Regime Switching in Dynamics of Risk Premium: Evidence from SHIBOR ABSTRACT

Transcription

1 Regime Swiching in Dynamic of Rik Premium: Evidence from SHIBOR Baochen Yang College of Managemen and Economic, Tianjin Univeriy Tianjin 372, China Tel: , Yunpeng Su College of Managemen and Economic, Tianjin Univeriy Tianjin 372, China Tel: ABSTRACT In he ligh of regime wiching and volailiy cluering in he dynamic of SHIBOR, regime-wiching CIR model (RSCIR) and regime-wiching GARCH CIR model (RSCIR-GARCH) are eablihed by inroducing regime-wiching and GARCH pecificaion ino CIR model ucceively. Then, a conra udy among CIR, RSCIR and RSCIR-GARCH model i performed baed on SHIBOR ample daa, which indicae ha he regime-wiching and GARCH pecificaion can improve he model fine ignificanly and eliminae he ARCH effec in he volailiy dynamic. Furhermore, an empirical reearch i carried ou on he rik premium dynamic of SHIBOR baed on he RSCIR-GARCH model, and i i found ha here are wo regime, ha i, he regime of higher inere rae wih higher volailiy and he regime of lower inere rae wih lower volailiy, in he dynamic of he erm rucure and rik premium of SHIBOR. Keyword: SHIBOR; rik premium; regime wiching; GARCH; Cox-Ingeroll-Ro model; Kim filer JEL claificaion: C58; E43; G2 Thi reearch i uppored by Naional Naural Science Foundaion of China (Gran No , 7744) and MOE Projec of Humaniie and Social Science (Gran No. YJCZH47).

2 Regime Swiching in Dynamic of Rik Premium: Evidence from SHIBOR Inroducion Modeling he erm rucure of inere rae play a ignifican role in pricing fixed income derivaive, in rik managemen and in deigning macroeconomic policie. So a wide range of inere rae model, uch a Vaicek (977) model and Cox-Ingeroll-Ro (985) model, have been pu forward o capure he dynamic behavior of he hor-erm rikle rae. However, here i increaing evidence in he lieraure ha change in he buine cycle condiion or moneary policy may caue inere rae o behave quie differenly in differen ime period, boh in erm of level and volailiy, o regime-wiching model are uggeed inead of ingle-regime model in capuring he dynamic of he hor rae, which i juified by exenive empirical lieraure on bond yield (Sola and Driffill, 994, Garcia and Perron, 996, Dahlqui and Gray, 2, ec.). The regime-wiching model in he lieraure are olved under differen aumpion abou he evoluion of he hor rae. Naik and Lee (997) inroduce regime-wiching ino he Vaicek model wih he aumpion ha only he volailiy erm i allowed o wich, while Banal and Zhou (22) adop a regime-wiching CIR model (hereafer RSCIR model) in capuring he dynamic of he hor rae, wih he aumpion ha he full e of parameer i allowed o wich. Dahlqui and Gray (2) ue a regime-wiching CKLS model o characerize hor rae in he European Moneary Syem. Then Naik and Lee (997), Landen (2), and Dai and Singleon (23, 27) furher develop hor rae model ino coninuou-ime regime-wiching dynamic erm rucure model (DTSM) ha yield cloed-form oluion for zero-coupon bond price. In addiion o regime wiching, characeriic of volailiy cluering and lepokuroi are alo oberved in he dynamic of he hor rae. So, cholar combine regime wiching wih GARCH or ochaic volailiy o improve he fine of he model o he hiorical inere rae daa. Among oher, Gray (996) and Bauwen e al. (2) propoe a Markov-wiching GARCH model, while Smih (22) a Markov-wiching ochaic volailiy model, boh of which are applied o capure he dynamic of he hor rae of US, leading o he concluion ha he inroducion of GARCH or ochaic volailiy ignificanly reduce he volailiy perience of he hor rae and hu improve he fine of regime-wiching model. However, he majoriy of he udie above focu on he model pecificaion and fine o he hiorical daa and mainly ue US daae. Recen aemp o apply regime-wiching hor rae model o counrie oher han he US include Dahlqui and Gray (2) uing European daa, Erlandon (22) uing Swedih daa, Ang and Bekaer (22) and Chriianen (28) uing UK and German daa. Thee udie juify regime-wiching hor rae model and inveigae he parameer inabiliy of ingle-regime inere rae model for differen counrie. Thi inabiliy in he inere rae proce could have imporan implicaion no only for capuring he dynamic of he hor rae, bu alo for valuing fixed income ecuriie and inere rae derivaive and for hedging inere rae rik. For example, he level of inere rae i imporan o he

3 valuaion of ock, bond, index fuure and opion, ec. In addiion, he volailiy of inere rae i a fundamenal deerminan in he valuaion of inere rae derivaive. Moreover, opimal hedging raegie for rik-avere inveor are very eniive o change in inere rae volailiy. In hi paper, we apply a regime-wiching GARCH CIR model (hereafer RSCIR-GARCH) o decribe he behavior of he rik premium implied in he hor-erm inere rae in China, uing one-monh Shanghai Inerbank Offered Rae (SHIBOR) a ample daa. Fir, we inveigae he regime-wiching and lepokuroi characeriic of he hor rae dynamic, baed on whoe reul we eablih regime-wiching CIR model (RSCIR) and regime-wiching GARCH CIR model (RSCIR-GARCH) by inroducing regime-wiching and GARCH pecificaion ino CIR model ucceively. Then, a conra udy among CIR, RSCIR and RSCIR-GARCH model i performed baed on he one-monh SHIBOR ample daa, which how ha he regime-wiching and GARCH pecificaion boh improve he model fine ignificanly and eliminae he ARCH effec in he volailiy dynamic. In he ligh of he above, we perform an empirical reearch on he rik premium dynamic of SHIBOR uing he RSCIR-GARCH model eimaed by he Kim filer baed maximum likelihood mehod, finding ha here are wo diinc regime, ha i, he regime of higher inere rae wih higher volailiy and he regime of lower inere rae wih lower volailiy, in he dynamic of boh he erm rucure and rik premium of SHIBOR. The high-volailiy regime i characerized by period of exremely high and volaile inere rae, and he correponding rik premium alo ha a high level and flucuaion, while in he low-volailiy regime, in conra, boh he hor rae and he rik premium diplay a low mean and volailiy. However, he mean-reverion of he hor rae in he low-volailiy regime i much ronger han ha in he high-volailiy regime. Beide, i i alo found ha he ARCH effec in he volailiy dynamic i only ignifican in he high-volailiy regime bu no in he low-volailiy regime. Furhermore, we make a comparion beween he CIR model and he RSCIR-GARCH model in capuring he dynamic of he rik premium implied in he hor rae, leading o he concluion ha he RSCIR-GARCH model by far ouperform he CIR model in capuring regime wiching in he dynamic of rik premium and hu in reflecing change in marke condiion, which i criical for pricing and hedging of ae. The remainder of he preen paper i organized in he following manner. Secion 2 preen he RSCIR model and he RSCIR-GARCH model. Secion 3 dicue he Kim filer (Kim, 994) baed maximum likelihood eimaor for he model. Secion 4 dicue he empirical reul and Secion 5 preen concluding commen. 2 Shor Rae Model In hi ecion, we preen he RSCIR model wih and wihou GARCH pecificaion for he hor rae volailiy baed on he regime-wiching model propoed by Banal and Zhou (22). Shanghai Inerbank Offered Rae (SHIBOR) i a daily reference rae baed on he inere rae a which bank offer o lend unecured fund o oher bank in he Shanghai wholeale (or "inerbank") money marke. There are eigh SHIBOR rae, wih mauriie ranging from overnigh o a year. They are calculaed from rae quoed by ixeen bank, eliminaing he wo highe and he wo lowe rae, and hen averaging he remaining welve.

4 Following Banal and Zhou (22), we aume he dynamic of he hor rae governed by a regime-wiching quare roo diffuion proce: r r = k ( θ r ) + σ r u () where r i he hor rae a ime. k, θ and σ are he regime-dependen mean reverion, long-run mean, and volailiy parameer repecively and u i he innovaion which i condiionally normal given r and. Given Φ he informaion e a ime, he condiional diribuion of r i a mixure of diribuion in all regime, wih he regime probabiliie a he weigh. Here, we aume here are wo ae for he hor rae regime, ha i = and =. Then he condiional diribuion of r can be wrien a where μ r k ( θ r ) r Φ = + and ( μ, ).. ( μ, ).. ( ) N V w p p N V w p p (2) V = σ r,, 2 =, Pr ( ) p = = Φ. In order o inveigae he conribuion of regime-wiching and GARCH pecificaion repecively o he improvemen of model fine o SHIBOR daa, we aume he volailiy parameer σ in Equaion () ha wo form a follow: σ () = c and σ () ( ) = c + a η + bσ r, where = + ( ) 2, ( ) ( ) σ p μ σ r p μ 2 η = r p μ, and p i he priori probabiliy of he regime a ime,. We name by RSCIR he model wih he former volailiy pecificaion and by RSCIR-GARCH he model wih he laer one. Suppoe he evoluion of he regime i governed by he raniional probabiliy marix of a Markov chain π π Π= π π, where = and < π ik <. Then he yield o mauriy of a π k =, ik zero-coupon bond wih a mauriy of τ period a ime i given a A ( τ ) B ( ) τ R (, τ ) = r ε τ + τ +,, = (3)

5 where ( ) ( ) ( ) + ( ) ( ) + ( ) A τ π π A τ kθb τ = A τ π π, A τ kθ B τ 2 2 ( k λ) B ( ) ( τ ) σb ( τ ) + B τ π π 2 = B( τ ) π π, 2 2 ( k λ) B( τ ) σb ( τ ) + 2 wih iniial condiion A() = A() = B() = B() =, λ i he marke price of rik in he regime and he meauremen noie ε N (, Ω). Equaion () and (3) make he ae-pace form of boh he RSCIR model and RSCIR-GARCH model, wih differen volailiy pecificaion, where () erve a he ae equaion, while (3) erve a he meauremen equaion. σ B τ r i he expoure of he yield o he andardized hock u in regime. Noe ha ( ) Furher, λ / r σ i he expoure of he pricing kernel o u in regime. The covariance beween hee expoure deermine he compenaion for rik in regime. Hence, he rik premium for regime i he produc ( ) / ( ) σ B τ r λ r σ = B τ λ r (4) Given he informaion regarding, r, and he regime raniion probabiliie, he expeced rik premium of he RSCIR model and he RSCIR-GARCH model a ime i given a follow, ( ) E B τ λ, ( ) r r = r E B τ λ In he abence of regime hif, ha i under he CIR model, he rik premium in (5), would imply be ( ) r B τ λ. (5) 3 The Kim Filer Baed Maximum Likelihood Eimaor In hi paper, we apply he maximum likelihood eimaor baed on he uncened Kalman filer (Julier and Uhlmann, 997) o eimae he CIR model, and he one baed on he Kim filer o eimae he RSCIR model and RSCIR-GARCH model. The Kim filer i an exenion of he Kalman filer, which combine he Kalman filer wih he Hamilon filer (Hamilon, 989) propoed for Markov wiching model and a collaping procedure. The Kim filer i an opimal eimaor in he ene ha no oher eimaor baed on a linear funcion of he informaion e yield a maller mean quare error. A deailed dicuion of he Kim

6 filer can be found in Kim (994) and Kim & Nelon (999). The Kim filer i ued for generalized ae pace model where model parameer are allowed o depend on an unoberved, dicree-valued S -regime Markov wiching variable, {,, S}, wih a q q homogeneou raniion probabiliy marix Π. The generalized pae pace model i hen r α ( α = + ) r + v R β ( β = + ) r + ε (6) where m r i he ae vecor, n R i he meauremen vecor, α, α, β, β are conan wihin each regime, v and ε are he ae error vecor and meauremen error vecor v Q repecively, and N,. ε H For he convenience of decripion, we define he noaion a follow, r and ql, Ψ ql, : Eimae of r and correponding mean quare error given R and = q, = l ; r and ql, Ψ ql, : Eimae of r and correponding mean quare error given R and = q, = l ; q q r and Ψ : Eimae of r and correponding mean quare error given R and = q, = q. The baic ep of he Kim filer are hen:. Run he Kalman filer aking ino accoun he differen regime: ( α ) ql, l l q r α = + r Ψ = α Ψ ( α ) + Q ql, ql, l ql, K ( β =Ψ ) ( P ) ql, ql, ql, ql, r = r + K v ql, ql, l ql, Ψ = ( I K β ) Ψ ql, l q l l (7) where v = R ( β + β r ), ql, l l ql, P = Ψ +. ql, l ql, l l β ( β) H 2. Run he Hamilon filer o compue condiional ae probabiliie:

7 S S f( R R ) = f( R = q, = l, R )Pr( = q, = l R ) l= q= f( R = q, = l, R )Pr( = q, = l R ) Pr( = q, = l R) = f( R R ) S Pr( = l R) = Pr( = q, = l R) q= (8) where Pr ( = q, = l R ) = Pr ( = l = q) Pr ( = j, = q R ) 2 j= S, S ( ) ( ) ( ) 2 ql, 2 ql, ql, ql,,, 2 π exp 2 ( )( ) f R = q = l R = P v P v. 3. Collape he poerior: S ql, Pr ( = q, = l R) r l q= r = Pr ( = l R) S Pr ( = q, = l R) l q= q, l l q, l l q, l ( r r )( r r Ψ = Ψ + ) Pr ( = l R) (9) l The collaping procedure in Sep 3 make he new poerior eimae r depend no more on he regime in ime and hu convenienly coniue he inpu of he Kalman filer for he nex ieraion. Then we conruc he log-likelihood funcion baed on he informaion he Kim filer provide above: J ( ) L ( θ ) = ln f R R () R = where J i he number of obervaion and f ( R R ) i given in (8). The maximum likelihood eimaor of θ i hen 2 ˆ θ = max ( θ ) () θ L R 4 Empirical Reul 4. Daa Decripion We ue he one-monh SHIBOR rae a he proxy for he hor-erm inere rae in China o eimae 2 In hi paper, we maximize he likelihood funcion by a geneic algorihm (Holland (975)), which ue he evoluionary principle o olve difficul problem wih objecive funcion ha do no poe nice properie uch a coninuiy and differeniabiliy. The algorihm earche he oluion pace of a funcion, and implemen a urvival of he fie raegy o improve he oluion.

8 he parameer of he CIR, RSCIR and RSCIR-GARCH model. The daa e ranging from Ocober 8, 26 o February 27, 29, i obained from China Foreign Exchange Trading Syem & Naional Inerbank Funding Cener (CFETS). There are oal 25 weekly obervaion. The iniial SHIBOR rae i fir convered ino i coninuou compounding form and hen i ime erie and fir difference plo are repored in Figure, and ummary aiic are given in Table. Figure One-Monh SHIBOR and Fir Difference Table Summary Saiic of One-Monh SHIBOR Mean Sd Dev Skewne Kuroi Jarque-Bera One-Monh SHIBOR R *** Fir Difference Correlaion Coefficien Δ R *** Corr( R, Δ R ) = Quand-Andrew Breakpoin Te H : No breakpoin Saiic Dae Value p -value Max LR F -aiic /29/27.46 ***.247 Max Wald F -aiic.46 ***.247 Noe: *, ** and *** denoe ha he null hypohei H i rejeced a he ignificance level of %, 5% and % repecively. The opimal lag order for he ARCH LM e i deermined by aking ino accoun AIC, SBC and HQC all ogeher, and p -value in he Quand-Andrew breakpoin e are calculaed by he mehod inroduced in Hanen (997). A hown in Table, he negaive correlaion coefficien beween one-monh SHIBOR rae R and i fir difference Δ R implie mean reverion in he behavior of he hor rae. Beide, he kuroi coefficien of he wo variable are boh ignificanly larger han 3, and Jarque-Bera aiic how ha he normal diribuion hypohei i rejeced a he ignificance level of % for boh of he wo variable, which implie ha he diribuion of one-monh SHIBOR rae and i fir difference have ignificanly lepokuroi characeriic. Furhermore, i i eay o ee from Figure ha here i a ignifican volailiy cluering phenomenon in he ime erie of he one-monh SHIBOR rae R. In he ligh of he analyi

9 above, here i ignificanly ARCH effec in he dynamic of he one-monh SHIBOR rae R. Figure alo how ha he behavior of he hor-erm inere rae varie dramaically during he daa period. During Sepember hrough December in 27, he cenral bank purued a igh moneary policy and frequenly raied he benchmark inere rae o curb inflaion, which ignificanly affeced he behavior of SHIBOR. The period aw a harp increae in boh he level and volailiy of SHIBOR, and he reul of he Quand-Andrew breakpoin e in Table how ha here i a rucure break on he dae of Ocober 29, 27, which indicae ha here may be regime hif during hi period. To furher inveigae wheher here i regime wiching in he dynamic of he one-monh SHIBOR rae R, we apply he likelihood raio e propoed in Hanen (992) o i ime erie, leading o he following reul. * LRT - Saiic Table 2 Hanen Likelihood Raio Te Reul for One-Monh SHIBOR * Saiic of upq ( α ) Minimum Maximum Mean Median Sd Dev α T p -value *** Noe: The hypohei i H : β =, H : β <, where β = π. The grid conidered here i π =.2,.4,.6,.8, k =.2,.4,.6,.8, θ =.2,.4,.6,.8, σ =.2,.6,.,.4, λ =-, 6, 2,2,6,, π =.2,.4,.6,.8, and he likelihood raio aiic * * LR and aociaed p -value are gained uing he diribuion of up Q ( ) T T α α calculaed by Mone-Carlo imulaion. *** denoe ignificance a he % level. A hown in Table 2, he null hypohei i rejeced a % ignificance level, which implie ha here are ignificanly regime hif in he dynamic of he one-monh SHIBOR rae during he ample period. 4.2 Model Eimaion and Conra Wih he eleced daa, we eimae he CIR model uing he UKF baed maximum likelihood mehod, while he RSCIR model and RSCIR-GARCH model uing Kim filer baed mehod, whoe reul i repored in Table 3. A repored in Table 3, he value of he regime claificaion meaure (RCM) 3 for he RSCIR model and RSCIR-GARCH model are 6.3 and 4.9 repecively, which implie ha here definiely exi wo regime in he dynamic of one-monh SHIBOR rae. Parameer eimaion reul for he RSCIR model and 3 A defined in Ang and Bekaer (22), he RCM for wo ae i T RCM = 4 p ( p ) / T, where p i he moohed probabiliy for one of he wo ae. RCM i beween and. A good regime claificaion i aociaed wih low RCM aiic value. A value of mean perfec claificaion and a value of implie no informaion abou he regime i revealed. =

10 Table 3 Parameer Eimae of Model CIR RSCIR RSCIR-GARCH Value Sd Dev Value Sd Dev Value Sd Dev k.9529 *** *** ***.4 k.8 ***.32 θ.29 *** ***.72 θ.285 ***.7.9 *** *** ***.55 c.683 ***.8.3 ***..22 ***.8 c.2956 *** ***.62 a a b b ** *.6 λ λ π π Iniial Parameer *** *** *** *** *** ***.7.89 *** ***..75 ***.8 r Q r Q r Q RCM Noe: '*', '**' and '***' denoe ignificance a he level of %, 5% and % repecively. RSCIR-GARCH model provide furher informaion ha θ θ and σ σ hold under boh he wo model, which ugge ha he wo regime are he low-volailiy and high-volailiy regime repecively.

11 Then, we make a daa fine conra among he CIR, RSCIR and RSCIR-GARCH model in order o deermine he opimal hor rae model for he SHIBOR marke 4. The reul i repored in Table 4. Table 4 Model Fine Conra among CIR, RSCIR and RSCIR-GARCH CIR RSCIR RSCIR-GARCH MAE SAE RMSE Skewne Kuroi Jarque-Bera *** *** * ARCH LM *** *** * Log likelihood L R Noe: The 'MAE' and 'SAE' and for he mean and andard deviaion of he abolue error repecively. '*', '**' and '***' denoe ignificance a he level of %, 5% and % repecively. Fir, we carry ou a conra beween he CIR model and RSCIR model o inveigae he conribuion of he regime-wiching pecificaion o he improvemen of model fine. A hown in Table 4, he inroducion of regime wiching grealy improve he daa fine of he hor rae model, which can be inferred from he fac ha he log likelihood dramaically increae from of he CIR model o of he RSCIR model, while MAE and RMSE drop harply from.4829 and.4985 under he CIR model o.2676 and.273 under he RSCIR model repecively. Meanime, he inroducion of regime wiching alo ignificanly improve he fiing abiliy of he hor rae model, which i refleced in he dramaic decline in he SAE from.462 under he CIR model o.24 under he RSCIR model. Beide, he regime-wiching pecificaion parly capure he lepokuroi and volailiy cluering characeriic and hu reduce he ARCH effec in he dynamic of he hor rae ignificanly, which can be een from he fac ha he kuroi and ARCH LM aiic of he reidual erie decreae dramaically from and under he CIR model o and under he RSCIR model repecively. Thu he diribuion of he reidual i made cloer o normal, which i refleced in he harp drop in he Jarque-Bera aiic of he reidual erie from under he CIR model o under he RSCIR model. However, ARCH LM and Jarque-Bera aiic of he reidual erie are ill ignifican a he % level, which indicae ha here i ill ome par of he ARCH effec i unexplained, making he diribuion of he reidual ignificanly differen from normal. So i i neceary o inroduce he GARCH pecificaion of volailiy ino he model. Then, we carry ou a conra beween he RSCIR model and RSCIR-GARCH model o inveigae he 4 Following Banal and Zhou (22), we fir deermine he regime ae a ime by he moohed probabiliy for each regime, and hen ake he eimae of he hor rae in he dominae regime a he hor rae eimae under he RSCIR model or RSCIR-GARCH model.

12 conribuion of GARCH pecificaion o he improvemen of model fine. A hown in Table 4, he inroducion of he GARCH eing of volailiy furher improve he daa fine of he hor rae model, which can be inferred from he fac ha he log likelihood furher increae from of he RSCIR model o 6.39 of he RSCIR-GARCH model, while MAE and RMSE drop furher from.2676 and.273 under he RSCIR model o.2237 and.2385 under he RSCIR-GARCH model repecively. Meanime, he inroducion of GARCH pecificaion alo furher improve he fiing abiliy of he hor rae model, which i refleced in he furher decline in he SAE from.24 under he RSCIR model o.68 under he RSCIR-GARCH model. Beide, he GARCH pecificaion of volailiy alo help capure he lepokuroi and volailiy cluering characeriic and hu furher reduce he ARCH effec in he dynamic of he hor rae, which can be een from he fac ha he kuroi and ARCH LM aiic of he reidual erie decreae furher from and under he RSCIR model o 3.48 and under he RSCIR-GARCH model repecively. Thu he diribuion of he reidual i made furher cloer o normal, which i refleced in he furher drop in he Jarque-Bera aiic of he reidual erie from under he RSCIR model o under he RSCIR-GARCH model. Wih he inroducion of he GARCH pecificaion, ARCH LM and Jarque-Bera aiic of he reidual erie become inignifican a he 5% level, which indicae ha he GARCH pecificaion help explain mo of he reidual ARCH effec, making he diribuion of he reidual approximaely normal. Finally, we ue he likelihood raio e o inveigae he uperioriy of he RSCIR-GARCH model over he RSCIR model. The reul how ha he likelihood raio aiic LR GARCH = i ignifican a he level of %, which indicae ha he inroducion of he GARCH pecificaion ignificanly improve he daa fine of he model. Specially, he reul of model eimaion in Table 3 how ha he GARCH eing i ignifican in he high-volailiy regime bu no in he low-volailiy regime, which indicae ha he inroducion of he GARCH pecificaion mainly improve he model fine in he high-volailiy regime, bu help lile in he low-volailiy regime. Tha may be becaue he ARCH effec mainly exi in he high-volailiy regime. 4.3 Rik Premium Analyi The model conra above reveal ha RSCIR-GARCH model provide a much beer performance han he CIR model and RSCIR model in fiing he daa of he one-monh SHIBOR rae, o we ue he RSCIR-GARCH model o analyze he dynamic of he rik premium in he SHIBOR marke. Fir, we deermine he regime ae a each ample poin under he RSCIR-GARCH model. A hown in Table 3, here exi he low-volailiy regime ( = ) and he high-volailiy regime ( = ) in he dynamic of he one-monh SHIBOR rae under he RSCIR-GARCH model, and he wo regime are governed by he raniional probabiliy marix of a Markov chain Specially, he level and volailiy of he hor rae in he high-volailiy regime are 2.99 and.37 ime higher han hoe in he

13 low-volailiy regime repecively, while he marke price of rik in he high-volailiy regime i 2.56 ime higher han i counerpar in he low-volailiy regime. However, he mean reverion peed in he high-volailiy regime i ju a hiry-evenh of i counerpar in he low-volailiy regime, which implie ha he mean-reverion of he hor rae in he low-volailiy regime i much ronger han ha in he high-volailiy regime. The moohed probabiliy for he high-volailiy regime i diplayed in Figure 2, wih he ime erie plo of one-monh SHIBOR rae for beer inerpreaion. Figure 2 Smoohed Probabiliy for he High-Volailiy Regime The olid line and for he moohed probabiliy of he high-volailiy regime and he do line for one-monh SHIBOR rae. A hown in Figure 2, boh he level and volailiy of he hor rae oared during he period February May and Sepember December, 27, and ha may imply regime wiching in he dynamic of he SHIBOR erm rucure, which i furher proven by he high level of he moohed probabiliy for he high-volailiy regime. There are mainly wo reaon for he regime hif. Fir, he cenral bank purued a igh moneary policy and frequenly raied he benchmark inere rae o curb he increaing inflaion emerging a he end of 26, which ignificanly raied he level and volailiy of he hor rae. Second, hock from huge IPO alo make a caue. In he conex of he ock marke boom, huge IPO were iued frequenly, and he ubcriber bridged heir financing gap via he inerbank marke, which poed grea hock o he SHIBOR marke, furher increaing he volailiy of he hor rae. A ypical cae i he IPO of CNPC (China Naional Peroleum Corp) on Ocober 26, 27, which froze an acquiiion fund up o 3.37 rillion RMB, dramaically driving up he SHIBOR rae. Dramaic increae in he level and volailiy of he inere rae mean dramaic increae in invemen rik and hu i compenaion: rik premium, o he dynamic of he rik premium during he wo period are cerainly differen from hoe during he oher period. Figure 3 how he flucuaion of he acual rik premium 5 and i eimaion under he CIR model and he RSCIR-GARCH model. 5 We calculae he rik-free inere rae by conrucing he yield curve uing he exended Nelon-Siegel model baed on he price daa of reaury bond a each ample poin, and hen we obain he acual rik premium a he one-monh SHIBOR rae minu he rik-free inere rae.

14 Figure 3 Conra of Rik Premium Eimaion under CIR and RSCIR-GARCH model The olid line and for he acual rik premium, and he dah line and do line and for rik premium eimaion under he CIR model and RSCIR-GARCH model repecively. The haded area repreen he period for he high-volailiy regime. A hown in Figure 3, he rik premium ha a high level and flucuaion in he high-volailiy regime, while in he low-volailiy regime, in conra, he rik premium diplay a low mean and volailiy, o here exi ignifican regime hif in he dynamic of he rik premium. However, he CIR model, a a ingle-regime model, fail in capuring nonlinear change in he marke condiion uch a he long-erm expeced rae of reurn θ and he volailiy of he hor rae σ, and overeimae (undereimae) he marke price of rik in he low-volailiy (high-volailiy) regime, which reul in biaed eimaion of rik premium. A hown in Figure 3, he CIR model no only undereimae he rik premium in he high-volailiy regime bu overeimae he rik premium in he low-volailiy regime, which may caue mipricing of ae and hu failure in hedging. In conra, he RSCIR-GARCH model no only allow he parameer value hif beween regime bu e he volailiy in he GARCH form, capuring he regime-wiching and lepokuroi characeriic of he hor rae dynamic, which enure he eimaion accuracy and reliabiliy of he rik premium. The conra beween he aiic of rik premium eimaion error under he CIR model and RSCIR-GARCH model i repored in Table 5. Table 5 Conra beween Saiic of Rik Premium Eimaion Error Mean Sd Dev MAE SAE RMSE CIR RSCIR-GARCH Noe: The 'MAE' and 'SAE' and for he mean and andard deviaion of he abolue error repecively. A hown in Table 5, he mean, MAE and RMSE of rik premium eimaion error under he RSCIR-GARCH model are all ignificanly maller han heir counerpar under he CIR model 6, which indicae ha he RSCIR-GARCH model provide a beer eimaion accuracy of he rik premium han he CIR model. Meanime, he andard deviaion and SAE of rik premium eimaion error under he 6 Thi inference i aiically ignifican a he % level.

15 RSCIR-GARCH model are boh much maller han heir counerpar under he CIR model, which implie ha he RSCIR-GARCH model alo provide a beer eimaion reliabiliy of he rik premium han he CIR model. So he RSCIR-GARCH model i a preferred hor rae model which provide a beer model bai for ae pricing and rik hedging. 5 Concluding Remark Change in macroeconomic facor, uch a buine cycle condiion or moneary policy, may caue regime wif in he dynamic of inere rae, o regime-wiching model are uggeed inead of ingle-regime model in capuring he dynamic of he hor rae. In hi paper, we apply he RSCIR-GARCH model o decribe he behavior of he rik premium implied in he hor-erm inere rae in China, uing weekly one-monh SHIBOR rae a ample daa, whoe emphai i on capuring regime wiching in he dynamic of rik premium. Fir, we ue Hanen likelihood raio e and aiical mehod o inveigae he regime-wiching and lepokuroi characeriic of he hor rae dynamic, leading o he finding ha here exi ignificanly regime hif and ARCH effec in he dynamic of he one-monh SHIBOR rae during he ample period. Second, baed on he analyi above, we eablih he RSCIR model and RSCIR-GARCH model by inroducing regime-wiching and GARCH pecificaion ino CIR model ucceively. Then, a conra udy among CIR, RSCIR and RSCIR-GARCH model i performed baed on he one-monh SHIBOR ample daa, which how ha he regime-wiching and GARCH pecificaion boh improve he model fine ignificanly and eliminae he ARCH effec in he volailiy dynamic. In oher word, RSCIR-GARCH model provide a much beer performance han he CIR model and RSCIR model in fiing he daa of he one-monh SHIBOR rae. Third, in he ligh of he model conra reul above, we perform an empirical reearch on he rik premium dynamic of SHIBOR uing he RSCIR-GARCH model eimaed by he Kim filer baed maximum likelihood mehod, finding ha here are wo diinc regime, ha i, he regime of higher inere rae wih higher volailiy and he regime of lower inere rae wih lower volailiy, in he dynamic of boh he erm rucure and rik premium of SHIBOR. The high-volailiy regime i characerized by period of exremely high and volaile inere rae, and he correponding rik premium alo ha a high level and flucuaion, while in he low-volailiy regime, in conra, boh he hor rae and he rik premium diplay a low mean and volailiy. However, he mean-reverion of he hor rae in he low-volailiy regime i much ronger han ha in he high-volailiy regime. Beide, i i alo found ha he ARCH effec in he volailiy dynamic i only ignifican in he high-volailiy regime bu no in he low-volailiy regime. Furhermore, we make a comparion beween he CIR model and he RSCIR-GARCH model in capuring he dynamic of he rik premium implied in he hor rae, leading o he concluion ha he RSCIR-GARCH model by far ouperform he CIR model in capuring regime wiching in he dynamic of rik premium and hu in reflecing change in marke condiion, which i criical for pricing and hedging of ae.

16 Reference [] Ang and Bekaer. Regime Swiche in Inere Rae. Journal of Buine and Economic Saiic, 22, 2, [2] Banal and Zhou. Term Srucure of Inere Rae wih Regime Shif. Journal of Finance, 22, 57, [3] Bauwen L, Preminger A, Rombou J V K. Theory and inference for a Markov wiching GARCH model. Economeric Journal, 2, 3(2): [4] Chan, Karolyi, Longaff, Sander. An Empirical Comparion of Alernaive Model of he Shor-Term Inere Rae. Journal of Finance, 992, 47, [5] Chriianen. Level-ARCH Shor Rae Model wih Regime Swiching: Bivariae Modeling of US and European Shor Rae. Inernaional Review of Financial Analyi, 28, 7, [6] Cox, Ingeroll and Ro. A Theory of he Term Srucure of Inere Rae. Economerica, 985, 53, [7] Dahlqui and Gray. Regime-Swiching and Inere Rae in he European Moneary Syem. Journal of Inernaional Economic, 2, 5, [8] Dai and Singleon. Term Srucure Dynamic in Theory and Realiy. Review of Financial Sudie, 23, 6, [9] Dai, Singleon and Yang. Regime Shif in a Dynamic Term Srucure Model of U.S. Treaury Bond Yield. Review of Financial Sudie, 27, 2(5), [] Erlandon. Regime Swiche in Swedih Inere Rae. Deparmen of Economic, Lund Univeriy, 22, Working Paper. [] Garcia and Perron. An Analyi of he Real Inere Rae Under Regime Shif. Review of Economic Sudie, 996, 78, 25. [2] Gray. Modeling he Condiional Diribuion of Inere Rae a a Regime-Swiching Proce. Journal of Financial Economic, 996, 42, [3] Hamilon. Raional Expecaion Economeric Analyi of Change in Regime: An Inveigaion of he Term Srucure of Inere Rae. Journal of Economic Dynamic and Conrol, 988, 2, [4] Hanen. Approximae Aympoic P Value for Srucural-Change Te. Journal of Buine and Economic Saiic, 997, 5(), [5] Harrion and Seven. Bayeian forecaing. Journal of he Royal Saiical Sociey, 976, B 38, [6] Holland. Adapaion in Naural and Arificial Syem. Ann Arbor: The Univeriy of Michigan Pre, 975. [7] Julier and Uhlmann. A New Exenion of he Kalman Filer o Nonlinear Syem. The Proc of Aeropace, 997, 87(2), [8] Kim. Dynamic Linear Model wih Markov-Swiching. Journal of Economeric, 994, 6, -22.

17 [9] Kim and Nelon. Sae Space Model wih Regime Swiching. Cambridge: MIT Pre, 999. [2] Landen. Bond Pricing in a Hidden Markov Model of he Shor Rae. Finance and Sochaic, 2, 4, [2] Naik and Lee. Yield Curve Dynamic wih Dicree Shif in Economic Regime. Faculy of Commerce, Univeriy of Briih Columbia, 997, Working Paper. [22] Smih D R. Markov-wiching and ochaic volailiy diffuion model of hor-erm inere rae. Journal of Buine and Economic Saiic, 22, 2(2): [23] Sola and Driffill. Teing he Term Srucure of Inere Rae from a Saionary Swiching Regime VAR. Journal of Economic Dynamic and Conrol, 994, 8, [24] Vaicek. An Equilibrium Characerizaion of he Term Srucure. Journal of Financial Economic, 977, 5,