he condiional mean. The empirical resuls from hese models are presened in Table 3. As indicaed above, he erm spread is universally signi can in hese models, wih a posiive esimaed coe cien increasing in enor. The spread coe ciens are of similar magniude across quarerly and monhly holding periods. The in-mean erm is signi can in he one- and wo-year models for quarerly holding periods, and for he wo-year model a he monhly frequency. When hese models are esimaed wihou he erm spread, he in-mean erm is always signi can; hus, here seems o be some subsiuabiliy beween including he erm spread or he volailiy erm in A descripion of he generalized error disribuion and is properies is presened in Appendix C. 8

10 he condiional mean equaion Resuls for he Coupon Daase 9 We may also compare he erm spread coe ciens wih heir corresponding esimaes from OLSbased models of excess reurns. Modeling he excess reurns series condiional heerosedasiciy always diminishes he magniude of he coe cien on he erm spread, bu i remains signi canly posiive in all cases. In conras, Engle e al. (1987) and Brunner and Simon (1996) found ha when condiional heerosedasiciy was accouned for wih an ARCH srucure, he erm spread became much less signi can. Our ndings sugges ha he erm spread has played an imporan role in he excess reurns process, even in he presence of condiional heerosedasiciy and an explici ARCH-in-mean erm. Since excess reurns may be decomposed ino wo componens, he erm spread and a muliple of he change in yield on he longer-erm bond over he holding period, our ndings sugges ha boh componens have been responsible for ucuaions in excess reurns over our sample period. In conras, a sudy focusing on recen erm srucure behavior migh aribue a much larger role o yield volailiy (as do Brunner and Simon (1996) when sudying he period). In applying he GED as an alernaive o normaliy for he error process, he ail hicness parameer for all monhly holding periods may be disinguished from.0, he value corresponding o normally disribued errors, and signals excess urosis. All models errors are free of residual ARCH, and Ljung-Box ess for auocorrelaion in heir squares are insigni can. Thus, for he zero-coupon daase, i appears ha an appropriae model of excess reurns may be consruced wih he (G)ARCH-M mehodology. The erm spread plays a qualiaively similar role in each of he resuling models, irrespecive of choice of enor or he disincion beween monhly and quarerly holding periods. They appear o capure he dynamics of excess reurns over his period, including he sharp spie in volailiy in he period, as Figure 1 (for he quarerly volailiy series) and Figure (for he monhly volailiy series) indicaes. The coupon daase, described in Secion 3, conains quoaions on several enors of Treasury securiies generaed from he CRSP Fixed Term Indices. The daa are originally available a he daily frequency. For his analysis, we have examined weely, monhly and quarerly holding periods Appendix Tables A.1 and A. presen he resuls of esimaing each of he excess reurns models wih OLS, allowing for AR(1) errors. 10 Noe ha we should inerpre he comparison beween our resuls and oher sudies wih cauion. Since we use a di eren speci caion for he mean equaion for excess reurns, changes in he coe ciens for he yield spread may occur due o a change in he amoun of small sample bias. We han an anonymous reviewer for bringing his poin o our aenion. 9

11 for four enors: he one-, wo-, and ve-year enors considered above, and he 30-year enor. A summary of he speci caions chosen for hese models appears as Table Table 4. Excess Reurns Model Speci caions for he Coupon Daase Holding Period Tenor Quarerly Monhly Weely 1year ARCH(4)-M, GED AR1-GARCH(1,1)-M, GED AR4-GARCH(1,1)-M, GED year ARCH(4)-M, GED AR1-GARCH(1,1)-M, GED AR4-GARCH(1,1)-M, GED 5year ARCH(4)-M, GED AR1-ARCH(4)-M, GED AR4-GARCH(1,1)-M, GED 30 year ARCH(4)-M, GED ARCH(4)-M, GED AR4-GARCH(1,1)-M, GED Abbreviaions used in his summary are hose used in Table above. The empirical resuls from hese models are presened in Tables 5, 6, and 7 for quarerly, monhly and weely holding periods, respecively. 11 The erm spread is universally signi can a beer han he 5% level in hese speci caions. In conras o he zero-coupon esimaes, he magniude of he erm spread coe ciens is no always reduced by aing accoun of condiional heerosedasiciy. For he quarerly holding period, he (G)ARCH coe ciens on he erm spread are uniformly larger han heir OLS counerpars. In he monhly and weely holding periods, all bu he 30-year monhly reurns have smaller coe ciens on he erm spread when condiional heerosedasiciy is aen ino accoun. All models conain in-mean erms, so ha he e ecs of he erm spread are esimaed conrolling for he possible confounding of volailiy and erm spreads impac on excess reurns. The in-mean erm is no signi can in he quarerly esimaes, and only appears as an imporan deerminan of one-year excess reurns for he monhly holding period. For he weely esimaes, he in-mean erm is signi can for one- and wo-year enors, and marginally signi can for he ve-year enor. The GED ail-hicness parameer may be disinguished from.0 (denoing normaliy) for each of he esimaed models, indicaing a considerable degree of excess urosis for many of he series. Wih he excepion of he weely wo-year reurns series, all models errors are free of residual ARCH, wih insigni can auocorrelaion in heir squares. The models of excess reurns ed from coupon bonds quoaions appear o be largely similar in heir qualiaive characerisics. The erm spread plays an imporan role in explaining excess reurns behavior, even when condiional heerosedasiciy and in-mean erms are included. We also models including he erm spread as a poenial componen of he condiional variance equaion, as suggesed by Engle e al. (1987, p.403). The erm spread was signi can in only one of he many models esimaed. I does no appear ha he level of he erm spread has any explanaory power over and above pas values of he condiional variance. 10

12 Alhough he resuls for a weely holding period are more volaile han hose for longer holding periods, here are qualiaive similariies among he resuls for all hree holding periods for each enor sudied. The models appear o capure he dynamics of excess reurns over his period, and heir condiional heerosedasiciy componen accouns for he sizable changes in bond mares volailiy experienced in he moneary conrol episode (see Figures 3, 4, and 5a-5b for he esimaed volailiy series for quarerly, monhly and weely holding periods, respecively). 4.3 Summary offindings In summary, he model of excess reurns presened here, incorporaing he erm spread, condiional heerosedasiciy, and a more exible error disribuion, appears o be quie robus o variaions in enor, holding period, and he form of he underlying daa. This robus behavior holds up over a lenghy sample period encompassing srucural changes in he behavior of he Federal Reserve as a dominan force in he Treasury mares, wih concomian ucuaions in he volailiy of ineres raes and reurns from holding xed-income securiies. Alernaive sampling inervals and holding periods, ranging from a ve-day o a 90-day period, do no appear o have a qualiaive impac on hese ndings. To pu i anoher way, one should no draw di eren conclusions abou he imporance of he curren slope of he yield curve o he performance of xed-income porfolios wheher one sudies 5-, 30-, or 90-day holding period reurns. These qualiaive ndings appear o hold for boh shor-erm (one- and wo-year) enors as well as much longer-erm insrumens such as he 30-year long bond. These resuls also highligh he imporance of modeling he condiional heerosedasiciy of excess reurns series as an essenial componen of any characerizaion of heir sochasic properies. Alhough his message is no new, i plays an imporan role in he robusness of our ndings over holding periods and enors. However, inclusion of an in-mean erm in he condiional mean equaion, aen by Engle e al. (1987) as a clear indicaion of he ineracion of ris and reurn, is no consisenly suppored by our ndings. In-mean erms play an imporan role in some of our esimaed models, bu one canno generalize over enor, holding period, or daase on heir relaion o he model. 5 Suggesions for Furher Research In his sudy, we have evaluaed he robusness of he relaionshipbeween he erm spread and excess reurns on holding Treasury securiies of shor, medium and long enors. Our ndings ha 11

13 he erm spread is an imporan conribuor o he explanaory power of ARCH-based models of excess reurns appear o generalize across holding periods, from one wee o a calendar quarer, suggesing ha sysemaic sampling in his range does no have a qualiaive impac on an appropriae speci caion. These conclusions also sugges ha he form of he underlying daa, wheher he more racable zero-coupon quoaions or reurns derived from coupon securiies quoaions, is no responsible for he main feaures of he models presened. Our furher wor wih hese daa will focus on exending hese resuls by maing use of he Fixed Term Indices-based daa a heir original daily frequency, and in exploring a variey of advanced GARCH speci caions o invesigae he imporance of asymmeries and non-normal behavior in heir error processes. Use of he daily daa will also permi us o conduc subsample analysis wih more sizable samples han have been commonly analyzed in he empirical lieraure. Our preliminary ndings from weely hrough quarerly holding periods should be srenghened by he consideraion of daily holding periods. 1

14 Appendices A Excess Reurns series from zero-coupon quoaions The monhly excess reurns series mae use of he Fama-Bliss price quoaions on 1- o 5-year zerocoupon securiies for 1964:3 hrough 1996:1. Since only hese ve enors are available, one-monh (hree-monh) holding period reurns canno be consruced direcly from hese daa, since ha would require, for example, he price of 11-monh (9-monh) zero-coupon securiies. To consruc esimaes of holding period reurns for one-monh and hree-monh holding periods, he approximaion de ned by Campbell (1995, p.13) in erms of yields is expressed in erms of zero-coupon bond prices. He assumes ha he yield on he securiy does no change over he holding period in order ha is price a he end of he holding period may be inferred (cf. Campbell, fn.7). De ne p m as he logarihm of he bond price for a $1 par bond wih m periods o mauriy. Then he m 1,+1 m,+1 m,+1 [ ] m 3 m m,+3 m m, m, m one-year log holding period reurn for an m period bond is r = p p. Since he price and coninuously compounded (log) yield, y, are lined by he relaion p = my, he log holding period reurn may be expressed in erms of yields as y ( m 1)( y y ). m m 1,+1 m [ ] m 1 m m,+1 m In he absence of measures of a performance of an ( m 1) period securiy, we apply he approximaion y y. In price erms, his implies ha r = p p, which period, we calculae r = p p, where again m is measured in monhs. Excess m,+3 reurns are compued from hese series (expressed in per cen per annum by h = 100r and h = 400 r ) by subracing he CRSP Risfree Raes for one- and hree-monh enors, m,+3 m,+3 m m m maes i possible o calculae he (log) holding period reurn from available daa. For insance, he one-monh log holding period reurn on a one-year securiy will be 11 1 of nex monh s (log) price of he zero-coupon securiy minus his monh s (log) price of ha securiy. For a hree monh holding m,+1 m,+1 respecively. The average of bid and as yields provided in he Risfree Raes le are used. B Excess Reurns series from Fixed Term Indices quoaions We generae excess reurns series from he CRSP Fixed Term Indices quoaions from 14 June 1961 hrough 30 Sepember 1996, for a oal of 8,804 business days. Each quoaion is ideni ed wih he speci c Treasury securiy from which i is derived in he Fixed Term Index le; ha ideni er is used on he Daily Governmen Bond le o recover he yield of ha speci c securiy 1 In Campbell (1995, p.13), his formula incorrecly refers o pm 1, 1 and ym 1, 1 raher han heir counerpars daed ( + 1). 13

15 a he end of he holding period, be i one day, ve days, one monh, or hree monhs hence. Alhough he holding period reurn for ha speci c securiy could be compued exacly for ha inerval, we wan o remove he coupon e ec from he calculaion. We do so by employing he approximaion developed by Shiller (1990, p. 640), originally presened in Shiller e al. (1983), and widely employed in he erm srucure lieraure. He expresses he holding period rae or reurn from o on a bond mauring a ime T, T, as 1 e r(,t ) r(,t ) DT r,t DT D r,t h,,t = ( ) ( ) [ ( ) ( )] ( ) D ( ) where h (,,T) is he holding period reurn, r(,t) is he yield on a securiy held from o T, r,t ( ) is he yield on a securiy held from o T,D( T ) is he Macaulay duraion of he securiy, and D( ) is he duraion of a hypoheical securiy held from o, which is approximaed as D( m) = (Shiller, 1990, Table 13.1, p.640). Our applicaion of Shiller s approximaion requires hree pieces of informaion: he yield o mauriy of he securiy underlying he index on he quoaion dae, r(,t ), he yield o mauriy of ha securiy a he end of he holding period, r(,t), and Macaulay s duraion for he securiy as of he quoaion dae, D( T ). From hese elemens, we calculae daily holding period reurns series for each enor. The daily Fed Funds rae from he Federal Reserve Board s H.15 daaban is used o generae excess reurns series. 13 C The Generalized Error Disribuion In esimaing an ARCH or GARCH model, an assumpion mus be made regarding he disribuion of he condiional mean equaion s error process. We consider hree alernaives in esimaing he models repored in his sudy: normally disribued errors, Suden disribued errors, and errors following a Generalized Error Disribuion (henceforh GED). This disribuion, rs used in an ARCH modeling conex by Nelson (1991), is also nown as he exponenial power disribuion (Box and Tiao, 1973). As Nelson poins ou, he GED includes he normal as a special case, wih alernaives exhibiing greaer or lesser degrees of urosis han he normal. This maes he GED paricularly aracive, as i allows for boh posiive and negaive excess urosis in he error process (as opposed o he Suden ) wih a single ail-hicness parameer. The densiy of a normalized (zero mean, uni variance) GED random variae is given by (Nelson, 1991, p.35): [ ] 1 z/ f ()= z exp 1, < z <, < 1+ ( ) 0 v (1 /v) 13 Ideally, erm Federal funds raes for one wee, one monh, or one quarer would be used o calculae excess reurns for hose holding periods, bu hese daa are no publicly available. 14

16 where ( ) is he gamma funcion and 1/ ( /v) (1 /v) = (3 /v) The ail-hicness parameer aes on he value for a sandard normal disribuion, wih values less han denoing hicer ails han he normal, and vice versa. In he limi, as, he GED converges on a uniform disribuion. Nelson nds his disribuion paricularly useful in he conex of his Exponenial GARCH (E-GARCH) model, bu i may be used o advanage in any ARCH modeling conex. The GED has recenly been applied o he modeling of excess reurns from Treasuries by Brunner and Simon (1996). 15

17 References [1] Bollerslev, T., Chou, R.Y., and K.F. Kroner, 199, ARCH modeling in nance: A review of he heory and empirical evidence, Journal of Economerics, 5, [] Bollerslev, T. and J.M. Wooldridge, 199, Quasi-maximum lielihood esimaion and inference in dynamic models wih ime-varying covariances, Economeric Reviews, 11:, [3] Brunner, Allan D. and David P. Simon, 1996, Excess reurns and ris a he long end of he Treasury mare: An EGARCH-M approach, Journal of Financial Research, 19:3, [4] Box, G.E.P. and G. C. Tiao, 1973, Bayesian inference in saisical analysis. Reading, MA: Addison-Wesley Publishing Co. [5] Campbell, John Y., 1995, Some lessons from he yield curve, Journal of Economic Perspecives, 9:3, [6] Cener for Research on Securiies Prices, 1997, File Guide. Universiy of Chicago CRSP Monhly US Governmen Bond [7] Cener for Research on Securiies Prices, 1997, 1997 CRSP US Governmen Bond File Guide. Universiy of Chicago. [8] Engle, Rober F., Lilien, David M. and Russell P. Robins, 1987, Esimaing ime-varying ris premia in he erm srucure: The ARCH-M model, Economerica 55:, [9] Klemosy, R.C., and E.A. Piloe, 199, Time-varying erm premia on U.S. Treasury bills and bonds, Journal of Moneary Economics 30, [10] Nelson, Daniel B., 1991, Condiional heerosedasiciy in asse reurns: A new approach, Economerica 59:, [11] Shiller, Rober J., 1990, The erm srucure of ineres raes, Chaper 13 in Handboo of Moneary Economics, Volume 1, Benjamin M. Friedman and Fran H. Hahn, eds. Amserdam: Norh-Holland. [1] Shiller, Rober J., Campbell, John Y. and K. L. Schoenholz, 1983, Forward raes and fuure policy: Inerpreing he erm srucure of ineres raes, Aciviy, 1: Brooings Papers on Economic 16

18 [13] Tzavalis, E. and M.R. Wicens, 1997, Explaining he failures of he erm spread models of he raional expecaions hypohesis of he erm srucure, Journal of Money, Credi, and Baning 9:3,

19 Table 1: Descripive Saisics for Excess Reurn Series (A) Quarerly Excess Reurns SERIES Type NOBS AVE. VAR SKEW KURT MIN MAX Q-STAT (0 LAGS) SIGNIF LEVEL 1964:3-1996:4 1 Year Z Year Z Year Z Year Z Year Z Year C Year C Year C Year C Year C Year C Year C (B) Monhly Excess Reurns SERIES TYPE NOBS AVE. VAR SKEW KURT MIN MAX Q-STAT (0 LAGS) SIGNIF LEVEL 1964:3-1996:4 1 Year Z Year Z Year Z Year Z Year Z Year C Year C Year C Year C Year C Year C Year C (C) Five-Day Excess Reurns SERIES Type NOBS AVE. VAR SKEW KURT MIN MAX Q-STAT (0 LAGS) SIGNIF LEVEL Year C Year C Year C Year C Year C Year C Year C

20 Table 3 Parameer Esimaes for Zero Coupon models wih Spread a Quarerly: 1965:1-1996:4 Model : XR = a 0 + a XR + bspread + mh + ε, ε ~ ( 0, h ), 1 1 = 15,, year h = c + q i 4 i i ph = 1 + ε year year 5 year ARCH()-M ARCH()-M ARCH()-M/GED p-val p-val p-val a b c q q h ν (0.306) Sew Kurosis LB(4) LBsq(4) F-es for Arch(4) JB LogLi a Sandard errors are in parenheses.

21 Table 3 (con.) Parameer Esimaes for Zero Coupon models wih Spread a Monhly: 1964:8-1996:1 Model : XR = a 0 + a XR + bspread + mh + ε, ε ~ ( 0, h ), 1 1 = 15,, year h = c + q i 4 i i ph = 1 + ε year year 5 year AR1-GARCH(1,1)-M/GED AR1-ARCH(4)-M/GED AR1-ARCH(4)-M/GED p-val p-val p-val a a b c q q q q p h ν (0.148) (0.159) (0.15) Sew Kurosis LB(4) LBsq(4) F-es for Arch(4) JB LogLi

22 Table 5 Parameer Esimaes for coupon models wih Spread a Quarerly: = ε ε = Model : XR a bspread mh, ~( 0, h ), 1,, 5, 30 year 4 = + iε i i= 1 1 year year 5 year 30 year ARCH(4)-M/GED ARCH(3)-M/GED ARCH(4)-M/GED ARCH(4)-M/GED p-val p-val p-val p-val a b c q q q q h ν (0.16) (0.10) (0.37) (0.04) Sew Kurosis LB(4) LBsq(4) F-es for Arch(4) JB LogLi a Sandard errors are in parenheses. h c q

23 Table 6 Parameer Esimaes for coupon models wih Spread a Monhly: Model : XR = a + a XR + bspread + mh + ε, ε ~( 0, h ), = 1,, 5, 30 year year year 5 year 30 year AR1-GARCH(1,1)-M/GED AR1-GARCH(1,1)-M/GED AR1-ARCH(4)-M/GED ARCH(4)-M/GED p-val p-val p-val p-val a a b c q q q q p h ν (0.118) (0.10) (0.095) (0.1) Sew Kurosis LB(4) LBsq(4) F-es for Arch(4) JB LogLi a Sandard errors are in parenheses. h = c + qε + ph i i i=

24 Table 7 Parameer Esimaes for coupon models wih Spread a Weely: Model : XR = a + a XR + bspread + mh + ε, ε ~( 0, h ), = 1,, 5, 30 year 0 4 i= 1 i i 1 year year 5 year 30 year AR4-GARCH(1,1)-M/GED AR4-GARCH(1,1)-M/GED AR4-GARCH(1,1)-M/GED AR4-GARCH(1,1)-M/GED p-val p-val p-val p-val a a a a a b C q p h ν (0.000) (0.04) (0.049) (0.035) Sew Kurosis LB(4) LBsq(4) F-es for Arch(4) JB LogLi a Sandard errors are in parenheses. h = c + qε + ph

25 Figure 1. Esimaed Volailiy for Quarerly Holding Period Zero-Coupon daase year zero year zero 5-year zero 100 Esimaed Volailiy Sep-65 Sep-69 Sep-73 Sep-77 Sep-81 Sep-85 Sep-89 Sep-93

26 Figure. Esimaed Volailiy for Monhly Holding Period Zero-Coupon daase year zero year zero 5-year zero 4000 Esimaed Volailiy Aug-64 Aug-67 Aug-70 Aug-73 Aug-76 Aug-79 Aug-8 Aug-85 Aug-88 Aug-91 Aug-94

27 Figure 3. Esimaed Volailiy for Quarerly Holding Period 7000 Coupon daase 1-year year 5-year 30-year Esimaed volailiy Sep-61 Sep-64 Sep-67 Sep-70 Sep-73 Sep-76 Sep-79 Sep-8 Sep-85 Sep-88 Sep-91 Sep-94

28 Figure 4. Esimaed Volailiy for Monhly Holding Period Coupon daase 1-year -year 5-year 30-year Esimaed volailiy Jul-61 Jul-65 Jul-69 Jul-73 Jul-77 Jul-81 Jul-85 Jul-89 Jul-93

29 1000 Figure 5a. Esimaed Volailiy for Weely Holding Period Coupon daase 1-year year 8000 Esimaed volailiy

30 Figure 5b. Esimaed Volailiy for Weely Holding Period Coupon daase 5-year (lef scale) 30-year (righ scale)

31 Table A.1 Parameer Esimaes for zero coupon models wih spread a Model: ER = α + β (R r ) + u u = ρu + ε, ε ~N(0, σ ) (A) Quarerly: 64:3-96:3 Dependen Variable 1 Year Year 5 Year α (0.351) (0.688) (1.487) β.034**.48** 3.83** (0.571) (0.704) (0.971) ρ -0.57** -0.3* (0.086) (0.087) (0.088) S.E.E QSTAT b (0.455) (0.449) (0.455) QSTATSQ b (0.105) (0.030) (0.006) D. W CENTERED R (B) Monhly: 64:4-96:11 Dependen Variable 1 Year Year 5 Year α (0.591) (1.049) (1.993) β 1.641**.57** 3.107** (0.538) (0.740) (1.047) ρ 0.14* 0.165** (0.051) (0.050) (0.051) S.E.E QSTAT b (0.000) (0.000) (0.046) QSTATSQ b (0.000) (0.000) (0.000) D. W CENTERED R a Sandard errors are in parenheses. b P-values are in parenheses. * Significan a he 5 percen level. ** Significan a he 1 percen level. 1

32 Table A. Parameer Esimaes for coupon models wih spread a Model: ER = α + β (R r ) + u u = ρu + ε, ε ~N(0, σ ) 1 ε (A) Weely Dependen Variable 1 Year Year 5 Year 30 Year α (0.511) (0.859) (1.544) (.846) β 0.989** 1.14** 1.94** 3.354* (0.6) (0.403) (0.745) (.566) 0.1** 0.19** 0.066** (0.04) (0.04) (0.04) (1.307) S.E.E QSTAT b (0.000) (0.000) (0.000) (0.311) QSTATSQ b (0.000) (0.000) (0.000) (0.000) D. W CENTERED R (B) Monhly Dependen Variable 1 Year Year 5 Year 30 Year α (0.394) (0.671) (1.18) (.194) 1.056** 1.143** 1.613** 1.91 (0.175) (0.317) (0.585) (0.998) 0.137** * (0.053) (0.05) (0.051) (0.051) S.E.E QSTAT b (0.000) (0.000) (0.000) (0.001) QSTATSQ b (0.000) (0.000) (0.000) (0.000) D. W CENTERED R a ρ β ρ Sandard errors are in parenheses. b P-values are in parenheses. * Significan a he 5 percen level. ** Significan a he 1 percen level.

33 Table A. (con.) Parameer Esimaes for coupon models wih spread a Model: ER = α + β (R r ) + u u = ρu + ε, ε ~N(0, σ ) (C) Quarerly Dependen Variable 1 Year Year 5 Year 30 Year α (0.4) (0.459) (0.946) (1.888) β 0.856** 0.743** 0.970* 1.874* (0.118) (0.3) (0.473) (0.870) ρ -0.49** -0.65** -0.36** * (0.088) (0.087) (0.087) (0.087) S.E.E QSTAT b (0.584) (0.838) (0.739) (0.70) QSTATSQ b (0.017) (0.004) (0.010) (0.08) D. W CENTERED R a 1 Sandard errors are in parenheses. b P-values are in parenheses. * Significan a he 5 percen level. ** Significan a he 1 percen level.