Modelling the term structure of Japanese bond yields with the Nelson-Siegel model

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1 Edih Cowan Universiy Research Online ECU Publicaions Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model A. Tsui J.X. Wu Zhaoyong Zhang Edih Cowan Universiy, This aricle was originally published as: Tsui, A., Wu, J., & Zhang, Z. (2013). Modelling he erm srucure of Japanese bond yields wih he Nelson- Siegel model. Proceedings of Inernaional Congress on Modelling and Simulaion (pp ). Adelaide, Souh Ausralia. Modelling and Simulaion Sociey of Ausralia and New Zealand. Original aricle available here This Conference Proceeding is posed a Research Online. hp://ro.ecu.edu.au/ecuworks2013/339

2 20h Inernaional Congress on Modelling and Simulaion, Adelaide, Ausralia, 1 6 December Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model A.K. Tsui a, J. X. Wu a and Z.Y. Zhang b a Deparmen of Economics, Naional Universiy of Singapore, Singapore b School of Business, Edih Cowan Universiy, Ausralia zhaoyong.zhang@ecu.edu.au Absrac: The Nelson-Siegel (1987) (NS) model has been credied for is high efficacy in he in-sample fiing and ou-of-sample forecasing of he erm srucures of ineres raes. The erm srucure of ineres raes, popularly known as he yield curve, is a saic funcion ha relaes he ime-o-mauriy o he yield-omauriy for a sample of bonds a a given poin in ime. The convenional way of measuring he erm srucure is by means of he spo rae curve, or yield curve, on zero-coupon bonds. Ye in realiy, he enire erm srucure is no direcly observable, which gives rise o he need o esimae i using several approximaion echniques. Over he las hree decades, various mehods o esimae erm srucures from bond prices have been proposed. In recen years mos of he exising sudies (as well as major cenral banks around he globe) have been employing he class of NS models o esimae and consruc zero-coupon yield curves. This paper aims o sudy he erm srucure of he Japanese bond yields by employing he NS model vs oher non-ns models using five differen ses of zero-coupon bond yield raes daa obained from he Bank of Japan covering he period spanning from January 2000 o November This period has been chosen because i clearly exhibis he liquidiy rap problem, which forces all bond yields o remain close o zero for an exended period. We propose 18 differen NS models, each wih differen decay componens and ime series appendages, agains 14 oher non-ns models ranging from he simple random-walk model o complicaed specificaions like he VAR and VECM models. A h-period(s)-ahead ou-of-sample expanding window forecas is conduced for each of hese 32 differen models, using daily, weekly and monhly bond yields of 15 differen mauriies. This sudy has demonsraed ha due o he presence of liquidiy rap in Japan, ou-of-sample expanding window forecass in general perform inferiorly vis-à-vis oher non-ns models, and his is coupled wih he oher problem of obaining negaive yield forecass for bonds wih shorer mauriies. Moreover, he resuls show ha he NS class of models can be useful in forecasing shorer horizons like weeks and days, works beer wih a decay rae oher han he convenional way of reaing i as he value ha maximizes he loading on he medium-erm facor a exacly 30 monhs, and can work well wih ime series models such as GARCH and EGARCH in erms of volailiy forecasing. I is also found ha, when he NS models are used for yield forecass, he NS-VAR model should be considered since i is up o par agains he compeior models, even wih liquidiy rap a work. Keywords: In-sample fiing and ou-of-sample forecasing; Japanese bond yields; he Nelson-Siegel (NS) model 1392

3 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model 1. INTRODUCTION The erm srucure of ineres raes, popularly known as he yield curve, is a saic funcion ha relaes he ime-o-mauriy o he yield-o-mauriy for a sample of bonds a a given poin in ime. The convenional way of measuring he erm srucure is by means of he spo rae curve, or yield curve, on zero-coupon bonds. Ye in realiy, he enire erm srucure is no direcly observable, which gives rise o he need o esimae i using several approximaion echniques. There are a wide variey of diverse yield models in he lieraure. The major goal of hese models is o describe he fuure yield curve srucure as bes as possible. However, he modelling of a yield curve is more complicaed han he modelling of a sock derivaive, and he models for he yield curve srucure alone are more complex as well. In recen years, he Nelson-Siegel (1987) (NS) model and is exended models have been credied for is high efficacy in he in-sample fiing and ou-ofsample forecasing of he erm srucures of ineres raes. Many exising sudies (as well as some major cenral banks around he globe) have been employing he class of NS models including he Svensson- Soderlind model o esimae and consruc zero-coupon yield curves. The primary goal of his paper is o invesigae he Nelson-Siegel and he exended NS models on heir forecasing performances, and o examine he erm srucure of he Japanese bond yields by employing he NS model vs oher non-ns models using five differen ses of zero-coupon bond yield raes daa obained from he Bank of Japan covering he period spanning from January 2000 o November In paricular, we propose 18 differen NS models, each wih differen decay componens and ime series appendages, agains 14 oher non-ns models ranging from he simple random-walk model o complicaed specificaions like he VAR and VECM models. A h-period(s)-ahead ou-of-sample expanding window forecas is conduced for each of hese 32 differen models, using daily, weekly and monhly bond yields of 15 differen mauriies. The sample period was chosen because i clearly exhibis he liquidiy rap problem, which forces all bond yields o remain close o zero for an exended period. This sudy has demonsraed ha due o he presence of liquidiy rap in Japan, ou-of-sample expanding window forecass in general perform inferiorly vis-à-vis oher non-ns models, and his is coupled wih he oher problem of obaining negaive yield forecass for bonds wih shorer mauriies. Moreover, he resuls show ha he NS class of models can be useful in forecasing shorer horizons like weeks and days, works beer wih a decay rae oher han he convenional way of reaing i as he value ha maximizes he loading on he medium-erm facor a exacly 30 monhs, and can work well wih ime series models such as GARCH and EGARCH in erms of volailiy forecasing. I is also found ha, when he NS models are used for yield forecass, he NS-VAR model should be considered since i is up o par agains he compeior models, even wih liquidiy rap a work. The res of his sudy is organized as follows. Secion 2 discusses he NS model and he exended NS models as well as he mehodology used in his sudy. Secion 3 analyzes he daa ses, namely, he daily, weekly and monhly spo raes o be used in his sudy, and evaluaes he various forecas resuls. The las secion concludes wih implicaion drawn from our findings. 2. METHODOLOGY AND THE MODEL We discuss he Nelson and Siegel model in his secion. Assume ha spo raes are obained from a secondorder differenial equaion wih real and unequal roos. The insananeous forward rae funcion f a mauriy is defined as: / 1, / 2, f b b e b e (1) 1, 2, 3, Where 1, and 2, are ime consans associaed wih he equaion, and he parameers 1,, 2, and are deermined by he iniial condiions. Ye, his model was deemed o be over-parameerized by 3, Nelson and Siegel (1987), and having oo many parameers makes i difficul for any sandard nonlinear esimaion sofware o converge. For his reason, equaion (1) was furher sreamlined by seing 1, 2, and adding o he facor loading on b 3,, hereby ransforming he forward rae curve ino he following form: 1393

4 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model f e e / / 1, 2, 3, (2) Wih his ransformaion, he forward rae curve can now be seen as a consan plus a Laguerre funcion, which is basically a produc of a polynomial and an exponenial decay erm. Inuiively, his paricular forward rae curve can be assumed o be he soluion o a second-order differenial equaion wih equal roos for spo raes. By aking he average of forward raes over differen mauriies, he corresponding spo rae or yield curve can hen be obained: y / / 1e 1e / 1, 2, 3, e / / (3) As he NS yield curve corresponds o a discoun curve ha begins a one a zero mauriy and approaches zero a infinie mauriy, following Fabozzi e al. (2005), Diebold and Li (2006) and Diebold, Rudebusch, and Aruoba (2006b), we rea as a muliplier and hence ransform equaions (2) and (3) ino: f e e (4) 1, 2, 3, y 1e 1e e 1, 2, 3, (5) which are used for all in-sample fiing and ou-of-sample forecasing exercises in he sudy. In he NS model, he exponenial decay rae ( ) conrols he speed of decay for he NS yield funcion y. Theoreically speaking, a smaller value of is supposed o produce slow decay and can hus beer fi he yield curve a long mauriies; whereas a greaer exhibis he direc opposie of producing an acceleraed decay which resuls in he beer fiing of he curve a shor mauriies. However, a challenge faced is how o choose a suiable decay rae * for each single poin in ime. As regards he hree coefficiens 1,, 2, and 3,, which are called he laen level, slope and curvaure facors in Diebold and Li (2006), hey each have heir own idiosyncraic rais. The long-erm facor, 1,, governs he yield curve level since an increase in his coefficien raises all shor- and long-erm yields equally, hereby changing he level of he yield curve. 2, relaes o he shor-erm facor and is closely associaed wih he yield curve slope. 3, is closely linked o he yield curve curvaure and i may be viewed as a medium-erm facor, akin o is loading 1 e e. The NS model was lae exended by researchers including Lierman and Scheinkman (1991), Björk and Chrisensen (1999), Bliss (1997) and Svensson (1994) o explore more flexible NS specificaions, eiher hrough he use of addiional facors, furher decay parameers, or by a composie of boh. Some of hese exensions have been esed in oher papers, hough he resuls were mixed. I is generally found ha exensions like Lierman-Scheinkman s (1991) wo-facor model are overly simplisic and yield inaccurae resuls, whereas models such as he Svensson exension are currenly welcomed by major banks around he globe. We now urn o he discussion of general specificaion of he models. Le Y be a N 1vecor of observed spo raes or yields which is dependen on N differen mauriies, such ha Y y1 y N, a any one poin in ime. In order o faciliae erm srucure forecasing, we incorporae dynamics o creae imeseries models for he colleced facor esimaes, and hen assess he models performance. Models seleced for he facor forecas include RW, AR(p), VAR(p), GARCH and EGARCH specificaions. Given he 1394

5 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model voluminous lieraure on ARCH, models buil around his concep have become an indispensable ool for financial analyss, bankers and fund managers hroughou he world. McAleer (2005) gives a recen exposiion of a wide range of univariae and mulivariae, condiional and sochasic models of volailiy, and oher recen surveys have examined mulivariae exensions of he original GARCH framework (Bauwens e al., 2006; Maasoumi and McAleer, 2006; and Caporin and McAleer, 2012). Unlike Diebold and Li (2006) and de Pooer (2007) who convenienly assumed firs orders for all heir ime series models, in his sudy we adop a more cauious approach by firs finding he respecive opimum orders wih he help of various informaion crieria. Thus, we define he general specificaion of he NS model as follows: Y X (6) (7) 1 where X is a N K marix of facor loadings which are poenially ime-varying if he decay parameer(s) are esimaed ogeher wih he facors, is a K 1 vecor of facors and is order of inegraion is assumed o be I 0a his poin in ime, and is a N 1vecor of esimaion errors and each componen wihin i is assumed o be independen across mauriies and have differen variance 2 erms. is a K 1 vecor, while is a K K square marix, and is also a K 1 vecor of residuals for he ime series model. I is also assumed ha he measuremen equaion and sae equaion error vecors are boh orhogonal and normally disribued, such ha: 0N 1 H 0 ~, Ν 0 K1 0 Q (8) where H represens a diagonal N N measuremen equaion covariance marix, and Q denoes a sae equaion covariance marix, which may be assumed o be eiher a diagonal K K marix or a full marix, depending on he esimaion procedure. For insance, Q (and also ) are diagonal if we consider separae AR(p) models for each facor; on he conrary, Q (and ) are full marices when a join VAR(p) esimaion is carried ou insead. The NS GARCH (1,1) model used for forecas is specified as follows: ˆ ˆ 1e ˆ 1e yˆ h 1, h 2, h 3, h e (9) 2 i, h i h h, h i, ~ Ν 0,, 1,2,3 (10) (11) 2 2 2, h h1, h1 And he EGARCH(1,1,1) wih AR(1) in he mean equaion can be expressed as follows when combined wih he NS model: ˆ ˆ 1e ˆ 1e yˆ h 1, h 2, h 3, h e 2 i, hiii, h1 h, h~ Ν 0,, h, i 1,2,3 (13) (12) 2 2 h ln, h ln, h1 1 h1 h 1 h 1 (14) 1395

6 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model 3. EMPIRICAL ANALYSIS Figure 1: Japanese Zero-Coupon Yields for Various Mauriies for Monhly Daa 3.1 Daa Descripion For he purpose of in-sample fiing and ou-ofsample forecasing, we colleced five differen ses of zero-coupon bond yield raes daa from he Bank of Japan archives, including daily yield (1888 observaions), weekly yield (408), and monhly yield (94), spanning almos eigh years from January 2000 o November This period was chosen because i clearly exhibis he liquidiy rap problem, which forces all bond yields o remain close o zero for an exended period. Similar o he Unied Saes, Japan applies a paricular varian of he smoohing splines mehod in he esimaion of zero-coupon yield raes. I means ha he insananeous forward rae curves, expressed as a linear combinaion of cubic B-splines, are consruced from price quoes on seleced risk-free fixed income asses: 3-, 6-, 120- and 240-monh bonds. Following ha, he forward rae curves are inerpolaed by using smoohing splines, afer which he spo raes can hen be compued by aking he average over he forward raes. The purpose for us o use he daily and weekly yield raes is o es he robusness of he NS model a he daily and weekly levels. Also, uilizing yield raes on daily and weekly bases allow for a larger sample, which in urn makes he esimaes more saionary. We also pool all he daa ses ino fifeen fixed mauriies, i.e., 3, 6, 12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 180, 240, and 360 monhs, which will grealy simplifies our esimaion and forecasing sequences. To save space, we show in Figure 1 he ime-series of zero-coupon yield curves for 3-, 12-, 60-, 120- and 360-monh bonds, based on monhly raw daa from January 2000 o November Two observaions can be noiced: here is a low-yield period beween 24 o 36 monhs for all ypes of bonds from January 2000; and bonds wih mauriies of 12 monhs or less have yields close o zero from 12 monhs onwards unil 72 monhs. The former characerisics may sugges a possible srucural break, corresponding o he privaizaion of he Posal Savings Sysem as well as a complee overhaul of he exising financial srucure in Japan. The laer is a classic example of liquidiy rap. Table 1 repors he summary of he descripive saisics for monhly yields from January 2000 o November As i can be seen in Table 1, he mean values confirm ha he average yield curve is characerized as upward sloping and concave, and he sandard deviaion reveals ha i is mos volaile for medium and long-erm mauriies, bu more sable for he shorer-erm mauriies. One possible explanaion is ha he presence of liquidiy rap forces shor-erm yields o converge owards zero, hereby sabilizing i. I is also noed ha sample serial auocorrelaions a a displacemen of 1 monh for all mauriies, as well as pairwise correlaions beween yields ha have close mauriies are exremely high. In addiion, he descripive saisics on daily and weekly daa lead o more or less he same conclusions, excep for he fac ha serial auocorrelaions remained very high even up o a displacemen of 24 monhs for all mauriies, and he Jarque-Bera probabiliy converges o zero as he number of observaions ges larger. 3.2 Empirical Resuls Table 1: Descripive Saisics for Monhly Daa Mauriy Jarque-Bera Mean Median Maximum Minimum Sd. Dev. Skewness Kurosis (Monhs) Probabiliy We conduced in-sample fiing exercise by esimaing equaion (5) using differen values for. In paricular, following Diebold and Li (2006) we performed OLS esimaion of equaion (5) by assigning 1396

7 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model equal o and , and also NLS esimaion assuming is ime varying wih daily, weekly and monhly yield daa from January 2000 o November The resuls (no repored due o space limiaion bu available upon reques) show ha in general he long-erm facor ( ˆ OLS ) and he medium-erm facor 1, ( ˆ OLS ) are found more persisen han he shor-erm facor ( ˆ 3, OLS ), which is consisen wih he sylized 2, facs ha yield dynamics are persisen and longer raes are more persisen han he shor raes. I is also found ha he pairwise correlaion beween ˆ OLS 1, and ˆ OLS is he larges in all he hree cases and also 2, significanly negaive. We have conduced he h-period(s)-ahead expanding window forecasing for boh he NS and non-ns models, bu repor in Table 2 he resuls for one-week ahead expanding window only due o space limiaion. The res of he resuls are available upon reques. Table 2: Ou-of-Sample 1-Monh-Ahead and 3-Monhs-Ahead Expanding Window Forecasing Resuls) Yield Forecasing NELSON-SIEGEL MODELS RIVAL MODELS 1-Monh-Ahead Forecas 3-Monhs-Ahead Forecas MODEL RMSPE RMSPE ARMSPE TRMSPE ARMSPE TRMSPE 3m 6m 12m 60m 120m 360m 3m 6m 12m 60m 120m 360m NS-OLS(λ=0.0598)-RW-L NS-OLS(λ=0.0598)-AR-C NS-OLS(λ=0.0598)-VAR-C NS-OLS(λ=0.0299)-RW-L NS-OLS(λ=0.0299)-AR-C NS-OLS(λ=0.0299)-VAR-C NS-NLS(λ=0.0330)-RW-L NS-NLS(λ=0.0330)-AR-C NS-NLS(λ=0.0330)-VAR-C RW-L SR AR-L AR-C VAR-L VAR-C VECM: 1 Common Trend VECM: 2 Common Trends Yield Volailiy Forecasing NELSON-SIEGEL MODELS RIVAL MODELS 1-Monh-Ahead Forecas 3-Monhs-Ahead Forecas MODEL RMSPE RMSPE ARMSPE TRMSPE ARMSPE TRMSPE 3m 6m 12m 60m 120m 360m 3m 6m 12m 60m 120m 360m NS-OLS(λ=0.0598)-GARCH-C NS-OLS(λ=0.0598)-GARCH-AR-C NS-OLS(λ=0.0598)-EGARCH-C NS-OLS(λ=0.0299)-GARCH-C NS-OLS(λ=0.0299)-GARCH-AR-C NS-OLS(λ=0.0299)-EGARCH-C NS-NLS(λ=0.0330)-GARCH-C NS-NLS(λ=0.0330)-GARCH-AR-C NS-NLS(λ=0.0330)-EGARCH-C GARCH-L GARCH-C GARCH-AR-L GARCH-AR-C EGARCH-L EGARCH-C As regards he yield forecasing, he RW model seems work excepionally well in mos cases, and boh he AR and VAR models produced almos similar and relaively fine forecass. There is also a endency for hese wo models o converge owards he RW model, which means ha, when he selecion crieria of significan coefficiens becomes more sringen, many of hose coefficiens on he lagged regressors as well as he consan erm were found o be insignifican and he model ulimaely becomes a RW. In conras, for he weekly and daily forecass, he NS models (excep for NS-VAR) generally performed worse han he non-ns models (excep VECM). For monhly forecass, he NS models appear o funcion well when i comes o longer horizons, paricularly for 12-monhs-ahead forecass. On op of ha, he NS-VAR model performed roughly he same for all hree variaions of decay raes. I is also ineresing o noe ha he NS-RW and NS- AR specificaions when yields he lowes TRMSPE among he hree decay raes, which indicaes ha Diebold s choice of is no necessarily he bes one available. On he oher hand, he resuls indicae ha he non-ns models ouperform he NS models in volailiy forecass wih he presence of liquidiy rap. This finding acually is no a surprise as liquidiy rap has impelled bond yields owards zero and less volaile. As regards he various decay raes, i appears ha he lowes ARMSPEs and TRMSPEs are aained when in mos cases, while he NLS-esimaed model wih se a performed a lo worse. This resul confirms ha a faser decay rae allows forecased condiional volailiy o converge quickly o zero. As liquidiy rap has induced acual volailiy close o zero, he NS model wih he larges is likely o sand ou from he res. 1397

8 Tsui e al., Modelling he erm srucure of Japanese bond yields wih he Nelson-Siegel model 4. CONCLUDING REMARKS We have demonsraed ha due o he presence of liquidiy rap in Japan, ou-of-sample expanding window forecass for he NS models in general perform inferiorly vis-à-vis he non-ns models, and his is coupled wih he oher problem of obaining negaive yield forecass for bonds wih shorer mauriies. Moreover, he resuls show ha he NS class of models can be useful in forecasing shorer horizons like weeks and days, works beer wih a decay rae oher han he convenional way of reaing i as he value ha maximizes he loading on he medium-erm facor a exacly 30 monhs, and can work well wih ime series models such as GARCH and EGARCH in erms of volailiy forecasing. I is also found ha, when he NS models are used for yield forecass, he NS-VAR model should be considered since i is up o par agains he compeior models, even wih liquidiy rap a work ACKNOWLEDGEMENT The hird auhor wishes o acknowledge he financial suppor from he Sumiomo Foundaion. REFERENCES Bauwens, L., Lauren, S., & Rombous, J. (2006). Mulivariae GARCH models: a survey, Journal of Applied Economerics, 21, Bliss, R. (1997), Tesing Term Srucure Esimaion Mehods," Advances in Fuures and Opions Research, 9, Björk, T. & Chrisensen, B.J. (1999). "Ineres Rae Dynamics and Consisen Forward Rae Curves," Mahemaical Finance, Wiley Blackwell, vol. 9(4), pages Caporin, M., & McAleer, M. (2012). Do we really need boh BEKK and DCC? A ale of wo mulivariae GARCH models, Journal of Economic Surveys, 26, de Pooer, Michiel (2007), Examining he Nelson-Siegel Class of Term Srucure Models: In-Sample Fi versus Ou-of-Sample Forecasing Performance, Insiue Discussion paper, No /4. Diebold, F. X., Piazzesi, M. and Rudebusch, G.D. (2005). Modeling bond yields in finance and macroeconomics. American Economic Review 95, Diebold, F. X. and Li, C. (2006), Forecasing he Term Srucure of Governmen Bond Yields, Journal of Economerics, Vol. 130, 2006, pp Diebold, F. X., Li, C. and Yue, V. Z. (2008), Global yield curve dynamics and ineracions: A dynamic Nelson-Siegel approach. Journal of Economerics 146, Diebold, F.X., Rudebusch, G.D. and Aruoba, S.B. (2006). The macroeconomy and he yield curve: A dynamic laen facor approach. Journal of Economerics 131, Fabozzi FJ, Marellini L, Priaule P. (2005). Predicabiliy in he shape of he erm srucure of ineres raes. Journal of Fixed Income 15-1: Lierman, R., and J. Scheinkman, 1991, Common Facors A ecing Bond Reurns," Journal of Fixed Income, June, Maasoumi, E., & McAleer, M. (2006). Mulivariae sochasic volailiy: An overview, Economeric Reviews, 25, McAleer, M. (2005). Auomaed inference and learning in modeling financial volailiy, Economeric Theory, 21, Nelson, Charles R. and Siegel, Andrew F. (1987), Parsimonious Modeling of Yield Curves, Journal of Business, Vol. 60, No. 4, pp Nelson, Daniel B. (1991), Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Economerica, Vol. 59, No. 2, pp Svensson, L.E.O. (1994). Esimaing and inerpreing forward ineres raes. NBER Working Paper Series, 1398