The U.S. Treasury Yield Curve: 1961 to the Present

Transcription

1 Finance and Economics Discussion Series Divisions of Research & Saisics and Moneary Affairs Federal Reserve Board, Washingon, D.C. The U.S. Treasury Yield Curve: 1961 o he Presen Refe S. Gurkaynak, Brian Sack, and Jonahan H. Wrigh 6-8 NOTE: Saff working papers in he Finance and Economics Discussion Series (FEDS) are preliminary maerials circulaed o simulae discussion and criical commen. The analysis and conclusions se forh are hose of he auhors and do no indicae concurrence by oher members of he research saff or he Board of Governors. References in publicaions o he Finance and Economics Discussion Series (oher han acknowledgemen) should be cleared wih he auhor(s) o proec he enaive characer of hese papers.

2 The U.S. Treasury Yield Curve: 1961 o he Presen * Refe S. Gürkaynak Brian Sack and Jonahan H. Wrigh ** June 6 Absrac The discoun funcion, which deermines he value of all fuure nominal paymens, is he mos basic building block of finance and is usually inferred from he Treasury yield curve. I is herefore surprising ha researchers and praciioners do no have available o hem a long hisory of high-frequency yield curve esimaes. This paper fills ha void by making public he Treasury yield curve esimaes of he Federal Reserve Board a a daily frequency from 1961 o he presen. We use a well-known and simple smoohing mehod ha is shown o fi he daa very well. The resuling esimaes can be used o compue yields or forward raes for any horizon. We hope ha he daa, which are posed on he websie hp:// and which will be updaed periodically, will provide a benchmark yield curve ha will be useful o applied economiss. * We are graeful o Oliver Levine for superlaive research assisance and o Brian Madigan, Vincen Reinhar and Jennifer Roush for helpful commens. All remaining errors are our own. All of he auhors were involved in yield curve esimaion a he Federal Reserve Board when working a ha insiuion. The views expressed in his paper are solely he responsibiliy of he auhors and should no be inerpreed as reflecing he views of he Board of Governors of he Federal Reserve Sysem or of any oher employee of he Federal Reserve Sysem. ** Gürkaynak: Deparmen of Economics, Bilken Universiy, 68 Ankara, Turkey; [email protected] Sack: Macroeconomic Advisers, LLC, Washingon DC 6; [email protected] Wrigh: Federal Reserve Board, Washingon DC 551; () ; [email protected]

3 1. Inroducion The U.S. Treasury yield curve is of remendous imporance boh in concep and in pracice. From a concepual perspecive, he yield curve deermines he value ha invesors place oday on nominal paymens a all fuure daes a fundamenal deerminan of almos all asse prices and economic decisions. From a pracical perspecive, he U.S. Treasury marke is one of he larges and mos liquid markes in he global financial sysem. In par because of his liquidiy, U.S. Treasuries are exensively used o manage ineres rae risk, o hedge oher ineres rae exposures, and o provide a benchmark for he pricing of oher asses. Wih hese imporan funcions in mind, his paper akes up he issue of properly measuring he U.S. Treasury yield curve. The yield curve ha we measure is an off-herun Treasury yield curve based on a large se of ousanding Treasury noes and bonds. We presen daily esimaes of he yield curve from 1961 o 6 for he enire mauriy range spanned by ousanding Treasury securiies. The resuling yield curve can be expressed in erms of zero-coupon yields, par yields, insananeous forward raes, or n- by-m forward raes (ha is, he m-year rae beginning n years ahead) for any n and m. Secion of he paper reviews all of hese fundamenal conceps of he yield curve and demonsraes how hey are relaed o each oher. Secion 3 describes he specific mehodology ha we employ o esimae he yield curve, and Secion 4 discusses our daa and some of he deails of he esimaion. Secion 5 shows he resuls of our esimaion, including an assessmen of he fi of he curve, and secion 6 demonsraes how he esimaed yield curve can be used o calculae he yield on synheic Treasury securiies wih any desired mauriy dae and coupon rae. As an applicaion of his 1

4 approach, we creae a synheic off-he-run Treasury securiy ha exacly replicaes he paymens of he on-he-run en-year Treasury noe, allowing us accuraely o measure he liquidiy premium on ha issue. Secion 7 offers some concluding houghs. The daa are posed as an appendix o he paper on he FEDS websie.. Basic Definiions This secion begins by reviewing he fundamenal conceps of he yield curve, including he necessary bond mah. I hen describes he specific esimaion mehod employed in his paper..1 The Discoun Funcion and Zero-Coupon Yields The saring poin for pricing any fixed-income asse is he discoun funcion, or he price of a zero-coupon bond. This represens he value oday o an invesor of a $1 nominal paymen n years hence. We denoe his as d ( n ). The coninuously compounded yield on his zero-coupon bond can be wrien as y ( n) = ln( d ( n))/ n, (1) and conversely he discoun funcion can be wrien in erms of he yield as d ( n) = exp( y ( n) n). () Alhough he coninuously compounded basis may be he simples way o express yields, a widely used convenion is o insead express yields on a coupon-equivalen or bond-equivalen basis, in which case he compounding is assumed o be semi-annual insead of coninuous. For zero-coupon securiies, his involves wriing he discoun funcion as

5 d ( n) = 1 ce n (1 + y / ), (3) where ce y is he coupon-equivalen yield. One can easily verify ha he coninuously compounded yield and he coupon-equivalen yield are relaed o each oher by he following formula: y ce = ln(1 + y / ). (4) Thus, i is easy o move back and forh beween coninuously compounded and couponequivalen yields. The yield curve shows he yields across a variey of mauriies. Concepually, he easies way o express he curve is in erms of zero-coupon yields (eiher on a coninuously compounded basis or a bond-equivalen basis). However, praciioners insead usually focus on coupon-bearing bonds.. The Par-Yield Curve Given he discoun funcion, i is sraighforward o price any coupon-bearing bond by summing he value of is individual paymens. For example, he price of a couponbearing bond ha maures in exacly n years (paying $1) is as follows: n P( n) = ( c/ ) d ( i/ ) + d ( n), (5) i= 1 3

6 where c / is he semi-annual coupon paymen on he securiy ha is, i has a saed annual coupon rae of c. 1 Of course, for coupon-bearing bonds he yield will depend on he coupon raes ha are assumed. One popular way o express he yields on coupon-bearing bonds is hrough he concep of par yields. A par yield for a paricular mauriy is he coupon rae a which a securiy wih ha mauriy would rade a par (and hence have a coupon-equivalen yield equal o ha coupon rae). The yield can be deermined from an equaion similar o (5), only seing he price of he securiy equal o $1: n p y ( n) 1 = d ( i /) + d ( n ), (6) i= 1 p where we have replaced he coupon rae wih he variable y ( n ) o denoe he n-year par yield. Solving equaion (6), he par yield is hen given by: y p ( n) = (1 d ( n)). (7) n d (/) i i= 1 The par yields from equaion (7) are expressed on a coupon-equivalen basis. A coninuously compounded version of his can be derived by assuming a bond pays ou a coninuous coupon rae, in which case he par yield wih mauriy n, y pcc, ( n ), is given by: y pcc, 1 d ( n) ( n) =. (8) n d () i di 1 Because he bond maures in exacly n years, i is assumed o make is coupon paymen oday. Thus, he end-of-day price of he bond includes no accrued ineres. We will have o address accrued ineres in he pricing of individual Treasury securiies below. For simpliciy, his formula again assumes ha a coupon paymen has jus been made and he nex coupon is a full coupon period away, so ha here is no accrued ineres. 4

7 Zero-coupon yields are a mahemaically simpler and more fundamenal concep han par yields. However, one advanage of expressing he yield curve in erms of par yields is ha financial marke paricipans ypically quoe he yields on coupon-bearing bonds. Mos financial commenary focuses on individual Treasury securiies, mos ofen he on-he-run issues he mos recenly issued securiies a each mauriy. These securiies rade near par (a leas iniially) and have shorer duraion (owing o he posiive coupon) han zero-coupon yields wih he same mauriies. 3 Of course, he choice of wheher o focus on zero-coupon yields or par yields is simply a choice of he manner o presen he yield curve once esimaed; hese are alernaive ways of summarizing he informaion in he discoun funcion. In fac, he yield curve can be used o compue he yield for a securiy wih any specified coupon rae and mauriy dae an approach ha we will use below o analyze individual securiies..3 Forward Raes The yield curve can also be expressed in erms of forward raes raher han yields. A forward rae is he yield ha an invesor would agree o oday o make an invesmen over a specified period in he fuure for m-years beginning n years hence. These forward raes can be synhesized from he yield curve. Suppose ha an invesor buys one n+ m- year zero-coupon bond and sells d ( n+ m) / d ( n) n-year zero-coupon bonds. Consider he cash flow of his invesor. Today, he invesor pays d ( n+ m) for he bond being bough and receives d ( n+ m) d ( n ) = d ( n + m ) for he bond being sold. These cash d ( n) 3 We inroduce he concep of duraion in secion.4 below. The coupon rae for an on-he-run issue is se afer he aucion a he highes level a which he securiy rades below par. Because Treasury ses coupons in incremens of 1.5 basis poins, his process leaves he issues rading very near par immediaely afer he aucion. 5

8 flows, of course, cancel ou, so he sraegy does no cos he invesor anyhing oday. Afer n years, he invesor mus pay dn ( + m)/ dn ( ) as he n-year bond maures. Afer a furher m years, he invesor receives $1 as he n+ m-year bond maures. Thus, his invesor has effecively arranged oday o buy an m-year zero-coupon bond n years hence. The (coninuously compounded) reurn on ha invesmen, deermined by he amoun dn ( + m)/ dn ( ) ha he invesor mus pay a ime n o receive he $1 paymen a ime n+m, is wha we will refer o as he n-by-m forward rae, or he m-year rae beginning n years hence. The forward rae is given by he following formula: 1 d ( n+ m) 1 f( nm, ) = ln( ) = (( n+ my ) ( n+ m) ny( n) ), (9) m d ( n) m wih he las equaliy following from (). Taking he limi of (9) as m goes o zero gives he insananeous forward rae n years ahead, which represens he insananeous reurn for a fuure dae ha an invesor would demand oday: f( n,) = lim m f( n, m) = y( n) + ny ( n) = ln( d( n)), (1) n where he las equaliy again uses equaion (). Noice ha (1) implies ha he yield curve is upward (downward) sloping whenever he insananeous forward rae is above (below) he zero-coupon yield a a given mauriy. One can hink of a erm invesmen oday as a sring of forward rae agreemens over he horizon of he invesmen, and he yield herefore has o equal he average of n hose forward raes. Specifically, from equaion (1), ln( d( n)) = f( x,) dx, and so, from equaion (), he n-period zero-coupon yield (expressed on a coninuously compounded basis) is given by: 6

9 1 n y( n) = f( x,) dx n. (11) Likewise we can wrie y ( n ) as he average of one-year coninuously compounded forward raes: 1 n y( n) = Σi= 1 f( i 1,1). (1) n Thus, given a complee range of forward raes, one can calculae he complee yield curve from equaions (11) and (1), or, conversely, given he complee yield curve, one can calculae all he forward raes from equaions (9) and (1). Yields and forward raes are simply alernaive ways of describing he same curve. By using forward raes, we can summarize he yield curve in some poenially more informaive ways. For example, he en-year Treasury yield can be decomposed ino one-year forward raes over ha en-year horizon. As we will discuss below, nearerm forward raes end o be affeced by moneary policy expecaions and hence cyclical variables, while longer-erm forwards insead are deermined by facors seen as more persisen or by changes in risk preferences. The en-year yield meshes hese wo ypes of influences ogeher, whereas i may be easier o inerpre ha yield when one considers he near-erm and disan forward raes separaely. Indeed, former Fed Chairman Greenspan ofen parsed he yield curve ino is various forward componens (see for example his February and July 5 Moneary Policy Tesimonies). Similarly, Gürkaynak, Sack, and Swanson (5) frame heir discussion of he responsiveness of he yield curve o macroeconomic news in erms of forward raes, poining o he fac ha disan forward raes appear o respond o incoming daa (which hey associae wih movemens in long-erm inflaion expecaions). 7

10 Lasly, one can also compue forward raes for fuure invesmens ha have coupon paymens. A par forward rae is he coupon rae ha one would demand oday o make a $1 invesmen a ime n and o receive back $1 in principal a ime n+m along wih semiannual coupons from ime n+½ o ime n+m, assuming ha n and n+m are p coupon daes. Le f ( nm, ) denoe his n-by-m par forward rae (expressed wih semiannual compounding). An invesor can synhesize his par forward rae agreemen p by selling one n-year zero coupon bond and buying f ( n, m ) / of n+ 1/, n+ 1,... n+ m-year zero-coupon bonds and one more n+m year zero-coupon p bond, where f ( nm, ) is se so as o ensure ha he ne cash flow oday is zero. This implies ha p m f ( n, m) d( n) Σ i= 1d( n+ i/) d( n+ m) =. (13) Solving his equaion gives he formula for he n-by-m par forward rae: f ( d ( n) d ( n+ m)) ( n, m) =. (14) d ( n+ i/) p m i= 1 Whereas coninuously compounded zero-coupon yields can be wrien as he average of he corresponding coninuously compounded forward raes, as in equaions (11) and (1), we canno simply wrie par yields as averages of he consiuen par forward raes. However, Campbell, Lo and Mackinlay (1997) and Shiller, Campbell and Schoenholz (1983) show, using a loglinear approximaion, ha p 1 ρ n i p y ( n) Σi 1ρ f ( i 1,1) n =, (15) 1 ρ p where ρ = 1/(1 + y ( n)), he analog of equaion (1) for par raes. 8

11 .4 Duraion and Convexiy Before moving on o yield curve modeling and esimaion, we inroduce a couple of key conceps for he yield curve: duraion and convexiy. Duraion is a fundamenal concep in fixed-income analysis. Much of he value of a coupon-bearing securiy comes from coupon paymens ha are being made before mauriy, so he effecive ime ha invesors mus wai o receive heir money is shorer han he mauriy of he bond. The Macaulay duraion of a bond is a weighed average of he ime ha he invesor mus wai o receive he cash flows on a coupon-bearing bond (in years): n 1 ic [ ( /) ( )] ( ) i 1 D = d i + nd n. (16) P n = Zero-coupon bonds have duraion equal o he mauriy of he bond, bu coupon-bearing securiies have shorer duraion. For a given mauriy, he higher he coupon rae is, he shorer he duraion. Closely relaed o his concep is he modified duraion of a bond, D MOD, which is defined as he Macaulay duraion divided by one plus he yield on he bond (assuming semi-annual compounding): D D =. (17) + MOD y 1 ce I can be shown ha he derivaive of he log price of a bond wih respec o is yield is simply DMOD. Thus, modified duraion provides he sensiiviy (in percen) of he value of a bond o small changes in is yield. A relaed concep is ha of convexiy. Modified duraion measures he sensiiviy of he log price of a bond o changes in yield, bu i is accurae only for small changes in 9

12 yield. The reason i is no accurae for large changes in yield is ha he relaionship beween prices and yields is nonlinear: he capial gain induced by a decline in he yield is larger han he capial loss induced by an equal-sized increase in he yield. Convexiy capures his nonlineariy. To a second-order approximaion, he change in he log price of he bond is given by: 1 dlog( P) D dy ( dy) = mod + κ, (18) 1 d P where κ = is he convexiy of he bond. Convexiy has implicaions for he shape P dy of he yield curve ha will be an imporan consideraion in he choice of our mehodology for esimaing he yield curve. In paricular, convexiy ends o pull down longer-erm yields and forward raes, an effec ha increases wih uncerainy abou changes in yields. Consider, for example, an increase in he uncerainy abou a long-erm ineres rae ha is symmeric in erms of he possible basis-poin increase or decrease in yield. For a given level of he yield, his ends o increase he expeced one-period reurn on he bond because of he asymmery noed above ha he capial gain from a fall in he yield is greaer han he capial loss from a rise in he yield. Formally, consider he expeced value of he n-period zerocoupon bond one period ahead, which we can wrie as E[ d+ 1( n)] = E[exp( y+ 1( n) n)]. By Jensen's inequaliy, we have: E [ d ( n)] = E [exp( y ( n) n)] > exp( E [ y ( n)] n). (19) In oher words, he expeced value is higher han he value ha would be associaed wih he expeced yield nex period. 1

13 Markes, of course, recognize his effec and incorporae i ino he pricing of he yield curve. In paricular, he convexiy effec ends o push down yields, as invesors recognize he boos o expeced reurn from he convexiy erm and hence are willing o pay more for a given bond. This effec ends o be larger for bonds wih longer mauriies, giving he yield curve a hump shape ha is discussed a greaer lengh below. 3. Yield Curve Esimaion If he Treasury issued a full specrum of zero coupon securiies every day, hen we could simply observe he yield curve and have a complee se of he yields and forward raes described in he previous secion. Tha, unforunaely, is no he case. Treasury has insead issued a limied number of securiies wih differen mauriies and coupons. Hence, we usually have o infer wha he yields would be across he mauriy specrum from he prices of exising securiies. For each dae, we know he prices (and herefore yields) of a number of Treasury securiies wih differen mauriies and coupon paymens. Accouning for he differences in mauriies and coupons is no a problem; he esimaion will simply view couponbearing bonds as baskes of zero-coupon securiies, one for each coupon paymen and he principal paymen (as described above). 4 The more significan problem is he fac ha we do no have securiies a all mauriies. To come up wih yields across he complee mauriy specrum, we have o inerpolae beween he exising securiies. This exercise is wha consiues yield curve esimaion. 4 Coupon securiies simply bundle ogeher all of hese individual paymens. Unbundling hese paymens is precisely he purpose of he Treasury STRIPS program, in which each coupon and he principal can be individually raded. See Sack () for an overview. 11

14 In embarking on his exercise, one is immediaely confroned by an imporan issue: how much flexibiliy o allow in he yield curve. Pu differenly, one has o decide wheher all observed prices of Treasury securiies exacly reflec he same underlying discoun funcion. This is surely no he case: Idiosyncraic issues arise for specific securiies, such as liquidiy premia, hedging demand, demand for deliverabiliy ino fuures conracs, or repo marke specialness (which is ofen relaed o he oher facors). Moreover, some variaion across securiies could arise from bid-ask spreads and nonsynchronous quoe imes, hough we believe ha hese effecs are quie small in our daa (described below). In any case, i is desirable (and in fac necessary) o impose some srucure on he yield curve o smooh hrough some of his idiosyncraic variaion. However, one can choose differen mehods ha vary in erms of how much flexibiliy is allowed. One can esimae a very flexible yield curve which would fi well in erms of pricing he exising securiies correcly, bu do so wih considerable variabiliy in he forward raes. Or, one could impose more smoohness on he shape of he forward raes while sacrificing some of he fi of he curve. The more flexible approaches end o be spline-based mehods ha involve a large number of esimaed parameers, while he more rigid mehods end o be parameric forms ha involve a smaller number of parameers. The choice in his dimension depends on he purpose ha he yield curve is inended o serve. A rader looking for small pricing anomalies may be very concerned wih how a specific securiy is priced relaive o hose securiies immediaely around i. Suppose, for example, ha he yield curve has a dip in forward raes beginning, say, in year eigh ha is associaed wih he fac ha securiies in ha secor are he cheapes-o- 1

15 deliver ino he Treasury fuures conrac (an example we will show below). The rader, in assessing he value of an individual securiy in ha secor, would probably wan o incorporae ha facor ino his relaive value assessmen, and hence he would wan o use a yield curve flexible enough o capure his variaion in he forwards. By conras, a macroeconomis may be more ineresed in undersanding he fundamenal deerminans of he yield curve. Because i is difficul o envision a macroeconomic facor ha would produce a brief dip in he forward rae curve eigh years ahead, he may wish o use a more rigid yield curve ha smoohes hrough such variaion. Our primary purpose in esimaing he yield curve is o undersand is fundamenal deerminans such as macroeconomic condiions, moneary policy prospecs, perceived risks, and invesors risk preferences. Considering his purpose, we will employ a parameric yield curve specificaion. As will be seen below, his specificaion will allow for very rich shapes of he forward curve while largely ruling ou variaion resuling from a small number of securiies a a given mauriy. Our approach follows he exension by Svensson (1994) of he funcional form ha was iniially proposed by Nelson and Siegel (1987). The Nelson-Siegel approach assumes ha insananeous forward raes n years ahead are characerized by a coninuous funcion wih only four parameers: f ( n,) = β + β exp( n/ τ ) + β ( n/ τ )exp( n/ τ ). () Wih his funcion, insananeous forward raes begin a horizon zero a he level β + β1 and evenually asympoe o he level β. In beween, forward raes can have a hump, wih he magniude and sign of he hump deermined by he parameer β and he locaion of he hump deermined by he parameer τ 1. 13

16 Below we will show some resuls ha allow us o inerpre he shape of he forwards ha resul from his funcional form. Bu a his poin, i is worh making a few noes. We can always inerpre forward raes as having wo componens: expeced fuure shor-erm ineres raes and a erm premium. Under Nelson-Siegel, he forward raes will end o sar a he curren shor-erm rae ha is largely deermined by he curren moneary policy seing (he saring poin), will be governed a inermediae-horizons by expecaions of he business cycle, inflaion, and corresponding moneary policy decisions (he hump), and will end up a a seady-sae level (he asympoe). I urns ou, however, ha his yield curve has difficuly fiing he enire erm srucure, especially hose securiies wih mauriies of weny years or more. The reason is convexiy. As discussed in secion above, convexiy ends o pull down he yields on longer-erm securiies, giving he yield curve a concave shape a longer mauriies (as will be seen below). The Nelson-Siegel specificaion, while fiing shorer mauriies quie well, ends o have he forward raes asympoe oo quickly o be able o capure he convexiy effecs a longer mauriies. For ha reason, we insead use he more flexible approach described in Svensson (1994). This approach assumes ha he forward raes are governed by six parameers according o he following funcional form: f ( n,) = β + β exp( n/ τ ) + β ( n/ τ )exp( n/ τ ) + β ( n/ τ )exp( n/ τ ). (1) In effec, his specificaion adds wo new parameers (he las erm in he equaion) ha allow a second hump in he forward rae curve. The yield curve collapses o Nelson- Siegel when β 3 is se o zero. However, as we will see below, he yield curve ypically needs a second hump, one ha usually occurs a long mauriies, o capure he convexiy 14

17 effecs in he yield curve. Inegraing hese forward raes gives us he corresponding zero-coupon yields: n n n 1 exp( ) 1 exp( ) 1 exp( ) τ τ n τ n y ( n) = β + β + β [ exp( )] + β [ exp( )], () n n τ n 1 τ τ τ τ 1 1 and from hese yields one can compue he discoun funcion a any horizon. Thus, for a given se of parameers, he Svensson specificaion characerizes he yield curve and discoun funcion a all mauriies. The discoun funcion can hen be used o price any ousanding Treasury securiy wih specific coupon raes and mauriy daes. In esimaing he yield curve, we choose he parameers o minimize he weighed sum of he squared deviaions beween he acual prices of Treasury securiies and he prediced prices. The weighs chosen are he inverse of he duraion of each individual securiy. To a rough approximaion, he deviaion beween he acual and prediced prices of an individual securiy will equal is duraion muliplied by he deviaion beween he acual and prediced yields. Thus, his procedure is approximaely equal o minimizing he (unweighed) sum of he squared deviaions beween he acual and prediced yields on all of he securiies. Of course, his is jus one of many specificaions ha we could have chosen. A number of oher papers insead use spline-based mehods, including Fisher, Nychka, and Zervos (1995), Waggoner (1997), and McCulloch (1975, 199). These mehods generally allow for more variaion in he forward rae curve (hough he degree of flexibiliy can be conrolled in he specificaions). However, ha flexibiliy may come wih some coss. Bliss (1996) compares a number of esimaion mehods and finds ha parsimonious specificaions such as he Nelson-Siegel mehod perform favorably relaive 15

18 o some of he more flexible mehods. In addiion, Sack () demonsraes some of he esimaion difficulies ha can arise under more flexible approaches for esimaing he U.S. erm srucure. 4. Daa and Esimaion Issues We employ he Svensson mehodology for esimaing our benchmark yield curve. In our view, his mehod srikes an appealing balance beween being flexible enough o fi he U.S. yield curve well and being parsimonious enough o avoid over-fiing he idiosyncraic variaion in he yields of individual securiies. As described above, we esimae he six parameers, using maximum likelihood, o minimize he sum of he squared deviaions beween he acual prices of Treasury securiies and he prediced prices, where he prices are weighed by he inverse of he duraion of he securiies. Our underlying quoes on Treasury securiies come from wo primary sources. For he period from 14 June 1961 o he end of November 1987, we rely on he CRSP daily Treasury file, which provides end-of-day quoes on all ousanding Treasury securiies. Since December 1987, we use Treasury quoes provided by he Federal Reserve Bank of New York (FRBNY), which is a proprieary daabase consruced from several sources of marke informaion. 5 An immediae issue ha arises is deermining he se of securiies o be included in he esimaion. The Treasury securiies ousanding a any poin in ime can differ in many dimensions, including heir liquidiy and heir callable feaures. Our goal is o use a se of securiies ha are similar in erms of heir liquidiy and ha do no have special 5 We are no permied o release eiher he underlying CRSP daa or he FRBNY daa. 16

19 feaures (such as being callable) ha would affec heir prices. In oher words, we would ideally have securiies ha only differ in erms of heir coupons and mauriies. To ha end, we include in he esimaion all ousanding Treasury noes and bonds, wih he following excepions: (i) We exclude all securiies wih opion-like feaures, including callable bonds and flower bonds. 6 (ii) We exclude all securiies wih less han hree monhs o mauriy, since he yields on hese securiies ofen seem o behave oddly. This behavior may parly reflec he lack of liquidiy for hose issues and segmened demand for shor-erm securiies by paricular invesor classes. (iii) We also exclude all Treasury bills ou of concern abou segmened markes. Indeed, Duffee (1996) showed ha bill raes are ofen disconneced from he res of he Treasury yield curve, perhaps owing o segmened demand from money marke funds and oher shor-erm invesors. (iv) We begin o exclude weny-year bonds in 1996, because hose securiies ofen appeared cheap relaive o en-year noes wih comparable duraion. This cheapness could reflec heir lower liquidiy or he fac ha heir high coupon raes made hem unaracive o hold for ax-relaed reasons. 7 (v) We exclude he wo mos recenly issued securiies wih mauriies of wo, hree, four, five, seven, en, weny, and hiry years for securiies issued in 198 or laer. These are he on-he-run and firs off-he-run issues ha ofen rade a a premium o 6 Flower bonds were securiies wih low coupons ha could be redeemed a par for he paymen of esae axes. 7 To avoid an abrup change o he sample, we allow heir weighs o linearly decay from 1 o over he year ending on January,

20 oher Treasury securiies, owing o heir greaer liquidiy and heir frequen specialness in he repo marke. 8 Earlier in he sample, he concep of an on-he-run issue was no well defined, since he Treasury did no conduc regular aucions and he repo marke was no well developed (as discussed by Garbade (4)). Our cu-off poin for excluding onhe-run and firs off-he-run issues is somewha arbirary bu is a conservaive choice (in he sense of poenially erring on he side of being oo early). (vi) Oher issues ha we judgmenally exclude on an ad hoc basis. For example, here were large and persisen fails-o-deliver in he May 13 3⅝ percen en-year noe, well afer i ceased o be eiher he on-he-run or firs off-he-run en-year securiy. Wih he securiy persisenly rading around he fails rae in he overnigh repo marke, he yield on he securiy in he cash marke was driven down. Thus, we dropped he securiy for some ime o avoid having our yield curve disored by he idiosyncraic dislocaion of his issue. These resricions imply ha we are esimaing an off-he-run Treasury yield curve, one for which he liquidiy of he included securiies should be relaively uniform. The liquidiy implici in our curve should be regarded as adequae, hough far shor of he remarkable liquidiy of on-he-run issues. The ranges of mauriies available for esimaion over our sample are shown graphically in Figure 1, which akes he same form as a figure repored by Bliss (1996). The dae is shown on he horizonal axis, he remaining mauriy is shown on he verical axis, and each ousanding Treasury coupon securiy is represened by a do showing is 8 Some simple saisics on rading volume highligh jus how differen he on-he-run issues are from oher Treasury issues. According o Sack and Elsasser (4), he weekly urnover rae for off-he-run Treasury securiies in 3 (ha is, weekly rading volume as a percen of ousanding deb) was abou percen, while i was a remarkable 14% for on-he-run issues. 18

21 remaining mauriy on ha dae. For example, a do a a en-year mauriy in 1985 denoes a securiy ha is o maure en years laer, in Tha same securiy will be represened by a do a he nine-year mauriy in In he resuls below, we repor esimaes of he yield curve a horizons ha go as far ou as possible wihou exrapolaing far beyond he range of mauriies ha are acually ousanding. The maximum mauriy ha we repor is shown by he horizonal line segmens in Figure 1. Specifically, we repor yield curve esimaes ou o seven years for he early par of he sample, exend hem o en years on 16 Augus 1971, o fifeen years on 15 November 1971, o weny years on July 1981, and o hiry years since 5 November Of course, wih he esimaed parameers from he Svensson yield curve, one could compue yields and forward raes a any horizon; however, we would srongly recommend only focusing on hose measures a he horizons for which ousanding securiies were available for esimaion. A final issue is wheher o make adjusmens for ax effecs, a opic discussed by McCulloch (1975). In our view, he correc adjusmen is difficul o deermine, in par because Treasury securiies are held by a wide range of invesors wih differen ax brackes. In fac, a large se of Treasury invesors, including pension funds and ohers, are exemp from axes. Moreover, Treasury securiies provide a ax advanage on sae and local axes ha depends on he sae of residence of he invesor. Given hese consideraions, we chose no o make adjusmens for ax effecs. Neverheless, ax consideraions may be imporan in some applicaions, in which case users migh wan o make ax adjusmens o our smoohed yield curve. 9 Noe ha from 1985 o 1995, here was a en-year gap a longer mauriies, reflecing he Treasury's decision o sop issuing weny-year bonds and o sar issuing hiry-year bonds in However, he yield curve esimaion appears o smooh hrough his gap wihou much problem. 19

22 5. Resuls Using he above mehodology, we esimae he U.S. Treasury yield curve from June 1961 o he presen. As an example of he resuls, Figure shows he esimaed yield curve on May 9, 6. The solid line is he coninuously compounded par yield curve, he open circles are he acual quoes on all ousanding coupon securiies included in he esimaion, and he crosses are he prediced yields for hose issues. As can be seen, he yield curve does an impressive job fiing he enire crosssecion of Treasury coupon issues wih a funcion of only six parameers. The larges misses are for very shor-erm issues, which we aribue o he idiosyncraic naure of hose securiies, and for several securiies in he wo- o hree-year mauriy range ha appear divorced from oher yields on ha day. The success a fiing Treasury yields on his dae is repeaed hroughou he sample. Figure 3 shows he average absolue yield predicion error in differen mauriy buckes over ime. As can be seen, all of he errors are quie small over he enire sample. The larges fiing errors end o be seen a he longes mauriies ha are being fied. In addiion, he fi of he yield curve esimaion has generally improved over ime and is paricularly good in he laer par of he sample. One possible explanaion of his fac is ha he marke has become more acive and liquid, which has reduced pricing anomalies across various securiies. Under ha inerpreaion, i is ineresing o noe ha he fi of he esimaion emporarily worsened slighly afer he financial marke urmoil in he fall of Fleming () has suggesed ha Treasury securiies may have

23 become less conneced o one anoher a ha ime because many of he arbirage desks ha assure he close relaionships across securiies became less acive. Figure 4 reurns o he specific dae considered in Figure, only now showing he insananeous forward raes and he zero-coupon yield curve. To assess he performance of he Svensson mehod, i is useful o add some inerpreaion o he shape of he yield curve ha day. A he shor horizons, forward raes and yields decline slighly, apparenly reflecing marke expecaions for sligh easing of moneary policy in 7 and beyond. Beyond his range, boh curves urn up, reflecing he normal upwardsloping paern of he yield curve associaed wih erm premia. The upward slope of he yield curve apers off a long horizons, however, and evenually urns down. Accordingly, he forward raes urn down earlier and much more sharply. This dae provides a good example of he way in which he Svensson mehod ypically fis he U.S. yield curve. Figure 5 provides more deails by showing he decomposiion of his yield curve ino is componens on his dae. Loosely speaking, as discussed in secion 3, he success of his mehod comes from allowing wo humps ha seem o serve very differen purposes. The firs hump (he β erm) is ofen locaed a relaively shor horizons, which in many cases is needed o capure he effecs of nearerm moneary policy expecaions (along wih he decay componen, he β 1 erm). The second hump (he β 3 erm) is ypically locaed a much longer horizons long enough so ha he forward rae schedule is sill downward sloping even afer 3 years, even hough he specificaion assumes ha forward raes evenually asympoe o a consan. The downward il o forward raes a long horizons is an imporan characerisic of he U.S. yield curve; for example, he insananeous forward rae ending 5 years 1

24 ahead has coninuously been below he insananeous forward rae ending years ahead for he pas decade. This paern mos likely reflecs he convexiy of longer-erm securiies, which pulls down heir yields for he reasons discussed above. 1 The asympoe for he forward rae curve (he β erm) is esimaed o be zero on his dae, which arises because he convexiy effec leaves forward raes sloping downward a he longes mauriies and here are no perpeuiies issued by he Treasury o accuraely pin down he infinie horizon yield. For his reason, he asympoe is usually no srongly idenified and is esimaed o be zero a imes. As was seen in Figure, his underidenificaion of he asympoe has no implicaions for he fi of he yield curve over he range of exising mauriies. Because he second hump is ypically locaed a such long horizons, i canno be reliably idenified in he earlier pars of our sample, when he longes mauriy available was fifeen, en, or even seven years. This consideraion forces us o impose some resricions on he esimaion mehod earlier in our sample. Specifically, we use he Nelson-Siegel specificaion for he period before 198; pu differenly, we resric he parameer β 3 o be zero. This is a resricion ha probably would no hold if longer-erm securiies were available over ha period, bu imposing i does no significanly affec he fi of he yield curve a he shor mauriies available. While he Svensson specificaion is sufficienly rich o capure he shape of he yield curve associaed wih policy expecaions and convexiy, i is no so flexible as o be significanly influenced by he idiosyncraic behavior of a small number of securiies. A useful example ook place in he spring of 5, when he markes reporedly became 1 Sack () shows ha his characerisic is also eviden in he prices of Treasury STRIPS.

25 concerned ha he supply of en-year noes ha were cheapes-o-deliver ino he Chicago Board of Trade s Treasury fuures conrac was no sufficien o mee he required delivery (Whiehouse, Lucchei and McKay (5)). As shown in Figure 6, he marke began o place a significan premium on he Augus 1 noe, which was cheapes-odeliver ino he Sepember 5 en-year fuures conrac. Oher securiies in he June and Sepember deliverable baske also araced a premium. Some fiing mehods would capure his paern wih sizable swings in forward raes abou seven years ahead ha are unrelaed o macroeconomic fundamenals. The Svensson yield curve, however, is rigid enough ha i does no give a dip a ha paricular mauriy, bu insead fis he general shape of he yield curve. One imporan advanage of our yield curve esimaes is ha hey are available over a long hisory. The full hisory of esimaes for a seleced se of Treasury yields and forward raes is shown in Figures 7 and 8. Yields and forward raes generally drifed higher over he lae 196s and 197s and hen drifed lower over mos of he 198s and 199s, following he general paern of inflaion and longer-run inflaion expecaions over he sample. Of course, here is also much variaion associaed wih he business cycle and oher facors, especially a shorer mauriies. One issue ha we confron when reviewing he hisorical yield curve esimaes is ha he esimaed parameers demonsrae some insabiliy, in ha hey someimes jump discreely from one day o he nex, ofen wih lile acual movemen in underlying bond prices. Anderson and Sleah (1999) illusrae clearly ha changing a single daa poin in he se of prices used o fi he Svensson yield curve can produce a noable shif in parameers and also in fied yields. This is a drawback of parameerized yield curves, 3

26 and forward raes can be paricularly affeced. However, i is imporan o noe ha alhough hese jumps in parameers can be large, he changes in prediced yields over mos of he mauriy range considered are quie mued. In effec, here is an idenificaion issue and he esimaion is arriving a fairly similar yield curve shapes over he mos of he mauriy range considered hrough differen combinaions of parameers. 11 This parameer insabiliy is a drawback o he Svensson mehodology, bu he smoohness of he Svensson yield curve, is ease of economic inerpreaion, and is relaive insensiiviy o he parameer shifs over he range of horizons ha are mos relevan o moneary policymakers leads us o noneheless prefer he Svensson curve o alernaives such as splines. 1 The appendix ha accompanies his paper provides daa on zero-coupon yields (coninuously compounded), insananeous forward raes (coninuously compounded) and par yields (coupon-equivalen) for he full range of mauriies ha we consider. The daa are daily and are available back o June 14, The appendix also includes one-year par forward raes (coupon-equivalen) beginning one, four, and nine years ahead, as well as he esimaes of he parameers of he Svensson yield curve. This appendix is posed on he websie hp:// using he mnemonics described in Table 1. We inend o updae he daa regularly, as a resource for academic researchers and financial marke praciioners. 11 For example, averaging over he days where he absolue change in β was in he op percenile, he maximum absolue change in he zero-coupon yield across all mauriies from one o weny years was 1 basis poins. Averaging over hese days, he maximum absolue change in he insananeous forward rae across all hese mauriies was basis poins. Meanwhile β, which is he asympoe of he insananeous forward curve jumped by over percenage poins on all hese days. Yields, and especially forward raes, were more sensiive o hese parameer jumps a mauriies beyond weny years. 1 The insabiliy in parameers could in principle owe o he numerical opimizaion algorihm for minimizing he crierion funcion failing o find he global minimum on some days. However, he insensiiviy of our resuls o saring values leads us o hink ha his is no he case. 4

27 We only repor he yield curve a horizons for which here are ousanding Treasury securiies (as discussed above). Alhough here is nohing o sop a researcher from consrucing yields a any horizons from hese parameer esimaes, we srongly advise agains doing so, since here is no reason o expec he exrapolaion o yield reasonable resuls. One mauriy poin for which his consideraion applies is zero. The risk-free shor-erm ineres rae (usually an insananeous ineres rae or perhaps an overnigh ineres rae) plays a criical role in many financial models. However, he yield curve esimaed above is no designed o fi well a he very shores mauriies. 13 A hose mauriies, he erm srucure of Treasury yields will be affeced by facors ha our parameerizaion will no capure, including seasonal paerns in Treasury bill issuance and he iming of FOMC meeings. Recall ha we excluded all bills and hose coupon securiies wih less han hree monhs of remaining mauriy from he esimaion, parly for his reason Synheic Treasury Securiies There are many advanages o having a smoohed Treasury yield curve. Here we will highligh one applicaion he use of he yield curve o consruc a synheic off-herun Treasury securiy wih any mauriy dae and coupon rae desired. Such securiies can be very useful for assessing he relaive value of oher securiies. 13 Indeed, on a few days he exrapolaed zero mauriy ineres rae in our daase would even be negaive. 14 If one insead waned o include an insananeous or overnigh ineres rae, one possibiliy would be o use he effecive federal funds rae series ha is published on he Federal Reserve s H.15 daa release, hough some adjusmen would have o be made for he small credi risk premium embedded in his rae and perhaps for is differen ax reamen. 5

28 The approach is sraighforward. The esimaed yield curve provides us wih he value ha invesors place on a known, risk-free paymen a any dae in he fuure. Thus, for a hypoheical coupon rae and mauriy dae, we can compue he value of each paymen on he securiy and sum hem up o arrive a a prediced price of he synheic securiy, which can hen be convered o is yield. This represens he yield ha one would expec on an off-he-run Treasury securiy wih hose exac paymens. This procedure can be used for compuing a benchmark agains which o measure spreads on various securiies. For example, i could be applied o measure he credi risk spread for a corporae bond. Whereas mos applicaions measure corporae spreads relaive o individual Treasury securiies, he paymens and mauriies of he wo securiies will differ some. By using a synheic Treasury wih idenical paymens, one eliminaes any disorions coming from mismached paymens and he resuling differences in duraion and convexiy. Moreover, since i is an off-he-run synheic Treasury securiy, i may have liquidiy ha is closer o ha of he corporae bond han would an on-he-run issue. Here we choose o demonsrae his echnique by considering he liquidiy premium for he on-he-run en-year Treasury noe. On-he-run Treasury issues were excluded from he esimaion of he yield curve. As noed above, hese securiies ypically rade a a premium o off-he-run Treasuries, as invesors are willing o pay a higher price for he greaer liquidiy offered by hese securiies. Many marke paricipans measure he on-he-run premia relaive o he firs or second off-he-run securiy, bu hose issues will have shorer duraion. The difference in duraion can disor he measure, especially when he Treasury yield curve is seep. 6

29 We insead measure he liquidiy premium for an on-he-run issue by comparing i o a synheic off-he-run Treasury securiy wih he same coupon rae and mauriy dae. The resuling measure for he en-year Treasury noe is shown in Figure 9 for he period since As can be seen, here is considerable variaion in he measure a high frequency. In par, his reflecs he aucion paern of he on-he-run premium: The premium end o jump higher a he aucion of a new securiy, when he liquidiy advanage of he on-he-run issue will be realized for he longes ime, and hen diminishes gradually up o he aucion of a new securiy. Bu here is also more gradual and persisen variaion over ime ha is of ineres. The liquidiy premium appears o move up during periods of financial urmoil, including he sock marke crash of 1987 and he seizing up of markes in he fall of Moreover, he premium remained relaively high from o, a period during which he supply of on-he-run issues was curailed. 15 More recenly, he premium has moved back o abou 1 basis poins, a level ha appears relaively normal for ranquil periods. The variaion in his premium highlighs a shorcoming of using he yields on onhe-run issues o measure of he shape of he yield curve. Movemens in hose yields can be associaed wih changes in he liquidiy premium, paricularly around Treasury aucions. This is why we exclude hese securiies from our yield curve esimaion. This concern also applies o he Treasury consan mauriy series (repored in he Federal Reserve s H.15 release), which are esimaed from a blend of on-he-run and off-he-run issues. 15 The on-he-run premium has been elevaed a oher imes as well. Indeed, he larges premium under our measure occurred in While he source of ha premium is unclear, i may have been driven in par by special demand for on-he-run securiies from foreign invesors. 7

30 7. Conclusion In his paper we have esimaed he U.S. Treasury yield curve using an approach ha is simple and parsimonious. The approach is quie effecive a capuring he general shape of he yield curve while smoohing hrough he idiosyncraic variaion in he yields on individual securiies. As such, he resuls should prove useful for undersanding he general macroeconomic and oher facors ha have broad effecs on he shape of he yield curve. The esimaed yield curve can be expressed in a variey of ways, including zerocoupon yields, par yields, and forward raes. Our yield curve fills a void in he academic lieraure. To our knowledge, no esimaed yield curve is available on a daily basis back o he early 196s. The daase of Fama and Bliss (1987) (which has been updaed) is monhly, and only provides esimaes ou o five-year mauriies, while he daase of McCulloch and Kwon (1993) is also monhly and only provides esimaes ou o en-year mauriies. Our daa se has he advanages of being available on a daily basis, exending back o 1961, providing esimaes for all mauriies ha are feasible given he disribuion of ousanding securiies, and being updaed on a regular basis. Given he imporance of he yield curve in boh he macroeconomics and finance lieraures, we hope ha our yield curve will serve as a valuable benchmark o be used in applied research. I is o his end ha we have made he full daase available o be downloaded from hp:// and will updae i regularly. 8

31 References Anderson, Nicola and John Sleah (1999), New Esimaes of he U.K. Real and Nominal Yield Curves, Bank of England Quarerly Bullein, November 1999, Bliss, Rober R. (1996), Tesing Term Srucure Esimaion Mehods, Advances in Fuures and Opions Research, 9, Campbell, John Y., Andrew W. Lo, and A. Craig MacKinlay (1997), The Economerics of Financial Markes, Princeon Universiy Press, Princeon. Duffee, Gregory (1996), Idiosyncraic Variaion of Treasury Bill Yields, Journal of Finance, 51, Fisher, Mark, Douglas Nychka and David Zervos (1995), Fiing he Term Srucure of Ineres Raes wih Smoohing Splines, Finance and Economics Discussion Series, Fleming, Michael J. (), The Benchmark U.S. Treasury Marke: Recen Performance and Possible Alernaives, Federal Reserve Bank of New York Economic Policy Review, 6, Garbade, Kenneh D. (4), The Insiuionalizaion of Treasury Noe and Bond Aucions, , Federal Reserve Bank of New York Economic Policy Review, 1, Gürkaynak, Refe S., Brian Sack and Eric T. Swanson (5), The Sensiiviy of Long- Term Ineres Raes o Economic News: Evidence and Implicaions for Macroeconomic Models, American Economic Review, 95, McCulloch, J. Huson (1975), The Tax-Adjused Yield Curve, Journal of Finance 3, McCulloch, J. Huson (199), U.S. Term Srucure Daa, , Handbook of Moneary Economics, Volume I, McCulloch, J. Huson and Heon-Chul Kwon (1993), U.S. Term Srucure Daa, , Ohio Sae Working Paper #93-6. Nelson, C. R. and A. F. Siegel (1987), Parsimonious Modeling of Yield Curves, Journal of Business 6, Sack, Brian (), Using Treasury STRIPS o Measure he Yield Curve, Finance and Economics Discussion Series Working Paper #-4. 9

32 Sack, Brian and Rober Elsasser (4), Treasury Inflaion-Indexed Deb: A Review of he U.S. Experience, Federal Reserve Bank of New York Economic Policy Review 1, Shiller, Rober, John Y. Campbell and Kermi L. Schoenholz (1983), Forward Raes and Fuure Policy: Inerpreing he Term Srucure of Ineres Raes, Brookings Papers on Economic Aciviy, 1, Svensson, Lars E. O. (1994), Esimaing and Inerpreing Forward Raes: Sweden 199-4, Naional Bureau of Economic Research Working Paper #4871. Waggoner, Daniel (1997), Spline Mehods for Exracing Ineres Rae Curves from Coupon Bond Prices, Federal Reserve Bank of Alana Working Paper Whiehouse, Mark, Aaron Lucchei and Peer A. McKay (5), Shor-Bond Shorage Isn Over, Wall Sree Journal, Augus 11, 5. 3

33 Table 1: Descripion of he series in he daa appendix Series Compounding Convenion Mnemonics Mauriies Repored (max) Zero-coupon yield Coninuously Comp. SVENYXX All inegers 1-3 Par yield Coupon-Equivalen SVENPYXX All inegers1-3 Insananeous forward rae Coninuously Comp. SVENFXX All inegers 1-3 One-year forward rae Coupon-Equivalen SVEN1FXX 1, 4, and 9 Parameers N/A BETA o N/A TAU Noes: XX in each case denoes he mauriy in years. For example, SVENY1 denoes he en-year zero-coupon yield. Mauriies repored are limied before 1985 as described in he ex. The one-year forward raes XX years hence denoe he one-year forward raes beginning XX years hence. For example, SVEN1F9 is he one-year forward rae from nine o en years hence. The parameers are labeled BETA, BETA1, BETA, BETA3, TAU1, and TAU, corresponding o he equaions in he ex. Noe ha he parameers BETA3 and TAU are resriced o zero in he earlier par of he sample, as discussed in he ex. 31

34 Figure 1: Ousanding Treasury Securiies 35 3 Years o Mauriy

35 Figure : Par Yield Curve on May 9, Yield Acual Yield 4.95 Prediced Yield Par Yield Mauriy in Years

36 1 Figure 3: Average Absolue Yield Predicion Errors by Indicaed Mauriy Bin years 5 years Basis Poins.6.4 Basis Poins years years.8.8 Basis Poins.6.4 Basis Poins years 1 3 years.8.8 Basis Poins.6.4 Basis Poins

37 Figure 4: Zero Coupon Yield Curve and Forward Raes on May 9, Yield Insananeous Forward Rae Zero Coupon Yield Mauriy in Years

38 Figure 5: Decomposiion of he Yield Curve on May 9, β β 1 erm β erm β 3 erm 4 Yield Mauriy in Years

39 Figure 6: Premium for he Cheapes o Deliver Issue on May 4, Aug 1 Yield Acual Yield.8 Prediced Yield Par Yield Mauriy in Years

40 Figure 7: Zero Coupon Yields Percen year Percen year Percen year Percen year Percen year Percen year

41 Figure 8: Insananeous Forward Raes Percen year Percen year Percen year Percen year Percen year Percen year

42 6 Figure 9: Premium for he Onherun Tenyear Treasury Noe