and Sun (14) and Due and Singlen (19) apply he maximum likelihd mehd while Singh (15), and Lngsa and Schwarz (12) respecively emply he hreesage leas s

Transcription

1 A MONTE CARLO FILTERING APPROACH FOR ESTIMATING THE TERM STRUCTURE OF INTEREST RATES Akihik Takahashi 1 and Seish Sa 2 1 The Universiy f Tky, Kmaba, Megur-ku, Tky Japan 2 The Insiue f Saisical Mahemaics, Minami-Azabu, Mina-ku, Tky Japan (March, 2000) Absrac. We develp new mehdlgy fr esimain f he erm srucure f ineres raes based n a Mne Carl lering apprach. As an example, we apply he mehd LIBORs (Lndn InerBank Oered Raes) and ineres raes swaps in he Japanese inerbank marke. Key wrds and phrases: Generalized Sae Sapce Mdel, Mne Carl Filer, Term Srucure f Ineres Rae, Mne Carl Simulain, Term Srucure Mdel 1. Inrducin We prpse a new framewrk f he esimain f he erm srucure f ineres raes based n he sae space mdel. In paricular, ur knwledge, i is he rs applicain f he Mne Carl ler he esimain f he erm srucure mdels described by sae variables fllwing muli-dimensinal Markv prcesses. Fr example, ur mehd can be applied he erm srucure mdels based n he dynamic general equilibrium hery f Cx, Ingersll and Rss (15a,b)(CIR) which includes muli-facr CIR mdels used by Chen and Sc (13), Pearsn and Sun (14), Singh (15) and Due and Singlen (19), and he schasic vlailiy mdel develped by Lngsa and Schwarz (12). I is well-knwn ha a leas w sae variables are necessary explain he dynamics f he erm srucure in he real wrld. Chen and Sc (13), Pearsn and Sun (14), Singh (15), and Due and Singlen (19) cncluded ha ne-facr mdels are n enugh describe he variain f he erm srucure by analyzing reasury r swap markes in he Unied Saes. Hwever, muli-facr mdels fen esed in he analysis are n necessarily amng he bes candidaes. Fr insance, in he muli-facr CIR mdel where he sp ineres rae is described by he sum f several sae variables independenly fllwing square-r prcesses, he inuiive inerpreain f he sae variables is n clear, and smeimes he esimaed parameers d n saisfy he nnnegaive cndiin f he sp rae. One f he reasns why he mdels which d n explain he daa very well are fen emplyed in empirical analyses is ha hey allw analyic sluins f zer cupn bnds' prices. I is mainly due he limiain f he mehds applied he esimain. In paricular, Chen and Sc (13), Pearsn 1

2 and Sun (14) and Due and Singlen (19) apply he maximum likelihd mehd while Singh (15), and Lngsa and Schwarz (12) respecively emply he hreesage leas square mehd wih he principal cmpnen analysis and he generalized mmen mehd(gmm) wih GARCH. Hwever, i is subsanially dicul apply hse mehds wihu analyic sluins f he zer-cupn bnds' prices. Mrever, exising researches end replace unbservable sae variables such as he sp rae and he vlailiy by sme bservable variables (Chan e al. (12), Lngsa and Schwarz (12)), bu hey may subsanially suer frm he measuremen errrs. While sme f hem ake he measuremen errrs in accun explicily, he ways f he cnsiderain are n naural and smewha ad hc. (Chen and Sc (13), Due and Singlen (19)) We prpse a Mne Carl lering apprach based n he generalized sae space mdel vercme he prblems f exising researches. The sae space mdel cnsiss f he sysem mdel describing he prcesses f sae variables and he bservain mdel represening he funcinal relain beween he sae variables and he bservainal daa in he real wrld, which implies ha he mehd can be naurally applied he esimain f he erm srucure mdels based n he Markv sae prcesses. I is als pssible inrduce measuremen errrs in he general way wihu any bias. Mrever, he Mne Carl ler can be applied much brader class f he erm srucure mdels, especially even he mdels in which he zer-cupn bnds' prices can n be analyically bained. The paper is rganized as fllws. In he nex chaper, we will rs summarize he sae space mdel and he erm srucure mdels based n Markv sae prcesses. Then, we will clarify he relain beween ineres rae mdels as well as bservainal daa and he sae space mdel. Nex, we will give a cncree algrihm f he Mne Carl ler applied he empirical analysis. In chaper hree, we will es he validiy f he Mne Carl ler using LIBORs (Lndn InerBank Oered Raes) and ineres rae swaps in he Japanese inerbank marke. In chaper fur, we will make cncluding remarks. 2. The Esimain f he Term Srucure Based n he Sae Space Mdeling We explain in his chaper he esimain mehd fr he sae variables and he parameers f erm srucure mdels based n he sae space mdeling. Firs, we give he general frm f sae space mdels. (See Kiagawa and Gersh(16) fr he deail.) A sae space mdel cnsiss f he fllwing sysem mdel and he bservain mdel. Tha is, 8 >< Y = F (Y, ;v ) sysem mdel (2.1) >: Z = H(Y ;u ) bservain mdel 2

3 where Y, Z and dene a N dimensinal sae vecr, a M dimensinal bservain vecr a ime and he ime inerval f bservainal daa respecively while v and u dene he sysem nise and he bservainal nise whse densiy funcins are given respecively by q(v) and (u). F and H are in general nn-linear funcins f R N R N 7! R N and R N R M 7! R M, and he iniial sae vecr Y 0 is assumed be a randm variable whse densiy funcin is given by p 0 (Y ). Befre describing hw sae space mdels are uilized in ur analysis, we briey explain erm srucure mdels based n Markv sae prcesses. We rs assume ha he ime hrizn f ecnmic aciviies is [0;T ], T < 1. Given he lered prbabiliy space (;F;fF g;p), we suppse ha a k dimensinal vecr f sae variables dened by Y fllws a k dimensinal Markv prcess, (2.2) dy = (Y; )d + S(Y; )db ; where B is he n dimensinal sandard Brwnian min under he lered prbabiliy space, and (Y; ) and S(Y; ) dene real-valued funcins f R k [0;T ] 7! R k, and R k [0;T ] 7! R kn, respecively. Suppse als ha he insananeus shr-erm ineres rae a ime dened by r() and a zer cupn bnd's price a wih he mauriy T dened by P (; T ) are sme funcins f Y where 2 [0;T ] and T 2 [; T ]. We ne ha he se f he zer cupn bnds' prices fp (; T )g T 2[0;T ] represens he erm srucure f ineres raes a ime. Then, based n he arbirage-free argumen f nancial ecnmics, P (; T ) saises a parial dierenial equain(pde) 1 2 race(ss0 P YY )+[, ]P Y + P, rp =0; wih he erminal bundary cndiin, P (T;T)=1. Here, (Y; ) denes s called he risk premium which is a funcin f Y and, and fr insance, i can be deermined by he general equilibrium asse pricing hery f ecnmics such as CIR(15a). Mrever, i is well knwn ha he sluin f his PDE is represened by he cndiinal expecain given infrmain a ime, (2.3) R P (; T )=E Q [e, T r(u)du ]; where E Q [] denes he cndiinal expecain perar a ime under s called he risk-neural prbabiliy measure Q. I is als knwn ha under he measure Q, he vecr f sae variables Y fllws a schasic dierenial equain, (2.4) dy = f(y; ), (Y; )gd + S(Y; )db where B denes he n dimensinal sandard Brwnian min under he measure Q. Nex, we clarify hw he sae space mdels can be applied he esimain f erm srucure mdels. When is sucienly small, he Euler apprximain he equain 3

4 (2:2) can be used fr he sysem mdel, Y = F (Y, ;v ). Tha is, (2.5) Y = Y, + (Y, ;, ) + S(Y, ;, )v p where he sysem nise v fllws he N dimensinal sandard nrmal disribuin. Of curse, he her apprximain schemes culd be applied he descreizain f (2:2). (Fr example, see secin D and nes f chaper 11 in Due(16).) Mrever, when Y is explicily slved given Y, as in he case f a linear schasic dierenial equain, i is beer use ha represenain. Tha is, in (2:2), suppse Y is represened by a linear schasic dierenial equain: (2.6) dy =(AY + )d + SdB where and A dene R k -valued funcins f he ime parameer and k k cnsan marices respecively, and S denes a k n cnsan marix s ha = SS 0 is psiive semidenie. Then, given Y,, Y is slved by (2.7) Y = e A Y, + Z, e (,s)a ds + v : Here, v fllws he nrmal disribuin wih he mean zer and he cvariance marix where Z 0 e sa e sa0 ds: In his case, he sysem mdel is given by (2:7) and he densiy funcin q(v) f he sysem nise v is ha f he nrmal disribuin. In he bservain mdel, he bservain vecr a ime, Z is expressed by n( 1) unis f zer-cupn bnds' prices and he nise vecr u. Tha is, Z = H(fP (; + T i )g n i=1 ;u ): In he equain abve, P (; + T i ) can be evaluaed by he cmpuain f he equain (2:3) under he prcess (2:4). In addiin, we assume hereafer ha he bservain mdel is f he frm, (2.8) Z = H(fP (; + T i )g n i=1)+u ; u N(0; u ); where he densiy funcin (u) f he nise vecr u is given by ha f he mulidimensinal nrmal disribuin wih he mean 0, and he variance-cvariance marix u. Here, u denes a M M diagnal marix wih psiive diagnal elemens where M is dened be he dimensin f he bservainal daa a each ime. LIBORs and ineres rae swap raes a ime in he inerbank marke are he ypical examples f he bservain vecr Z. In his case, H() is deermined by he hereical relain 4

5 beween LIBORs/ineres rae swap raes and he zer-cupn bnds' prices. Tha is, LIBOR wih he erm T n dened by L (T n ) and swap raes wih he erm T n dened by S (T n ) are expressed by he zer-cupn bnds' prices as fllws. (2.9) (2.10)! 1 1 L (T n ) = P (; + T n ), 1 T n S (T n ) = 1 P, P (; + T n) n i=1 P (; + i); where denes he inerval f cash ws and T n = n. Fr example, =0:5 is sandard in he Japanese yen swap marke. We will explain he acual daa f LIBORs and swap raes used fr he empirical analysis in he nex chaper. Frm he discussin abve, I can be seen ha he framewrk f he sae space mdel is naurally applied he esimain prblem f he erm srucure f ineres raes. Finally, we disucuss abu ur esimain mehd in deail. We ne ha he sandard Kalman ler can n be applied he esimain as bh he sysem mdel and he bservain mdel described abve are generally nn-linear, and hence we uilize he Mne Carl ler. While several appraches are prpsed fr he Mne Carl ler (see Duce, Bara, and Duvau(15), Durbin, and Kpman(19), Grdn, Salmnd, and Smih(13), Tanizaki(13), fr insance.), we adp he apprach develped by Kiagawa(16). In he fllwing, we describe he uline f he algrihm f he Mne Carl ler applied he empirical analysis in he nex chaper. Firs, we summarize he nain fllwing Kiagawa(16). p(y jz, ), called \ne sep ahead predicin" denes he cndiinal densiy funcin f Y given Z, where is he inerval f ime series. p(y jz ), called \ler" denes he cndiinal densiy funcin f Y given Z. fp (1) ; ;p (m) g and ff (1) ; ;f (m) g represen he vecrs f he realizain f m rials f Mne Carl frm p(y jz, ) and p(y jz ), respecively. Then, if we se ff (1) 0 ; ;f (m) 0 g as he realizain f Mne Carl frm p 0 (Y ), he densiy funcin f he iniial sae vecr Y 0, he algrihm f he Mne Carl ler is as fllws. [The summary f he algrihm f he Mne Carl ler] (i) Generae he iniial sae vecr ff (1) 0 ; ;f (m) 0 g. (ii) Apply he fllwing seps (a)(d) each ime = 0; ; 2; ; (T, );T where T denes he nal ime pin fhedaa. (a) Generae he sysem nise v (j), j =1; ;m accrding he densiy funcin q(v). (b) Cmpue fr each j =1; ;m p (j) = F (f (j), ;v(j) ): 5

6 (c) Evaluae n[x;0; u ], he densiy funcin f N(0; u )ax = Z,H(p (j) ),j = 1; ;m and dene hse as (j), j = 1; ;m. Here, H() represen fr insance, he equain (2:9) and (2:10) which are regarded as funcins f he sae vecr Y. The prices f zer-cupn bnds in hse equains are cmpued hrugh he equain (2:3) by using he prcess (2:4), and if i is n analyically slvable, sme numerical mehd such as Mne Carl simulain is implemened. g. Mre pre- g wih (d) Implemen resampling f ff (1) ; ;f (m) cisely, implemen resampling f each f (i) he prbabiliy Prb.(f (i) = p (j) jz )= (j) P m k=1 (k) g frm fp (1) ; ;p (m), i =1; ;mfrm fp (1) ; ;p (m) ; j =1; ;m; i =1; ;m: The esimain f unknwn parameers is based n he maximum likelihd mehd. If denes he vecr represening whle unknwn parameers, he lg-likelihd l() is given by l() =p(z ; ;Z T j) = T k=1 p(z kjz ; ;Z (k,1) ;) where p(z jz 0 ) = p 0 (Z ). The lg-likelihd l() is cmpued apprximaely wihin he framewrk f he Mne Carl ler by l() = T X k=1 lg m X j=1 (j) k 1 T A, lg m: Then, maximize l() wih respec bain he maximum likelihd esimar ^. Fr pimizain, grid search and a self-rganizing mehd are applied. (See Kiagawa(19) fr deails f a self-rganizing sae-space mdel.) Finally, we uilize AIC(Akaike's Infrmain Crierin) as a crierin selec he erm srucure mdels if here are several candidaes. mdel. Tha is, he mdel wih he smaller AIC can be regarded as he beer 3. An Applicain he Time Series f he Japanese Ineres Raes In his chaper, we examine he validiy f ur mehd using he ime series f ineres raes in he Japaneses inerbank marke. As an example f ineres rae mdels, we use Hull and Whie(14) in which he dynamics f a sae vecr Y can be represened by a linear schasic dierenial equain (2:6), and he resuling sysem mdel is described by he equain (2:7); given, he sysem mdel can be expressed as a linear mdel in he sae space seing: (3.1) Y = FY, + ()+v ; 6

7 where Y = (Y i ());i = 1; 2 denes a w dimensinal sae vecr, () is a R 2 -valued funcin f he ime parameer, F is a 2 2 marix wih cnsan elemens, and v fllws a w dimensinal nrmal disribuin wih he mean zer. They als assume ha he sp rae r is expressed as a funcin f Y 1, r = g(y 1 ) where g() is sme realvalued funcin. Fr a funcinal frm f r = g(y 1 ), if we ake g(y 1 ) = Y 1, we allw negaive ineres raes because f he nrmaliy f he sp rae while we can bain an analyic sluin f P (; T ), which subsanially reduce he cmpuainal burden. In paricular, he mdel implies relaively high prbabiliy f he negaive ineres raes in such lw ineres raes envirnmen f he recen Japan and his seems inapprpriae. Hence, we specify g(y 1 ) based n Hull(19) (chaper 21, pp.588) such ha g(y 1 ) = Y 1 fr Y 1 ", and g(y 1 ) = "e (Y 1,") " fr Y 1 < ", where " is sme predeermined psiive cnsan. Clearly, g() is psiive, mnnically increasing, and lim Y1!,1 g(y 1 ) = 0. We nex deermine he bservain mdel as he equain (2:8) wih he equains (2:9) and (2:10). Mrever, he zer-cupn bnds' prices in H() f (2:8) are cmpued by he equain, (3.2) P (; T )=E Q [e,r T g(y 1 (u))du jy ()]: We ne ha his is n analyically slvable and hence is evaluaed by sme numerical mehd such as Mne Carl simulains. Nex, we explain he resul f esimains fr he Japanese inerbank marke. The daa used fr he analysis is summarized as fllws. he perid and he frequency f he daa: daily daa f 19/1/1-19/7/22(662 series) Japanese yen LIBOR: six-mnh, welve-mnh Japanese yen swap raes: w-year, hree-year, fur-year, ve-year, seven-year, enyear Figures 1(1)(3) shw he bservainal daa fr he perid f he analysis; (1) and (2) respecively shw he series f LIBORs and hse f swap raes while (3) shws he spread beween en-year and w-year. We apply he algrihm f he Mne Carl ler described in he previus chaper he esimain. The dimensin f he bservainal daa is M =8,he equains (2:9) and (2:10) are applied H() in he equain (2:8), and zer-cupn bnds' prices in he equain (3:2) are cmpued by using Mne Carl simulains. The numbers f rials in he Mne Carl ler and in he Mne Carl simulains are 5000 and 300 respecively where he randm numbers are generaed by ran2 in Numerical Recipes (Secnd Ediin), and he aniheic variables mehd is uilized in he Mne Carl simulains. Mrever, we implemen parallel cmpuain 7

8 wih 40 prcessrs by IBM RS/6000-SP. In addiin, we suppse " be 0:0005(5 basis pin) fr r = g(y 1 ). Firs we invesigae he ime series f he esimaed facrs. Figures 2(1)(2) shw he relain beween he esimaed facrs Y i, i = 1; 2 bained by averaging m = 5000 samples f lers and he frward raes cmpued frm he daa. The crrelain beween he level f each facr and he crrespnding frward rae is als lised belw each gure. Frm he graphical bservain, here seems be srng relain beween Y 2 and enyear frward rae which is dened be he raes wih he erm 0.5 year saring frm 9.5 years frward, and he variain f Y 1 is similar ha f six-mnh LIBOR. We nex see he ing f he mdel he daa. The ime series f each bservain and crrespnding esimae are shwn in gures 3(1)(8), and he explanain pwer f each esimae is lised abve crrespnding gure. We ne ha he esimaes f LIBORs and swap raes are cmpued by Mne Carl simulains based n esimaed parameers and esimaed facrs, where he number f rials f he simulains are They shw ha he mdel s very well he swap raes while i des n LIBORs: The explanain pwers are mre han.5% fr all he swap raes while hey are 39.4% and 77.3% fr six-mnh and welve-mnh LIBORs respecively. Finally, gures 4(1)(4) shw he bservains and esimaes f erm srucures a fur daes (/6/24, /12/4, /9/30, /5/14), and hese indicae ha he shapes f he erm srucure f he mdel are very clse he real nes excep fr he LIBORs a ember 4h f 19 as well as a 14h f 19. Hence, we can cnclude ha he mdel explains he variains f he swap raes very well while i des n explain hse f LIBORs. This implies ha anher facr may be necessary imprve he ing LIBORs. Hence, we nex implemen he case in which he sae vecr Y = (Y i ()), i =1; 2; 3 is hree dimensinal, and has he same frm f (3:1). As in he previus case, we rs invesigae he ime series f he esimaed facrs. Figures 5(1)(3) shw he relains beween he esimaed facrs Y i, i =1; 2; 3 and he frward raes as well as heir crrelains. In his case, here is much clearer relain beween facrs and frward raes han in he previus case. Tha is, here seems exis srng relains beween Y 1 and six-mnh LIBOR, and Y 2 and he w--en-year frward raes' spread as well as beween Y 3 and en-year frward rae. The ime series f each bservain and crrespnding esimae as well as he explanain pwers fr his case are shwn in gures 6(1)(8). Clearly, he explanain pwers are mre han.5% fr all he raes and mre han % excep fr welve-mnh LIBOR and w-year swap rae. Especially, we ne ha he mdel remarkably imprves he ing LIBORs. Figures 7(1)(4) shw he bservains and esimaes f he erm srucures a fur daes(/6/24, /12/4, /9/30, /5/14), which implies ha he mdel can replicaes he real erm srucures including LIBORs very clsely. 8

9 Finally, we ne ha AIC in his case(aic= ) is mre imprved han in he previus case(aic= ), and hus, we can cnclude ha he mdel explains he variains f all he raes fairly well. Frm he abve analysis f he Japanese inerbank marke, we cnrm ha ur apprach uilizing he Mne Carl ler is valid fr he esimain f muli-facr erm srucure mdels which is usually cnsidered ugh ask. 4. Cncluding Remarks We develp a new framewrk fr he empirical analysis f he erm srucure f ineres raes based n he generalized sae space mdel. As an example, we apply he Mne Carl ler he ime series f LIBORs and f ineres rae swaps in he Japanese inerbank marke, and cnrm he validiy f ur mehd. References Chan, K.C., G.A. Karlyi, F.A. Lngsa, and A.B. Sanders(12), \An Empirical Cmparisn f Alernaive Mdels f he Shr-Term Ineres Rae," The Jurnal f Finance 47, pp Chen, R., and L. Sc(13), \Maximum Likelihd Esimain fr a Mulifacr Equilibrium Mdel f he Term Srucure f Ineres Raes," Jurnal f Fixed Incme, ember, pp Cx, J.C., J.E. Ingersll, and S.A. Rss(15a), \An Inerempral General Equilibrium Mdel Asse Prices," Ecnmerica 53, pp Cx, J.C., J.E. Ingersll, and S.A. Rss(15b), \A Thery f he Term Srucure f Ineres Raes," Ecnmerica 53, pp Duce, A., E. Bara, and P. Duvau(15), \A Mne Carl Apprach Recursive Bayesian Sae Esimain," Prceedings IEEE Signal Prcessing/Ahs Wrkshp n Higher Order Saisics, Girna, Spain. Due, D.(16), Dynamic Asse Pricing Thery(secnd ediin), Princen Universiy Press. Due, D., and K.J. Singlen(19), \An Ecnmeric Mdel f he Term Srucure f Ineres-Rae Swap Yields," The Jurnal f Finance, 52, ember pp Durbin, J., and S.J. Kpman(19), \Mne Carl Maximum Likelihd Esimain fr Nn-Gaussian Sae Space Mdels," Bimerika 84, pp Grdn, N., D.J. Salmnd, and A.F.M. Smih(13), \Nvel Apprach Nnlinear/Nn-Gaussian Bayesian Sae Esimain," IEE Prceedings-F, 140, N.2, pp Hull, J.(19), Opins, Fuures, and Oher Derivaives (furh ediin), Prenice-Hall. Hull, J., and A. Whie(14), \Numerical Prcedures fr Implemening Term Srucure Mdels II:Tw- Facr Mdels," Jurnal f Derivaives 2, Winer pp Kiagawa, G.(16), \Mne Carl Filer and Smher fr Nn-Gaussian Nnlinear Sae Space Mdels," Jurnal f Cmpuainal and Graphical Saisics 5, N.1, pp Kiagawa, G.(19), \A Self-Organizing Sae-Space Mdel," Jurnal f he American Saisical Assciain, 93, N.443, pp Kiagawa, G. and W. Gersch(16), Smhness Prir Analysis f Time Series, Lecure Nes in Saisics N. 116, Springer-Verlag. Lngsa,F., and E.S. Schwarz(12), \Ineres Rae Vlailiy and he Term Srucure:A Tw-Facr General Equilibrium Mdel," The Jurnal f Finance, 47, pp

10 Pearsn, N.D., and T.S. Sun(14), \Expliing he Cndiinal Densiy in Esimaing he Term Srucure: An Applicain he Cx, Ingersll, and Rss Mdel," The Jurnal f Finance 49, pp Press, W.H., B.P. Flannery, S.A. Teuklsky and W.T. Veerling (12), Numerical Recipes in C (2nd Ediin), Cambridge Universiy Press. Singh, M.K.(15)\Esimain f Mulifacr Cx, Ingersll, and Rss Term Srucure Mdel," Jurnal f Fixed Incme, ember,pp Tanizaki, H.(13), Nnlinear Filers, Lecure Nes in Ecnmics and Mahemaical Sysems, N. 400, Springer-Verlag. 10

11 (1) LIBORs (3) Spread (Swap10Y - Swap2Y) LIBOR-1Y LIBOR-6M (BP) Aug Nv Jul Mar Aug Nv Jul (2) SWAP raes SWAP-10Y SWAP-7Y SWAP-5Y SWAP-4Y SWAP-3Y SWAP-2Y Aug Nv Fig. 1. Jul Mar ime series f bserved LIBORs and swap raes (1) Libr6M(slid), Y1(ds) (Scale:lef) (Scale:righ) (2) Frward10Y(slid), Y2(ds) (Scale:lef) (Scale:righ) Crr= 0.90 Crr= 0. (3) Frward 2Y-10Y(slid), Y1(ds) (Scale:lef) (Scale:righ) Crr= 0.67 Fig. 2. esimaed facrs (w-facr case) 11

12 (1) Libr6M (2) Libr1Y R2= 39.4 % R2= 77.3 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (3) Swap2Y (4) Swap3Y R2=.7 % R2=.5 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (5) Swap4Y (6) Swap5Y R2=.6 % R2=.7 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (7) Swap7Y (8) Swap10Y R2=.8 % R2=.6 % h Observains (slid line), Esimaes (ds line) i Observains (slid line), Esimaes (ds line) Fig. 3. bservains and esimaes (w-facr case); he explanain pwer (R2) is dened by max 1, he variance f residuals ; he variance f bservains 0. 12

13 (1) 24,19 (2). 4, (Yr) Observain(), Esimaes() (Yr) Observain(), Esimaes() (3). 30, 19 (4) 14, (Yr) (Yr) Observain(), Esimaes() Observain(), Esimaes() Fig. 4. he erm srucures(w-facr case) 13

14 (1) Libr6M(slid), Y1(ds) (Scale:lef) (Scale:righ) (2) Frward10Y(slid), Y3(ds) (Scale:lef) (Scale:righ) Crr= 0.90 Crr= 0. (3) Frward 2Y-10Y(slid), Y2(ds) (Scale:lef) (Scale:righ) Crr= 0.90 Fig. 5. esimaed facrs (hree-facr case) 14

15 (1) Libr6M (2) Libr1Y R2=.6 % R2=.6 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (3) Swap2Y (4) Swap3Y R2=.6 % R2=.8 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (5) Swap4Y (6) Swap5Y R2= % R2=.9 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) (7) Swap7Y (8) Swap10Y R2=.9 % R2=.6 % Observains (slid line), Esimaes (ds line) Observains (slid line), Esimaes (ds line) Fig. 6. bservains and esimaes (hree-facr case) 15

16 (1) 24,19 (2). 4, (Yr) Observain(), Esimaes() (Yr) Observain(), Esimaes() (3). 30, 19 (4) 14, (Yr) (Yr) Observain(), Esimaes() Observain(), Esimaes() Fig. 7. he erm srucures (hree-facr case) 16