New esimaes of he UK real and nominal yield curves Nicola Anderson and John Sleah The views expressed are hose of he auhors and do no necessarily reflec hose of he Bank of England. The auhors wish o hank Rober Bliss and Dan Waggoner for many helpful commens and suggesions a he ouse of his work. Thanks also go, wihou implicaion, o Creon Buler, Roger Clews, John Lumsden and Jim Seeley. Copies of working papers may be obained from Publicaions Group, Bank of England, Threadneedle Sree, London, EC2R 8AH; elephone 020-760 4030, fax 020-760 3298, e-mail mapublicaions@bankofengland.co.uk. Working papers are also available a www.bankofengland.co.uk/workingpapers/index.hm The Bank of England s working paper series is exernally refereed. Bank of England 200 ISSN 368-5562
Conens Absrac 5 Inroducion 7 2 Model selecion and overview 2. Parameric models 2 2.2 Spline-based models 3 3 Esimaion and resuls 8 3. Smoohness and flexibiliy 8 3.2 Sabiliy of he curves 2 3.3 Summary 25 4 Improving esimaes of he shor end of he yield curve 26 4. Choice of addiional daa 26 4.2 Resuls of incorporaing GC repo raes 27 5 Esimaing he real and implied inflaion erm srucures 30 5. Evans heoreical framework 3 5.2 Two modificaions 32 5.3 Sabiliy of he esimaed real curve 35 5.4 Comparison of new approach wih he ITS echnique 38 6 Conclusions 4 Appendix: Smoohing cubic spline models 42 References 43 3
Absrac This paper presens some new esimaes of he UK real and nominal yield curves. These esimaes are derived using a spline-based echnique pu forward by Waggoner (997), modified for he UK governmen bond markes. A he shor end of he nominal yield curve, addiional daa are included from he GC repo marke. Esimaes of he real yield curve are derived from he prices of index-linked gils wihin a modified version of he framework pu forward by Evans (998). I is found ha he new yield curves ouperform exising mehods on a number of crieria ha are designed o examine he suiabiliy of esimaes for he purpose of assessing moneary condiions. In paricular, he esimaes are found o be smooh across mauriy while having sufficien flexibiliy o describe he shape of he curve a shorer mauriies where expecaions are relaively precise. The curves are also robus o small errors in he daa. 5
Inroducion In his paper, we presen some new esimaes of he UK real and nominal yield curves, for he purpose of assessing moneary condiions. These esimaes differ from hose presened in previous sudies in a number of ways. Firs, he yield curves are esimaed using a mehod pu forward by Waggoner (997) for he Unied Saes, adaped by us for he UK governmen bond marke. Second, daa from he generalised collaeral (GC) repo marke are used o provide improved esimaes of he nominal yield curve a shorer mauriies. Third, esimaes of he real yield curve are exraced from he prices of index-linked gils wihin a modified version of he framework suggesed by Evans (998). Before describing each of hese developmens in deail, i is worh briefly reviewing he main problems involved in exracing useful informaion from he prices of convenional and index-linked gils. The mos basic ype of informaion we are ineresed in esimaing is he implied forward raes of ineres a various horizons. These are imporan in heir own righ as hey reflec, albei imperfecly, he marke s expecaions abou he fuure pah of ineres raes. They also provide he building-blocks for calculaing oher erm srucure variables, such as zero-coupon yields, as well as enabling comparisons beween he reurns on governmen bonds and oher asses, for example, he credi spread embodied in he prices of corporae bonds. Implied forward raes of ineres are defined as he marginal raes of reurn ha invesors require in order o hold bonds of differen mauriies. The se of insananeous forward raes, f(m), are relaed o he price, B(τ), of a τ-mauriy zero-coupon bond by: () τ B( τ ) = exp f dm 0 ( m) () Equaion () shows ha o measure hese forward raes direcly from he marke requires a se of observable zero-coupon bond prices across a () We focus on insananeous forward raes of ineres as hese can be used o derive forward raes and yields beween any wo mauriies of ineres. For example, he s-period forward rae a some poin τ in he fuure is given by f ( τ, τ + s ) = f ( m ) dm. 7 τ + s τ
coninuum of mauriies (he discoun funcion ). Bu in pracice, he discoun funcion is no direcly observable we only observe he prices of coupon-bearing bonds. (2) We can, however, wrie he price of each observable bond in erms of his discoun funcion. So leing c i denoe he cash flow due on he bond a ime τ i and n refer o he number of such paymens ousanding, we express he price of he bond, P(c i,τ i, i=, n), as: n P( c, τ, i =, K, n) = c B( τ ) (2) i i i= i i Togeher wih equaion (), his shows ha here is a direc relaionship beween he bond prices we observe and he insananeous forward raes we wan o measure. Moreover, equaion (2) can be defined for cash flows denominaed in eiher real or nominal erms, wih he forward raes idenified accordingly. The idenificaion of real and nominal forward curves via his relaionship means overcoming hree ses of problems. These are: Boh convenional and index-linked bonds are issued for only a finie se of mauriies. We herefore need a mehod of disenangling he cash flows on he bond and filling in he gaps o give a coninuous curve. The real value of cash flows on index-linked bonds is no known wih cerainy. Hence, o esimae he real yield curve we need a mehod for approximaing he value of hese cash flows. For boh he real and nominal yield curves, here is a lack of daa a shorer mauriies. Forward raes along his porion of he curve canno herefore be idenified. (2) In fac, zero-coupon gils have exised since he inroducion of he srips marke in December 997. These separae he wo componens of a coupon-bearing gil o give a principal srip wih mauriy equal o is redempion dae and a series of coupon srips relaed o each paymen dae. The marke in srips is, however, sill small relaive o coupon-bearing gils, wih hin rading. We herefore do no use srips prices o esimae he yield curve. 8
Mehods for addressing he firs of hese problems have been available for more han 30 years, and hese have been used in he Bank of England o esimae he nominal yield curve. For he pas five years, in common wih many oher cenral banks, we have used he esimaion mehod proposed by Svensson (994, 995). This is a parameric mehod, wih he enire forward curve characerised by a single se of parameers represening he long-run level of ineres raes, he slope of he curve and humps in he curve. Oher compeing models in he lieraure include he more parsimonious funcional form of Nelson and Siegel (987), and he spline-based mehods of McCulloch (97, 975) and Fisher, Nychka and Zervos (995). Esimaion of he real yield curve is a more recen innovaion, made possible by he inroducion of index-linked bonds in he Unied Kingdom in 98. As noed above, addiional problems arise because he value of he cash flows on hese bonds is no known wih cerainy his is because hey are indexed only imperfecly o he price level. We herefore have o use informaion from he nominal yield curve o exrac he real risk-free raes of ineres embodied in heir prices. Unil now he Bank of England has been using an ieraive echnique developed by Deacon and Derry (994) in which he real yield curve is described by a resriced version of Svensson s model. For mauriies of less han wo years, esimaes of boh he real and nominal yield curves have no been hough reliable, and as a resul have no been used by he Bank s Moneary Policy Commiee, nor published in he Inflaion Repor or Quarerly Bullein. This is mainly because here are few gils a he shor end (wih erms o mauriy of wo years or less), where expecaions may be relaively precise, and where he curve may be expeced o have quie a lo of curvaure. This is a paricular problem wih he index-linked gil marke in which here are relaively few observaions across he enire curve. In his paper, we reassess each of hese problems in urn. In he firs case, experience has led us o quesion wheher he Svensson echnique we currenly use provides he bes esimae of he rue yield curve, even a he longer mauriies. We herefore examine four alernaive mehods of esimaion, according o he following crieria: 9
() Smoohness he echnique should give relaively smooh forward curves, raher han rying o fi every daa poin, since he aim is o supply a marke expecaion for moneary policy purposes, raher han a precise pricing of all bonds in he marke. Noneheless, subjec o he former, a beer fi o he daa would be preferred. (2) Flexibiliy he echnique should be sufficienly flexible o capure movemens in he underlying erm srucure. More flexibiliy is likely o be needed a shorer mauriies (where expecaions are beer informed and more subjec o revision as news reaches he marke) han a he longer end. (3) Sabiliy esimaes of he yield curve a any paricular mauriy should be sable in he sense ha small changes in he daa a one mauriy (such as a he long end) do no have a disproporionae effec on forward raes a oher mauriies. The aim is o find he yield curve model ha will provide us wih he mos reliable and useful esimaes, no only on any paricular day, bu also over ime. Based on hese crieria, we find ha a echnique developed by Waggoner (997), bu modified for he Unied Kingdom, ouperforms he compeing mehods. Having chosen his as our basic model, we hen urn our focus o he shor end of he yield curve, where here is a lack of daa in boh he convenional and index-linked gil markes. The challenge is o invesigae wheher here are alernaive sources of daa ha can feasibly be included o help fill he gaps. In he case of he real yield curve, here is very lile we can do index-linked gils are he only direc source of real ineres rae daa, a leas in he Unied Kingdom. For nominal yields, however, we find ha daa from he GC repo marke can be used o supplemen he daase a he shor end of he convenional gil marke. Finally, we re-examine esimaes of he real yield curve in wo ways. Firs, we use a modified version of he Waggoner echnique o fi a curve o index-linked gil prices. Second, we improve on he ieraive echnique used o exrac real raes of ineres from hese prices, wihin he framework suggesed by Evans (998). This also leads o improved measures of he inflaion erm srucure (ITS), which is he difference beween he real and nominal yield curves. 0
The res of he paper is srucured as follows. In Secion 2 we se ou he four mehods of yield curve esimaion ha form he basis of our comparison. In Secion 3 we describe he resuls of our comparison based on he se of crieria oulined previously. Secion 4 inroduces he use of GC repo raes o improve esimaes a he shor end of he curve, and Secion 5 illusraes how he Waggoner-based curve can be adaped o esimae real and inflaion erm srucures from he prices of index-linked bonds, specifically wihin he framework pu forward by Evans (998). Finally, Secion 6 concludes. 2 Model selecion and overview We invesigae four alernaive mehods for esimaing he erm srucure. Two of hese are parameric: Nelson and Siegel (987) hereafer NS, and Svensson (994, 995) hereafer SV; and wo are based on cubic splines: Fisher, Nychka and Zervos (995) hereafer FNZ, and a modified version of Waggoner s variable roughness penaly model (997) hereafer VRP. An overview of hese models, highlighing he main differences beween hem is given in Table A. Table A: Overview of he four models Parameric models Spline-based models Propery SV NS FNZ VRP Naure of forward rae curve Forward rae is single funcion defined a all mauriies Forward rae is piecewise cubic polynomial wih segmens joined a kno poins No. of parameers 6 4 Approx. 4 depending on number of bonds Objecive funcion Minimise residual sum of squares Minimise residual sum of squares Minimise residual sum of squares plus roughness penaly Minimise residual sum of squares plus roughness penaly Pre-specified parameers None None Number of kno poins Number of kno poins, smoohing funcion Consrains Long-run asympoe Long-run asympoe None None Noice ha he only propery all hese models have in common is ha forward raes are modelled as a funcion of a se of underlying parameers. The models hen differ in erms of he way in which his funcion is specified and he crierion used o derive opimal esimaes of he
parameers. These differences, which are discussed furher below, are paricularly imporan in undersanding he rade-off beween he flexibiliy of he funcional form, and he way in which he resuling yield curve esimaes are made o be smooh across mauriy. 2. Parameric models Parameric models offer a concepually simple and parsimonious descripion of he erm srucure of ineres raes. Smooh yield curve esimaes are ensured by modelling he insananeous forward rae curve as a funcion, f(m,b), of a relaively small vecor of parameers, b. The degree of flexibiliy of he curve is hen largely deermined by he number of parameers o be esimaed. Of he wo curves we esimae, he leas flexible is he NS model. In his case, forward raes are defined as: f m m ( m, ß ) = β + + 0 β exp β 2 exp τ τ τ m (3) so ha here are four parameers o be esimaed, b=(β 0,β,β 2,τ ). These parameers can be inerpreed as being relaed o he long-run level of ineres raes, he shor rae, he slope of he yield curve and a hump in he curve. The SV model hen exends his parameer se o b=(β 0,β,β 2,β 3,τ,τ 2 ), where he addiional parameers can be inerpreed as allowing an addiional hump in he curve. Forward raes are now given by: f m m m ( m, ß ) = β + + + 0 β exp β 2 exp β 3 exp τ τ τ τ 2 τ 2 m m (4) As noed in Table A, boh models are consrained o converge o a consan level. The raionale for his consrain is based on he assumpion ha forward raes reflec expecaions abou fuure shor ineres raes, or equivalenly ha he unbiased expecaions hypohesis holds. I hen seems implausible ha agens will perceive a differen pah for he fuure shor rae in 30 years ime compared wih, say, 25 years. Hence, we should expec o see consan expecaions and forward raes a he long end. To esimae he wo yield curves, he funcional forms can be used via equaions () and (2) o derive a fied value for each bond price, given he se of underlying parameers. The parameers are hen esimaed o 2
minimise an objecive funcion ha compares hese fied values wih observaions from he gil marke. A variey of objecive funcions is available o us; we choose o minimise: X P = N i= P i Π D i i ( ) 2 β (5) where P i is he observed price of he i h bond, D i is is modified duraion and Π i (β) is he fied price. This is approximaely equal o minimising he sum of squared yield residuals (alhough i is much quicker o calculae) and so implies roughly equal yield errors, irrespecive of mauriy. 2.2 Spline-based models The spline-based echniques model forward raes as a piecewise cubic polynomial, wih he segmens joined a so-called kno poins. The coefficiens of he individual polynomials are resriced so ha boh he curve and is firs derivaive are coninuous a all mauriies. Typically, his resuls in 4 parameers (wo more han he number of kno poins) ha have o be esimaed. (3) This approach allows for a much higher degree of flexibiliy han eiher of he wo parameric models, and accouns for he principal advanage of spline-based echniques over parameric mehods. Specifically, he individual curve segmens can move almos independenly of each oher (subjec o he coninuiy consrains), so ha separae regions of he curve are less affeced by movemens in nearby areas. This is in conras o he parameric forms for which a change in he daa a any one poin can affec he enire curve, as esimaes a any mauriy are a funcion of all he parameers o be esimaed. (3) To ensure numerical sabiliy, we choose o represen he splines using basis funcions, which means ha he esimaed parameers canno be relaed in a simple manner o he coefficiens of he polynomials see he appendix for deails. 3
Char : Svensson mehod versus cubic spline (a) (i) Original se of daa poins Daa SV Spline 0 5 0 5 20 25 30 6 5 4 3 2 0 (ii) Change of single daa poin Daa SV Spline 0 5 0 5 20 25 30 6 5 4 3 2 0 (a) The daa used in his example are purely illusraive chosen a random, and hence he curves should no be inerpreed as yield curves. This is clearly illusraed in Char, which shows a simple non-linear leas squares regression o an arbirary se of daa poins, using boh he Svensson funcional form and a cubic spline. (4) When a single daa poin is changed a he long end, he Svensson curve changes dramaically, paricularly a he (4) This is a much simpler problem han esimaing erm srucures from coupon bond prices, bu illusraes he poin more clearly. The spline has been chosen o have he same number of degrees of freedom as he Svensson curve. 4
shor end, whereas he spline moves only slighly o accommodae he new daa, and only a he long end. This is because he cubic spline is much more flexible han he parameric funcional form, and hence is beer able o accommodae differen paerns in he daa. The downside of his resul is ha an unconsrained spline, such as he one used in Char, would be far oo flexible o generae yield curves appropriae for moneary policy purposes. To achieve he required degree of smoohing, we herefore modify he objecive funcion, as in he FNZ and VRP models. As wih he parameric models, he main objecive is o choose he parameers so as o minimise he difference beween acual and fied values, as described by X P in equaion (5). Bu o conrol he rade-off beween goodness-of-fi (flexibiliy) and he smoohness of he curve, a roughness penaly is also included o penalise excessive curvaure (measured by he square of he second derivaive) of he forward curve. The size of his penaly is deermined by a smoohness parameer, λ (m), so ha he modified objecive funcion is: (5) X S = X P + M 2 λ [ ( m ) f ( m )] dm (6) 0 The wo mehods differ according o how he value of he smoohing parameer, λ (m), is chosen and how i varies across mauriy. FNZ assume ha he smoohing parameer is invarian o mauriy bu variable over ime, so ha λ (m)= λ. They hen use a procedure known as generalised cross-validaion o choose λ on a daily basis. In he VRP model, on he oher hand, he smoohing parameer is allowed o vary across mauriy, bu is chosen o be consan over ime, hence λ (m)=λ(m). In his case, as noed in Table A, he smoohing funcion λ(m), has o be pre-chosen. Waggoner chose a hree-iered sep funcion for his smoohing parameer, wih seps a one and en years o mauriy. This was based on he naural (5) Noe ha his modified objecive funcion represens he major difference beween he spline-based echniques considered in his paper and earlier mehods such as McCulloch (97,975). In hese models, he objecive funcion was specified in a similar way o hose for he parameric form, ha is wihou a roughness penaly. The degree of smoohing in he yield curve esimaes was hen deermined by he number and posiioning of he kno poins. As described in he nex secion, hese facors are less of a concern for he wo mehods considered in his paper as smoohing is guaraneed, a leas o some exen, via he roughness penaly described in he main ex. 5
segmenaion of he US marke ino bills, noes and bonds, wih he hree levels chosen roughly o maximise he ou-of-sample goodness-of-fi of he model. Unforunaely, he UK marke canno be naurally divided in he same way. Hence, o use Waggoner s sep funcion approach would have mean choosing five parameers (wo for he mauriies a which here was a sep in he funcion, and hree for he sep levels). We chose insead o define λ(m) as a coninuous funcion of only hree parameers. (6) Following Waggoner, he main crierion for choosing hese parameers was o maximise he ou-of-sample goodness-of-fi averaged over our sample period. However, i was found ha many combinaions of hese parameers gave similar goodness-of-fi measures. We herefore oped for he se of parameers ha corresponded o he highes level of smoohing among hese combinaions. The resuling funcion is ploed alongside Waggoner s sep funcion in Char 2. Noe ha we would no necessarily expec he levels o be he same (alhough hey are close), since he smoohing funcions are opimised for differen markes. Char 2: Smoohing funcions used by Waggoner (US marke) and his paper (UK marke) λ (m) 00,000 0,000,000 00 0 0. Our curve Waggoner 0.0 0 5 0 5 20 25 Mauriy (years) (6) In paricular, we specify he following funcion: log λ(m) = L - (L - S)exp(-m/µ), where L, S and µ are he hree parameers o be esimaed. 6
For boh FNZ and VRP models, we also have o choose he placemen of he kno poins. In principle, his can have a very influenial effec on he resuling yield curve esimaes: in paricular, when here is no roughness penaly, more kno poins imply a less smooh curve wih a greaer in-sample goodness-of-fi. Thus he choice of kno poins in his case deermines he rade-off beween he goodness-of-fi of he curve and he smoohness of he esimaes. Bu when, as in he wo spline-based models considered in his paper, excess curvaure in he curve is penalised, he choice of kno poins is significanly less crucial. However, in choosing heir placemen, we should clearly aim o ensure ha here are sufficien kno poins o allow he curve o reflec he underlying daa. Table B: Sensiiviy of VRP forward raes o he number of kno poins Basis poins 2 year 0 year 20 year Average difference from knos a mauriy of every hird bond Knos a mauriy of every bond 0.0-0.03 0.08 Knos a mauriy of every sixh bond 0.07-0.04.58 Max. absolue difference from knos a mauriy of every hird bond Knos a mauriy of every bond 0.38 0.22 0.29 Knos a mauriy of every sixh bond 0.99 0.60 7.06 FNZ found ha placing knos a he mauriy of roughly every hird bond gave he same resuls as placing a kno a he mauriy of every bond, bu ook considerably less compuaional ime. We adop a similar rule. As he resuls in Table B show for he VRP mehod, increasing he number of kno poins o every bond would have a negligible effec on he insananeous forward rae curve. Halving he number of kno poins, however, produces slighly differen resuls, paricularly a he long end. In fac, he long end of he curve is always higher han our defaul case, suggesing ha i is insufficienly flexible o capure any downward slope a longer mauriies. By placing a kno a he mauriy of every hird bond we ensure ha he degree of smoohing is deermined by he roughness penaly, and no he number of kno poins. Inuiively, here are a number of reasons o suspec ha he Waggoner curve will provide us wih more reliable esimaes of he yield curve. Firs, by consraining he smoohing parameer o be mauriy-invarian, he FNZ curve assumes ha here is he same degree of curvaure along he lengh of he erm srucure. Bu here are srong reasons o believe ha his is no he case. In paricular, invesors are likely o be more informed abou he 7
precise pah of he erm srucure a shorer and medium-erm mauriies (when ineres raes are deermined by moneary policy and business cycle condiions) han a longer mauriies. Thus he curve may be oo siff a he shor end and/or oo flexible a he long end. Furhermore, by allowing he smoohing parameer o vary over ime, FNZ esimaes may be unsable in he sense ha changes are driven no only by he underlying daa, bu also by movemens in his parameer. 3 Esimaion and resuls We esimaed each of he four curves over he sample period from May 996 o 3 December 998. (7) In his secion, we compare he resuls of his esimaion in wo ways. We begin by examining how well each mehod appears o fi he daa, focusing in paricular on he rade-off beween he smoohness of he yield curve esimaes and heir abiliy o fi he daa. We hen provide a rigorous assessmen of he sabiliy of he esimaes by looking a he condiion numbers associaed wih each mehod, as suggesed by Waggoner (996). 3. Smoohness and flexibiliy In he previous secion, we described how here was likely o be a rade-off beween he flexibiliy of he four yield curve models and he way in which esimaes were made o be smooh across mauriy. In his secion, we aemp o make hese ideas more concree by assessing how well each mehod capures he shape of he underlying erm srucure and fis he daa. We do his in wo ways: firs, we examine saisics for he goodness-of-fi of he four mehods; and second, we compare he shape of he esimaed curves wih daa from he srips marke. As Bliss (997) noes, he appropriae measure of goodness-of-fi is an ou-of-sample saisic. Each mehod will produce a high in-sample fi, bu his may no be indicaive of he underlying erm srucure. The imporan es is o see wheher or no he esimaed curve can accuraely price a bond which has no been used o esimae he curve. Bliss calculaes an (7) The beginning of he sample period is chosen o coincide wih changes o he axaion of coupon and capial gains income from gils. These changes effecively eliminaed he disorion which had previously exised as a resul of he differenial reamen of he wo ypes of income. See Anderson e al (994) for furher deail. 8
ou-of-sample saisic by fiing he curves using alernaive bonds and hen examining he residuals relaing o he bonds ha were no used in he esimaion. This approach, however, is unsuiable for he Unied Kingdom since we have a relaively small number of securiies in he se of daa used o esimae he curve (ypically 30-40). Hence, we adop insead an approach known as leave one ou cross-validaion (Davison and Hinkley (997)). This involves esimaing he curves many imes, leaving ou a differen bond on each occasion. The cross-validaion saisic is hen he average absolue residual of he bonds omied from he fi. The resuls are presened in Table C. Table C: Ou-of-sample goodness of fi saisics ( ) Mean Sd dev SV 0.090 0.024 NS 0.0 0.025 VRP 0.088 0.024 FNZ 0.6 0.043 We see ha he average price errors are small for all mehods (beween 9 and pence), alhough he NS and FNZ curves boh have a slighly worse fi han he Svensson mehod. The fi of he VRP and Svensson curves is almos idenical, wih he former appearing o perform slighly beer. On he basis of hese saisics, herefore, we would be indifferen beween he Svensson and VRP mehods. (8) Bu now consider he shape of he yield curves. In heory, we should be able o obain a direc reading on he rue shape of he underlying erm srucure using observaions from he srips marke. Char 3 compares each of he esimaed curves wih he yields on srips on a day chosen a random, 2 July 998. (8) We should noe, of course, ha ou-of-sample goodness-of-fi was acually used as a crierion for choosing he opimal smoohing parameers in he case of he VRP curve. Hence, we would expec i o do well in his es. However, i is worh noing ha when we calculaed he opimal smoohing parameers over a variey of differen sample periods we found hem o be very sable. We would herefore no expec o observe significanly differen resuls were we o conduc an enirely ou-of-sample es for he VRP mehod, ie by choosing he smoohing parameers on he basis of daa no used o calculae he es saisics in Table C. 9
Char 3: Comparison of all mehods wih srips prices (2 July 998) Per cen 7.5 7 6.5 SV NS VRP FNZ Srips 6 5.5 5 0 5 0 5 20 25 30 Mauriy (yr) The srips yields clearly indicae a downward-sloping erm srucure a he long end. And his shape is capured by boh of he spline-based models. As noed previously, however, boh he Nelson and Siegel and Svensson funcional forms are consrained o lie fla a longer mauriies hence here is some divergence beween hese esimaes and he srips prices hemselves. In he conex of our earlier discussion, one inerpreaion of his resul is ha, assuming ha expecaions abou fuure ineres raes do converge a longer mauriies, he unbiased expecaions hypohesis does no hold. In oher words, srips prices are deermined by facors oher han jus expecaions abou fuure ineres raes. Candidae explanaions include he presence of risk premia and convexiy erms (see, for example, Brown and Schaefer (2000)). Of course, he srips may be mispriced (alhough, noe ha since he srips are no included in he daase for esimaion, his would no explain how he wo spline-based mehods mach heir shape). Direc evidence from he gil marke (in he form of redempion yields), however, suggess ha a downward-sloping yield curve may be jusified, a leas over he mauriy range ha we consider. Moreover, i appears ha forcing he long end of he curve o converge o a consan level can produce a significan amoun of insabiliy in he esimaed yield curve. This is shown in Char 4, where we plo he redempion yields on he en-year benchmark bond and he 20
longes-mauriy bond (wih mauriy of 29 years), ogeher wih 20-year zero-coupon yield esimaes derived using Svensson. Char 4: Time series of redempion yields a he long end Per cen 5.5 5.75% 2009 6% 2028 Svensson 20-year 5.2 4.9 4.6 4.3 Jan Feb Mar Apr May Jun Jul Aug Sep 999 4 This illusraes he fac ha, as he observed bond yields have diverged more and more, he yield curve esimaes have been increasingly unsable. We aribue his o he parameerised naure of he Svensson curve. Esimaes a all mauriies rely on a single se of parameers, of which one is he long-run level, deermined largely by he yield on he longes bond. Bu he increasing divergence of he wo redempion yield series (he series marked 5.75% 2009 and 6% 2028 in Char 4) suggess ha he level of his asympoe is no well-defined, a leas in his mauriy range. As a resul, he asympoe iself is likely o be unsable. Furhermore, as we saw in Char, his volailiy may be ransmied ino esimaes of he enire yield curve. Thus i appears ha he cos of he Svensson curve providing a good fi o he underlying daa is a loss in sabiliy of he curve. 3.2 Sabiliy of he curves This idea of insabiliy is discussed more formally by Waggoner (996). He assesses he sabiliy of he yield curve in he ligh of small price changes in he underlying bond daa. The idea is ha, assuming bond prices are measured wih error, he curves should be sable in he sense ha changes in 2
his measuremen error do no have a significan influence on our esimaes. An obvious source of measuremen error is he fac ha prices are quoed in ineger muliples of he ick size. (9) This means ha here will generally be an error of up o plus or minus half a ick beween he observed price and ha implied by he underlying erm srucure. As a resul, a very small change in he erm srucure could be refleced by a change in he observed bond price of up o half a ick. We refer o an esimaion mehod as being sable if he effec of hese changes on he resuling yield curve is small. Waggoner quanifies his noion of sabiliy by compuing wha is known as a condiion number. This measures he sensiiviy of he oupu of a process o a change in one of is inpus. Suppose, for example, ha for a paricular yield curve model (or process ), when he se of bond prices (he inpu ) changes by 2%, he zero-coupon yield curve (he oupu ) varies by 20%. We say ha his process has a condiion number of en, his being he raio of he oupu variaion o he change in inpu. We are ineresed here in processes (we use he noaion G(x)) which, from a discree vecor of bond prices (x), compue a coninuous yield or forward rae curve (g(m;x), where m is mauriy). To formalise he noion of a change in he vecor of bond prices, x, we need o specify a norm over hese prices. This effecively measures he size of he vecor. We use he sandard Euclidean norm such ha if x=(x,x 2,,x N ) is our vecor of bond prices, hen he norm is defined as: N 2 = x X i i= x (7) Similarly, we need o describe he size of he process, G(x). In his case, he sandard Euclidean norm is no available, as he oupus are curves ha are coninuous raher han discree. Insead, we define wo differen norms (from an infinie family). The firs, he average norm (or L ) is given by: s M m ; d G( x) = g( s x) M m (8) (9) On November 998 he gil ick size was changed from (/32) o 0.0. Since our sample runs from May 996 o 3 December 998, we use he larger value for his analysis. 22
and he second, he max (or L ) norm, is defined by: G ( x) = max g( s; x) (9) [, ] s m M where [m, M] defines he mauriy range of ineres. Condiion numbers relaing o each of hese norms are hen defined respecively as: CN γ ( G, x) = sup 0< ε < γ N X G ( x + e) G( x) G( x ) e X x X (0) and CN γ ( G, x ) = sup 0< ε < γ N X G ( x + e) G( x) G ( x ) e X x X () In equaions (0) and (), he parameer γ represens he maximum perurbaion, which we se o be equal o half he ick size. The sup operaor indicaes ha we ake he larges value of he percenage change in he yield curve over all possible perurbaions (e). Table D: Zero-coupon yield curve condiion numbers CN CN SV NS VRP FNZ SV NS VRP FNZ Median 5.0 2.3.9.6 68.3 6.8 38.2 0.3 90h percenile 27.8 5.4 3.2 3. 524 55.2 83.6 28.6 95h percenile 37.7 6.9 3.6 3.9 996 79.7 95.9 35.3 Maximum 6 02 6.4 62. 500 27900 73 309 Table E: Forward curve condiion numbers CN CN SV NS VRP FNZ SV NS VRP FNZ Median 4.4 5.0 3.7 3.5 8. 8. 37.3.7 90h percenile 0 3.5 5.8 6.7 554 62.2 82.7 28.0 95h percenile 26 9.9 6.6 8.5 964 92.0 95.0 34.8 Maximum 49 70.4 44 58300 27800 73 496 23
Table D and Table E presen summary saisics for he disribuion of CN γ for each ype of norm and for he wo ypes of curve: zero-coupon yield and insananeous forward rae. The figures are calculaed as follows. On each day, he se of bond prices is perurbed by a vecor of random numbers wihin he range [-γ, γ]. For each model and for each ype of curve and norm, he sensiiviy of he esimaes o hese perurbaions are calculaed. This is repeaed seven imes, and he condiion number for ha paricular model is given by he maximum of hese sensiiviies. This is repeaed on a daily basis. Thus, given ha here are 676 days in he sample, he figures repored for each specificaion (model, curve and norm) are based on 4,732 simulaions. (0) To inerpre he absolue levels of he condiion numbers noe ha he average percenage change in he norm of he inpu bond prices was 0.0082% (wih a range of 0.0060% o 0.002%). Hence, a condiion number of 20 roughly corresponds o a % change in he oupu level while a condiion number of 600 corresponds approximaely o a 5% change in oupu. Since ineres raes averaged around 7% during he sample period, i follows ha hese numbers (20 and 600) correspond o shifs of 7 and 35 basis poins respecively in he esimaed erm srucure. Hence, we can broadly characerise condiion numbers of less han 20 or so as indicaing a sable process, and condiion numbers of greaer han abou 600 as characerising an unsable process. On his measure he VRP mehod is sable irrespecive of he norm used o calculae he change in he esimaed erm srucure. Moreover, i is more sable han all he oher mehods. We can also compue condiion numbers for segmens of he mauriy specrum. Table F gives resuls on hese segmens for he L norm calculaed for he forward curve. This confirms ha he VRP mehod is sable across he enire mauriy specrum. All of he oher mehods can occasionally exhibi subsanially larger disorions (as shown by he maxima of he condiion number disribuions) for he same inpu perurbaion. (0) Noe ha, o ensure comparabiliy, idenical perurbed prices are used as inpu for each mehod on any paricular day. 24
Table F: CN condiion numbers (forward curve) by mauriy segmen Range SV NS VRP FNZ 0-2 years Median 2.5 6.6.3 4.9 Maximum 775 825 49.4 68 2-5 years Median 0.8 4.5 3.4 3.7 Maximum 92 70 8.5 49 5-0 years Median 9.4 3.6 2. 2.7 Maximum 339 5 0. 82 0 20 years Median 5.3 5. 2.7 3.3 Maximum 644 272 9.7 80 3.3 Summary To summarise, we have conduced a number of ess, based on he crieria se ou in Secion. A sriking feaure of hese ess is ha no one alone is conclusive, hus demonsraing he poenial rade-off which exiss beween he hree properies we have chosen o focus on: smoohness, flexibiliy and sabiliy. The NS model, for example, appears o be much more sable han he Svensson echnique. Bu, as we saw in Secion 3., his is a he cos of a lower goodness-of-fi. A he same ime, we know ha he spline-based mehod of FNZ is beer able o capure he shape of he underlying erm srucure, as measured by srips, bu again is ou-of-sample goodness-of-fi is worse han ha of Svensson. In all cases, he VRP curve appears o perform well. I is by far he mos sable approach, defined on he basis of our condiion numbers, and is able o capure he shape of he underlying erm srucure, as measured by srips. The curves are also smooh across mauriy, while providing ou-of-sample goodness-of-fi resuls ha are a leas as good as each of he oher compeing mehods. The leas sable mehod appears o be he Svensson curve. As we have seen, one explanaion for his resul is ha, while i is fairly flexible, he fac ha i is consrained o lie fla a he long end means ha i is in some way inherenly unsable, a leas when his consrain is no saisfied by he underlying daa. Inuiively, i is unlikely ha hese resuls will change when we include addiional daa a he shor end, or apply similar crieria o esimaes of he real yield curve from index-linked gils. Problems highlighed wih he Svensson curve, for example, appear o be relaed o he parameric naure of he funcional form and he consrain applied a he long end, raher han 25
o specific problems in fiing daa from he convenional gil marke. For compleeness, however, we compare our VRP esimaes in wha follows wih resuls derived using he Svensson mehod. 4 Improving esimaes of he shor end of he yield curve In his secion, we focus on he shor end of he yield curve. As menioned in he inroducion, we expec he shor end of he yield curve o exhibi he greaes amoun of srucure expecaions of he fuure pah of ineres raes are beer informed a shorer mauriies, and are more likely o respond o shor-erm news. This can be confirmed by considering he forward curve derived, for example, from he price of shor serling fuures conracs. Ye a he shor end of he yield curve, here are relaively few gils in relaion o he expeced srucure in he shape of he curve, he sampling frequency is oo low. The quesion is wheher or no here are alernaive sources of daa ha we can use o supplemen observaions from he convenional gil marke a hese mauriies. 4. Choice of addiional daa By using gil prices o esimae he yield curve we aim o measure he risk-free (or defaul-free) erm srucure of ineres raes. Thus, alhough here is a wide range of shor-erm insrumens raded in he UK money marke, heir prices are no consisen wih gil prices as hey generally include a credi risk premium. As a resul, yield curves esimaed using asses of mixed credi raing would be inernally inconsisen. Even for Libor (where he paricipaing banks all have high credi raing), his premium can be subsanial (abou 25-40 basis poins for hree-monh Libor). Moreover, since he premium is likely o be boh ime-varying and a funcion of mauriy, we canno simply subrac a consan value o obain he risk-free rae. The requiremen ha daa used for esimaing he yield curve are (virually) defaul-free leaves us wih a choice of wo possible insrumens: reasury bills (T-bills) and general collaeral (GC) repo raes. Treasury bills are shor-erm zero-coupon bonds ha are issued by he governmen, and herefore have he same risk-free naure as gils. The T-bill discoun rae is, however, generally acceped as being unrepresenaive of he underlying fundamenal rae deermined by expecaions. This is because commercial 26
banks use T-bills for cash managemen purposes, and heir prices are largely deermined by he banks liquidiy requiremens. A GC repo agreemen is equivalen o a secured loan, and hus he credi risk is much lower han on unsecured Libor. In addiion, he repo is marked-o-marke on a daily basis, hereby limiing he exposure of eiher pary o large moves in he value of he collaeral. The risk premium is furher reduced since he collaeral comprises gils or similar insrumens, for which here is virually no chance ha he issuer will defaul during he erm of he repo. GC repo herefore provides us wih he only widely raded, virually riskless insrumen. The mauriies of GC repo conracs range from overnigh o as long as one year. The shores raes are quie volaile, however, and he longer raes are raded oo infrequenly for heir prices o be considered reliable. Hence we use one-week, wo-week, one-monh, wo-monh, hree-monh and six-monh repo raes in our esimaion. Since he repo rae is deermined by he difference beween he sale and re-purchase prices of he collaeral, here is no raded asse whose price is a direc reflecion of he rae. We herefore creae synheic zero-coupon bonds whose mauriy and yield mach he corresponding repo rae, and for which he erminal cash flow is 00, so ha repo raes are given exacly he same weigh in he esimaion as a shor-daed gil. As a resul of adding in he exra daa, we found i necessary o place kno poins a he mauriy of every securiy included in he esimaion process, raher han every hird, as discussed above. The addiional srucure implied by he repo raes could no oherwise be capured reliably; wih fewer kno poins he effecive siffness of he curve was being conrolled by he disribuion of he kno poins, and no, as required, wholly by he roughness penaly. 4.2 Resuls of incorporaing GC repo raes The effec of including GC repo raes can be seen mos clearly by focusing on a single day. Char 5 shows he shor end of he zero-coupon yield curve on a randomly chosen day esimaed using boh Svensson and VRP mehods, wih and wihou he inclusion of GC repo raes. Clearly, wihou he GC repo raes, he iniial upward slope of he yield curve was being missed by boh echniques. Bu we also see ha he VRP mehod is beer 27
able o capure he rue shape of he curve. In paricular noe ha he Svensson curve is significanly alered by he inclusion of GC repo raes a mauriies of around one year, alhough he longes repo rae has a ime o mauriy of only six monhs. Char 5: Shor end of zero-coupon yield curve on 9 June 998 Per cen 7.6 7.5 7.4 7.3 7.2 7. 7 VRP wih repo VRP, gils only Repo Srips SV wih repo SV gils only 6.9 0 0.2 0.4 0.6 0.8.2.4.6.8 2 Mauriy (yr) Indeed, he Svensson esimaes of he forward curve are significanly alered even a mauriies of 20 years or more (see Char 6). In conras, VRP esimaes of he one-year forward rae are very similar wih and wihou repo raes, and a wo years and longer are indisinguishable. This is imporan as i indicaes ha, even if here is reason o doub he reliabiliy of he GC repo daa, or if hese are no available, () we can sill have confidence in he VRP esimaes a longer mauriies. Once again his difference beween he mehods is a consequence of he relaive behaviour of spline and parameric curves illusraed in Secion 2.2 esimaes using parameric mehods can be significanly changed across he whole curve by small aleraions o he daa in one small region of he curve. () For example, he repo marke effecively ceased rading for a couple of weeks around he end of 999. 28
Char 6: Differences in 20-year insananeous forward raes, esimaed using SV and VRP, wih and wihou GC repo raes Basis Poins 60 50 SV VRP 40 30 20 0 Mar-97 Jun-97 Sep-97 Dec-97 Mar-98 Jun-98 Sep-98 Dec-98 0-0 -20-30 -40 To ensure ha including GC repo raes had no derimenal effec on he performance of he VRP mehod, we repeaed he ess described in he previous secion. Firs, we re-esimaed condiion numbers, reaing he repo raes as bonds o be perurbed in he same manner as gils. (2) We found ha he inclusion of repo raes does no in any way impair he performance of he mehod on his measure: he curve remains sable irrespecive of he norm, curve ype (forward or yield) or mauriy segmen. We hen re-calculaed he ou-of-sample goodness-of-fi of he VRP esimaes: a comparison of resuls, wih and wihou he inclusion of he GC repo daa, is given in Table G. This shows ha he ou-of-sample goodness-of-fi is acually improved subsanially when repo raes are included. Inuiively, his is because he grealy reduced mauriy spacing a he shor end means ha he abiliy of he fied curve o move when a repo rae is omied from he fi (and hus give a poor ou-of-sample fi) is curailed by he reduced disance o he neighbouring raes. (2) Alhough marke paricipans can in principle specify any value for repo raes, in pracice he minimum separaion beween observed repo raes is 0.005%. This is equivalen o he ick size for bonds. Thus we perurb each repo rae by a random amoun drawn from a random disribuion, cenred on zero and wih a half-widh of 0.0025%. This perurbed rae is hen convered o a perurbed price for he esimaion procedure. In his way, he repo raes are reaed in an idenical fashion o he gil prices. 29
Table G: VRP ou-of-sample goodness-of-fi ( ) Mean Sandard deviaion Gil only 0.088 0.024 Wih GC repo daa 0.04 0.006 5 Esimaing he real and implied inflaion erm srucures In his secion, we focus on he paricular problems associaed wih esimaing he real yield curve and erm srucure of inflaion expecaions, using he prices of boh convenional and index-linked gils (IGs). If paymens from IGs were perfecly indexed, and hence were fixed in real erms, we could esimae he real erm srucure in much he same way as we derive he nominal yield curve. Bu in pracice, IG cash flows are indexed o he level of he reail price index prevailing eigh monhs previously. This means ha coupon paymens from IGs ha occur wihin he nex eigh monhs are known exacly in nominal erms, bu are uncerain in real erms. The price of he bond is, herefore, a complicaed funcion of boh real and nominal ineres raes, and hence, in addiion o a curve-fiing mehodology, we need a heoreical framework o enable us o disenangle he real yield from he prices of boh convenional and index-linked gils. Unil recenly, he Bank of England used a mehod proposed by Deacon and Derry (994), referred o as he inflaion erm srucure or ITS model. To implemen his echnique, an iniial assumpion is made abou marke expecaions of fuure inflaion. Using his assumpion, a real forward rae curve is hen fied o he prices of he index-linked gils. (3) By comparing he real forward raes wih a nominal ineres rae curve (derived from he prices of convenional gils using he Svensson mehod described above), a revised esimae of he inflaion erm srucure can be derived, (4) which in urn is used o re-esimae he real forward curve. This process is repeaed unil he real forward rae curve converges. (3) In pracice, a very simple parameric form is used o fi he forward curve, namely f(m)=β 0+β exp(-m/τ). This is basically a runcaed version of he Svensson funcional form, using only he firs wo erms (consan and exponenial decay). The idea of using a simpler funcional form is o preven overfiing given he small number of IGs. (4) We implicily assume ha here is no inflaion risk premium so ha nominal forward raes equal he sum of real raes and inflaion expecaions. 30
The aim in his secion is o describe a new approach o esimaing he real yield curve and inflaion erm srucure. We have already seen he advanages offered by a spline-based echnique over parameric approaches. Recen work by Evans (998) has furher provided us wih a more elegan and ransparen framework for dealing wih he indexaion lag. The mehod described below incorporaes boh of hese advances, ogeher wih some modificaions o Evans approach necessary o allow us o esimae he real and inflaion erm srucures on a daily basis. 5. Evans heoreical framework Evans main innovaion was o make he relaionship beween IG prices, real yields and nominal ineres raes explici by inroducing he noion of an index-linked discoun funcion. To undersand his concep, we firs need o develop some noaion. In paricular, le M denoe he nominal sochasic discoun facor or pricing kernel. This is defined so ha he curren value, V, of a fuure nominal cash flow, V +h, payable a ime +h, is given by: [ M V ] V = E + h + h (2) Now le Q (h) denoe he ime price of a nominal zero-coupon bond, paying a cerain a ime +h. According o equaion (2), his is given by: ( h) = E [ M.] Q + h (3) Similarly he nominal price of a real (perfecly indexed) bond which pays (P +h /P ) a ime +h, where P is he general price level, is given by: Q * P + h ( h) = E M + h P (4) Bu, as discussed above, IGs are no perfecly indexed cash flows are linked o he price level wih a lag of lengh l. Hence, we also define he price of a zero-coupon index-linked bond as: Q + P + h l ( h) = E M + h P 3 (5)
Given hese definiions, i is reasonably sraighforward o show ha here is a unique relaionship beween he yields on he hree ypes of bond: nominal, real and index-linked (see Evans (998) for furher deails). Denoing hese by y, y * and y + respecively: * h + γ τ y ( τ ) = y ( h) [ hy ( h) τy ( τ )] +, (6) τ τ τ where τ =h-l, and γ (τ) is a risk premium erm, which we assume in wha follows o be equal o zero. This assumpion is suppored by Evans who found ha he risk premium erm conribued only approximaely.5 basis poins o he annualised real yield. All ha remains o do is o describe he relaionship beween he heoreical index-linked zero-coupon bonds and he prices of IGs ha we observe in pracice. So le Q c+ (H) denoe he ime price of an IG, mauring a ime H, and paying an annual (real) coupon rae of c. Assuming ha is price is equal o he presen discouned value of is fuure cash flows, we can wrie: ( ) Q c+ c l + h l + + ( H) = I ( h) Q ( h) + I( h) Q ( h) + Q ( H) 2 h= 0 P P i c P 2 P H i h= l P P i (7) Here I (h) is an indicaor funcion, which is equal o one if a cash flow occurs a ime h, and zero oherwise. Coupon paymens are indexed from a base price level, P i, which is se when he bond is issued. Noe ha he nominal values of cash flows occurring wihin he nex eigh monhs (he indexaion lag) are known, and are herefore discouned a he nominal rae. 5.2 Two modificaions In his paper, Evans esimaed he zero-coupon nominal and index-linked yield curves from he prices of convenional and index-linked gils respecively, and hen used equaion (6) o back ou he real yield curve. Bu by implemening his approach he implicily assumed ha he curren price level is known a all imes while, in pracice, his is never he case. (5) (5) Evans did some ess using a cerainy equivalen value of he price level, which he esimaed as an addiional parameer. He found, using end-of-monh daa, ha his was an unnecessary refinemen. We wish o esimae he curves on a daily basis, however, which means we have o ake he publicaion schedule of he RPI ino accoun. 32
The reail prices index (RPI), o which IGs are indexed, is published monhly, and he daa indicae he level of he index on he las day of he monh. This creaes wo problems: Since he daa are published wih an approximae wo-week delay, he curren price level, P, is never known wih cerainy a any ime. Cash flows are indexed o he level of he RPI prevailing eigh monhs previously. Bu monhs are no equal in lengh; moreover, wheher a cash flow is due on November or 30 November, say, i will be uplifed wih reference o he same RPI value (ha prevailing he previous March). Thus he indexaion lag, l, does no have a consan value. Boh of hese problems are bes illusraed wih acual examples. In he firs case, suppose ha corresponds o 29 February 2000. The price level P is no known; he laes RPI value available corresponds o 3 January 2000. Moreover, his remains he laes value unil mid-march 2000. Hence, if we le denoe he dae of he laes RPI value (in his case 3 January 2000), hen here can be up o around six weeks beween and. To accoun for his, we re-define he index-linked discoun facor wih respec o he laes known RPI value, such ha: Qˆ + P + h l ( h) = E M + h P (8) + Denoing he yield on his bond by yˆ ( h), equaion (6) hen becomes: y * h τ + ln( P ) ) ˆ τ ( τ) = y ( h) [ hy ( h) τy ( τ) ] + ln( P τ + ( ) γ τ (9) In his way we have made explici he fac ha we need o know P, and alhough we sill have o make an assumpion abou is value, we can now see exacly how ha assumpion impacs on our esimaed yields. One simple approach is o base he curren price level, P, on he laes available value, P, increasing i by he curren annual rae of inflaion, such ha: 33