First version: September 1997 This version: October On the Relevance of Modeling Volatility for Pricing Purposes

Transcription

1 First version: September 1997 This version: October 1999 On the Relevnce of Modeling Voltility for Pricing Purposes Abstrct: Mnuel Moreno 3 Deprtment of Economics nd Business Universitt Pompeu Fbr Crrer Rmon Tris Frgs, Brcelon, Spin. phone: (34-93) fx: (34-93) e-mil: mnuel.moreno@econ.upf.es. This pper presents two-fctor (Vsicek-CIR) model of the term structure of interest rtes nd develops its pricing nd empiricl properties. We ssume tht defult free discount bond prices re determined by the time to mturity nd two fctors, the long-term interest rte nd the spred. Assuming certin process for both fctors, generl bond pricing eqution is derived nd closed-form expression for bond prices is obtined. Empiricl evidence of the model's performnce in comprisson with double Vsicek model is presented. The min conclusion is tht the modeling of the voltility in the long-term rte process cn help (in lrge mount) to t the observed dt cn improve - in resonble quntity - the prediction of the future movements in the medium- nd long-term interest rtes. However, for shorter mturities, it is shown tht the pricing errors re, bsiclly, negligible nd it is not so cler which is the best model to be used. Keywords: Term Structure of Interest Rtes, Bond Pricing Eqution, Two- Fctor Models, Ornstein-Uhlenbeck Process, CIR Process. JEL clssiction: C51, E43, G13.

2 The evolution over time of interest rtes for defult-free zero-coupon bonds is topic tht hs been extensively nlyzed in the nncil literture. Initilly, the nlysis of this evolution ws performed by mens of one-fctor models which ssume tht movements in interest rtes re driven by chnges in the short-term (instntneous) riskless interest rte (see, mong others, Vsicek (1977), Cox et l (1985) or Chn et l (1992)). However, it is now widely ccepted tht interest rtes re ected by more thn one stte vrible. In this direction, severl ppers s Richrd (1978), Brennn nd Schwrtz (1979), Schefer nd Schwrtz (1984), Cox et l (1985), Longst nd Schwrtz (1992), Due nd Kn (1996), Chen (1996), Di nd Singleton (1997) nd Boudoukh et l (1999) use multiple fctors to explin the future movements tht interest rtes my show. There is substntil empiricl evidence 1 tht shows tht movements in interest rtes cn be decomposed in three types of \bsic" chnges relted to the level of interest rtes, the slope, nd the curvture of the yield curve. As the curvture is usully the less importnt explntory vrible when deling with spot interest rtes, we cn think tht movements in spot interest rtes my be resonbly well explined by the two rst fctors. In fct, this is the motivtion for the model previously presented nd developed in Moreno (1996) which uses the long-term interest rte nd the spred of interest rtes s stte vribles (tht is, the dierence between the long-term interest rte nd the short-term interest rte is used s rough mesure of the slope of the yield curve). In tht pper, both fctors re ssumed to follow Vsicek process nd, therefore, both vribles (1) show men reversion to certin long term vlue nd (2) their diusions reect constnt vrince term. Under these ssumptions, generl bond pricing eqution ws derived nd closed-form expression for zero-coupon bond nd for interest rte derivtives prices ws computed, This pper lso presented the empiricl performnce of this model in reltion to n lterntive onefctor model. It cn be rgued tht one of the ssumptions mde in Moreno (1996), nmely, the constnt vrince in the diusion of the processes followed by 1 See, for instnce, Jones (1991), Littermn nd Scheinkmn (1991), Zhng (1993) nd Knez, Littermn, nd Scheinkmn (1994). 1

3 both fctors is too restrictive from n empiricl point of view. 2 This restrictive feture leds to the present pper whose min objective is to nlyze if modeling the voltility improves the empiricl performnce of the Moreno (1996) model. Thus, the sme stte vribles will be used lthough we will ssume tht the long-term interest rte does not follow Vsicek process but root-squre (CIR-type) process. This lterntive model will be denoted herefter s the Vsicek-CIR model nd it cn be considered, from theoreticl point of view, s n specil cse of the Schefer nd Schwrtz (1984) model. The schedule of this pper is s follows. Section 2 presents the min ssumptions of the Vsicek-CIR model nd provides the bsic pricing eqution tht ny derivtive sset must stisfy. This eqution, with the pproprite terminl condition, llows us to obtin the price of ny sset tht, t mturity, pys certin pyo s indicted in such terminl condition. In this section we () compute the nlyticl expression tht indicte the price of ny discount bond under the ssumptions given by this two-fctor model nd (b) recll the nlogous formul tht ws obtined in Moreno (1996) (Vsicek-Vsicek model herefter). Section 3 nlyzes the empiricl behvior of both models by compring the usefulness of these lterntive formuls to t nd forecst bond prices, tht is, the in- nd out-of-smple performnce of such expressions. The dt nlyzed correspond to Spnish interest rtes nd bond prices for dierent mturities during the period Finlly, Section 4 summrizes nd concludes. In this section we present the two-fctor (Vsicek-CIR) model tht we will use to price defult-free discount bonds by deriving (nd solving) the pricing eqution which must be veried by the prices of these bonds. The min ssumption of this model is tht the price, t time t, of defult-free discount bond tht pys $1 t mturity T depends only on the current vlues of two stte vribles nd time to mturity, = T t. The 2 Interest rte voltility is usully incresing in interest rte level lthough there is no consensus bout the exct reltionship between voltility nd level. See Chn et l (1992), At-Shli (1996), Conley et l (1997) nd Stnton (1997) for this issue. 2

4 min motivtion for the fctors to be used is the empiricl evidence (see footnote 1) tht chnges in interest rtes re combintion of movements in () the level of interest rtes, (b) the slope nd (c) the curvture of the yield curve, which eect is usully negligible. Therefore, we cn use the long-term rte nd the spred s the vribles tht help us to explin the movements in the generl level of interest rtes nd chnges in the reltionship between the short nd the long end of the yield curve. With both vribles, we cn lso try to explin the intermedite movements of the yield curve. 3 Although most previous studies use the short-term interest rte s one of the stte vrible, we redene these vribles nd, nlogously to Schefer nd Schwrtz (1984), the fctors to be used re the long-term rte, denoted by L, nd the spred, denoted by s, the dierence between the long- nd the short-term rte, denoted by r. This selection of stte vribles llows us to use the ssumption of orthogonlity between them. 4 Once chosen these vribles, we ssume tht their evolution over time is given by the following stochstic dierentil equtions 5 : ( ds = 1 (s; L)dt + 1 (s; L)dw 1 dl = 2 (s; L)dt + 2 (s; L)dw 2 (1) where t denotes clendr time, nd dw 1 nd dw 2 re stndrd Brownin processes where E[dw 1 ]=E[dw 2 ]=,dw 2 = 1 dw2 2 = dt, nd (by the orthogonlity ssumption) it is veried tht E[dw 1 dw 2 ]=. 1 (:) nd 2 (:) re the expected instntneous rtes of chnge in the stte vribles nd 1(:) 2 nd 2(:) 2 re the instntneous vrinces of chnges in these fctors. Let P (s; L; t; T ) P (s; L; ) be the price, t time t, of defult-free discount bond tht pys $1 t mturity T = t +. We cn express the instntneous percentge chnge in the price of this bond s the sum of its expected rte of return nd the unexpected vritions in return due to the 3 Two lterntive couples of fctors to be used my be: () the long-term interest rte nd the short-term interest rte nd (b) the short-term interest rte nd the spred. However, the bove two vribles re chosen becuse of better nlyticl trctbility. 4 This ssumption simplies the nlyticl trctbility of the model. Empiricl evidence tht supports this ssumption cn been seen in Ayres nd Brry (198), Schefer (198), Nelson nd Schefer (1983) nd, for the Spnish cse, in Moreno (1996). 5 After presenting this generic model nd deriving the generl pricing eqution, we will prticulrize it to obtin the Vsicek-CIR model 3

5 rndom chnges in the fctors dp (s; L; t; T ) P (s; L; t; T ) = (s; L; t; T )dt + s 1(s; L; t; T )dw 1 + s 2 (s; L; t; T )dw 2 (2) The steps to be given to obtin the bond pricing eqution re very stndrd 6 nd cn be summrized s follows: 1. Appliction of It^o's Lemm 2. Setting up of (hedging) portfolio, composed of three bonds with different mturities, tht is instntneously riskless 3. Under no-rbitrge conditions, the expected rte of return of this portfolio must equte the instntneous riskless rte of interest These three steps jointly with little lgebr led us to the following prtil dierentil eqution 1 2 [2 1(:)P ss + 2(:)P 2 LL ]+[ 1 (:) 1 (:) 1 (:)]P s +[ 2 (:) 2 (:) 2 (:)]P L + P t rp = (3) where subscripts denote prtil derivtives. The coecients 1 (:) nd 2 (:) cn be interpreted s the mrket prices of the spred nd long-term rte risk, respectively. Therefore, given the stochstic process (1) we hve ssumed for both vribles, (3) is the fundmentl eqution for the pricing of defult-free discount bonds of dierent mturities which depend solely on the spred, the longterm interest rte, nd its time to mturity. In this eqution we del with the mrket prices of risk, i (:), becuse the only wy to tie down the bond prices in our (prtil equilibrium) model is by mens of these (exogenous) prmeters. The solution of the eqution (3), subject to the terminl condition given by the nl pyment of the bond, P (s; L; )=1; 8s; L, is the price of the discount bond we re looking for. 6 For more detils, see Moreno (1996). 4

6 The coecients of the bond pricing eqution (3) re the prmeters of the stochstic process (1) which ws ssumed for the two fctors nd the mrket prices of the risk relted to both stte vribles. As this eqution is too generl to be solved nlyticlly, we will mke the following ssumptions bout these coecients: Assumption 1 The mrket price of the spred risk is liner in this vrible, tht is 1 (:) =+bs Assumption 2 The mrket price of the long-term rte risk is proportionl to the squre root of this vrible, tht is p 2 (:) =d L Assumption 3 Ech of the stte vribles follow diusion process ( ds = k1 ( 1 s)dt + 1 dw p 1 (4) dl = k 2 ( 2 L)dt + 2 Ldw 2 The motivtion for the rst two ssumptions is tht constnt mrket price of risk is too restrictive nd quite unrelistic. The rst ssumption is the generliztion of the one presented in Vsicek (1977) while the second one is similr to the one obtined in Cox et l (1985). Regrding the third ssumption, the rst process, known s Ornstein-Uhlenbeck process, hs been used previously by Vsicek (1977) while the second one ws proposed in Cox et l (1985). Both processes show men reversion, n importnt stylized fct tht interest rtes usully show. In the process ssumed for the spred, we nd constnt vrince in the diusion term while the vrince of the longterm rte is proportionl to its level. For ech stte vrible, k i > isthe coecient of men reversion which reects the speed of djustment of the vrible towrds its long-run men vlue, i, nd dw i re stndrd Brownin motions. Under these three ssumptions, we cn rewrite the eqution (3) s P ss + q 1 (^ 1 s)p s LP LL + q 2 (^ 2 L)P L + P t (L + s)p = (5) 5

7 q subject to the terminl condition where P (s; L; )=1; 8s; L (6) q 1 = k 1 + b 1 ; ^ 1 =(k )=q 1 q 2 = k 2 + d 2 ; ^ 2 = k 2 2 =q 2 Solving the prtil dierentil eqution (5) we obtin the following proposition: Proposition 1 The vlue t time t of discount bond tht pys $1 t time T, P (s; L; t; T ) P (s; L; ), is given by where = T t nd P (s; L; t; T )=A()e B()sC()L (7) A() = A 1 ()A 2 () ( ) A 1 () = exp 2 1 B 2 ()+s 3 (B()) 4q 1 2 2exp n o 3 (q 2 + ) 2 A 2 () = 4 5 (q 2 + ) expfg +(q 2 ) B() = 1 eq 1 C() = q 1 2(expfg1) (q 2 + ) expfg +(q 2 ) 2k 2 2 = 2 2 (8) with q 1 = k 1 + b 1 ; ^ 1 =(k )=q 1 ; s 3 =^ =(2q2 1) q 2 = k 2 + d 2 ; ^ 2 = k 2 2 =q 2 ; = q (9) Proof: See Appendix. 6

8 The terms in eqution (8) verify <A i ()<1; 8>; A i ()=1; A i (1)=; i=1;2 <B()<; 8>; B() = ; B(1)=1=q 1 (1) <C()<; 8>; C() = ; C(1)=2=(q 2 +) Substituting t = T into (7), it is shown tht the terminl condition for the price bond, P (s; L; ) = 1; 8s; L, is stised. Moreover, it is lso derived tht P (; ;)=A()=A 1 ()A 2 ()<1; 8> It cn be checked tht the following relistic fetures re veried lim P (s; L; ) = lim P (s; L; ) = lim P (s; L; ) = s!1 L!1!1 tht is, when ny of the rguments included in the bond price formul tends to innity, the price converges to zero. It is lso esily shown tht the bond price function is decresing nd convex in both fctors nd decresing with the time to mturity. Once we hve obtined the expression (nd properties) for the bond price formul under the Vsicek-CIR model, we will recll the ssumptions mde in Moreno (1996) nd the corresponding pricing formul tht ws derived in tht pper: Assumption 1' (equl to Assumption 1) The mrket price of the spred risk is liner in this vrible, tht is 1 (:) =+bs Assumption 2' The mrket price of the long-term rte risk is liner in this vrible, tht is 2 (:) =c+dl Assumption 3' Ech of the stte vribles follow diusion process of Vsicek type ( ds = k 1 ( 1 s)dt + 1 dw 1 (11) dl = k 3 ( 3 L)dt + 3 dw 3 Under these ssumptions, we cn rewrite the eqution (3) s P ss + q 1 (^ 1 s)p s P LL + q 3 (^ 3 L)P L + P t (L + s)p = (12) 7

9 where q 1 = k 1 + b 1 ; ^ 1 =(k )=q 1 q 3 = k 3 + d 3 ; ^ 3 =(k 3 3 c 3 )=q 3 The solution of the dierentil eqution (12), subject to the terminl condition given by the pyo of the bond t mturity (see eqution (6)), ws estblished in the following proposition: Proposition 2 (Proposition 1 in Moreno (1996)) The vlue t time t of discount bond tht pys $1 t time T, P (s; L; t; T ) P (s; L; ), is given by where = T t nd P (s; L; ) =D()e E()sF()L (13) D() = D 1 ()D 3 () ( ) D 1 () = exp 2 1 B 2 ()+s 3 (B()) 4q 1 ( ) D 3 () = exp 2 3 C 2 ()+L 3 (C()) 4q 3 E() = 1eq 1 q 1 F() = 1eq 3 q 3 (14) with q 1 = k 1 + b 1 ; ^ 1 =(k )=q 1 ; s 3 =^ =(2q2 1) q 3 = k 3 + d 3 ; ^ 3 =(k 3 3 c 3 )=q 3 ; L 3 =^ =(2q2 3) (15) Proof: It is similr to the proof of Proposition 1 nd it is omitted for the ske of brevity. 8

10 In this section, we describe the empiricl ppliction in which we compre the tting nd forecsting behvior of the Vsicek-CIR nd the Vsicek-Vsicek models. This comprison is performed nlyzing the in- nd out-of-smple properties of both models. The dtset consists of dily Spnish interest rtes nd zero-coupon bond prices nd cover the period For ech dy of this period, we hve interest rtes (in nnulized form) nd bond prices for ten dierent mturities: 1, 7, nd 15 dys, 1, 3, nd 6 months, nd 1, 3, 5, nd 1 yers. The interest rtes corresponding to the shortest nd longest mturity (1 dy / 1 yers) re used s proxies of the short- nd long-term interest rte, respectively. The min descriptive chrcteristics of the stte vribles used in both two-fctor models re: 1. For both interest rte series, the unconditionl verge is lrger thn 1%. Short-term interest rtes re lrger thn this men vlue until October 1993 while the long-term interest rtes exceed this level in the whole period except from June 1993 through June On the other hnd, the spred hs men vlue very close to zero nd rnges between 4% nd 8%. 2. The short-term rte is more voltile nd moves into wider intervl thn long-term rtes do. 3. Both stte vribles show n uniformly high degree of seril correltion. 4. Most of the chnges in the short-term interest rtes re smller thn 1 bsis points while chnges in long-term rtes re much smoother. As consequence, chnges in the spred re quite similr to chnges in short-term interest rtes. 5. It is seen smll decrese - in men - in interest rtes through the smple period. 6. Evidence of men reversion in spred nd interest rtes is derived. 7 For more detils on these dt, see Nu~nez (1995) for technicl detils on the procedure used by the Bnk of Spin to estimte them nd Moreno (1996) for descriptive nd grphicl nlysis. 9

11 7. The theoreticl ssumption bout the orthogonlity between the stte vribles is empiriclly corroborted. Next we present the empiricl performnce of both models. We recll tht both models use the sme fctors nd the min dierence between them derives from the lterntive processes ssumed for the long-term rte, L. Ech stte vrible of the two competing models, s nd L, follows diusion process (see equtions (4) nd (11)). The diusion prmeters of these processes (k i ; i ; i ; i =1;2;3) re estimted by the Generlized Method of Moments presented in Hnsen (1982) 8. The econometric speciction in discrete time is s t s t1 = 1 + b 1 s t1 + " s t ; "s t IID (;1) 2 L t L t1 = 2 +b 2 L t1 +" L t ; "L t IID (; 2 2 r t1) L t L t1 = 3 +b 3 L t1 +" L t ; "L t IID (; 2) 3 so tht k i = b i ; i = i ; i =1;2;3 b i Tble I includes the estimtion results obtined for the smple period nd shows tht the prmeters b i of the discrete time speciction (nd hence, the diusion prmeters k i ) re signicntly dierent from zero. So, there is evidence of men reversion in ll the stte vribles. In both models, the long-term interest rte tends to men vlue close to 1% while the spred tends to men vlue close to zero. Compring the two processes ssumed for the long-term rte, it my be interesting to recognize tht, under the CIR model, the long rte reverts fster to its long-term vlue thn when considering the Vsicek ssumption. After estimting the prmeters of the diusion processes followed by the fctors in both models, these vlues re used to obtin the remining prmeters of equtions (7) nd (13). Thus, similrly to Moreno (1996), we use the specictions P = P (q1;q2;s 3 jk1;k2;1;2;1;2;s; L; )+" P =P(q1;q3;s 3 ;L 3 jk1;k3;1;3;1;3;s; L; )+" (16) 8 For detils on this technique nd its pplictions in the estimtion of continuous-time models, see Moreno nd Pe~n (1996). 1

12 where P is the observed price of the discount bonds vilble t time t, P (:) is the closed-form pricing eqution for ech model (see equtions (7) nd (13)) nd " is n error term. The prmeters of the equtions (16) (q i ;i =1;2;3;s 3 ;L 3 ) re estimted on dily bsis for the period by mens of pnel of dt where we hve dily yield curves contining cross-section of discount bond prices. Therefore, we hve mtrix with 123 rows nd 1 columns where ech row includes the (ten) zero-coupon bond prices vilble t ech dy nd ech column contins the bond prices for certin mturity. We estimte the non-liner equtions (16) for ech dy of the period The estimtion of the rst eqution provides the prmeters of the Vsicek-CIR model (tht is, q1; q2; s 3 ) while the estimtion procedure, when pplied to the second eqution, provides the prmeters of the Vsicek- Vsicek model, tht is, q1; q3;s 3 ;L 3. Estimtion results for the dily prmeters of the Vsicek-CIR model re included in Tble II. 9 This tble shows the verge of the estimted prmeters obtined for () the full smple period nd (b) the smple period divided yer by yer nd reects tht ll the prmeters re positive nd highly signicnt. The evolution over time of these prmeters cn be seen grphiclly in Figure 1. This gure shows tht the highest vlues re ttined in 1991 while the lowest (nd more stble) prmeters correspond to the period In the next step, we cn compute the vlues, dy by dy, of the mrket prices of risk relted to ech stte vrible using the estimted prmeters obtined from eqution (16) jointly with the expressions (9) nd (15) nd the Assumptions 1 nd 2. A grphicl representtion of these vlues, for the Vsicek-CIR model, cn be seen in Figure 2. The verge vlues of these prices - under both models - for the whole period nd for every yer, re included in Tble III. Anlyzing the two fctors of the Vsicek-CIR model, both mrket prices of risk re highly signicnt nd hve similr behvior cross the period : ech price hs lwys the sme sign during ll the period (for the long-term rte, the mrket price of risk is lwys positive nd 9 We do not show the results for the Vsicek-Vsicek model tht cn be seen in Tble VIII in Moreno (1996). Tht pper provided ll the results for the Vsicek-Vsicek model tht re included in the following tbles in this pper. 11

13 the risk relted to the spred hs lwys negtive price) nd both series of mrket prices re specilly low in 1992 nd the second hlf of the period In bsolute terms, the price of the spred risk is, t every moment, much higher thn the price of risk of the long-term rte. On the other hnd, for the Vsicek-Vsicek model, we cn observe very dierent behvior between both fctors nd with respect to the lterntive model. Thus, the two bsic fetures for the prices of risk in this model re: 1. For the full period, the mrket prices of risk for both stte vribles re positive nd signicntly dierent from zero. 2. Anlyzing this period yer by yer, the prmeters re lso signicntly dierent from zero but they show chnging sign. Thus, the men mrket price of risk of the spred is negtive in the lst two yers of the smple period while the verge of the mrket price of risk relted to the long-term rte is negtive in nd Finlly, we will use the vlues of the diusion prmeters jointly with the prmeters estimted by mens of the eqution (16) to nlyze the tting nd forecsting power of both two-fctor models. The within- nd out-of-smple periods re nd 1995, respectively. First, the in-smple estimted dt, for ech dy of the period nd for both models, re provided by the inclusion of the (dily) estimted prmeters nd the estimted prmeters of the diusion processes in the non-liner eqution (16). Next, we will compre the out-of-smple properties of both models by using the k-step-hed forecsts tht re generted for the bond prices. These t + k-time forecst vlues re built using the coecient estimted from time t. This procedure is repeted for ech dy of Once obtined the in- nd out-of-smple forecsts, the (within nd outof-smple) pricing errors of both models re computed to compre one ech other. Thus, we dene, for time t, the error, e t, nd the percentge error, PE t,s e t =P t ^P t ; PE t = P t ^Pt 21 P t where P t nd ^Pt re, respectively, the observed nd the estimted (tted or forecsted) price, for time t, of the zero-coupon bond with certin mturity. 12

14 For both models, the within-smple (bsolute nd percentge) pricing errors re shown in Figures 3 nd 4. For ll the mturities, it cn be seen tht the Vsicek-Vsicek model provides very lrge pricing error in My, This error coincides with shrp chnge in the short-term rte nd in the spred. For both models, neither gure suggests systemtic pttern in these pricing errors. Denoting by N the number of dys of the period to be nlyzed, we use the pricing errors to compute severl ccurcy mesures tht help us to compre the empiricl performnce of both models: 1. Men Error (ME). This mesure weights eqully the dily errors. Therefore, positive vlues cn be oset with negtive vlues nd, thus, this mesure my be smll even with lrge errors. Its expression is NX ME = 1 N t=1 e t = 1 N NX (P t ^P t ) t=1 2. Men Absolute Error (MAE). As the men error, this mesure gives n equl weight to the dily errors but positive nd negtive errors do not cncel out. It is dened s MAE = 1 NX N je t j = 1 NX jp t ^Pt j t=1 N t=1 3. Root Men Squred Error (RMSE). It is usully the most common mesure of ccurcy nd its denition is vu u RM SE = t 1 NX vu (e t ) 2 u = t 1 NX (P t N t=1 N ^P t ) 2 t=1 4. Men Percentge Absolute Error (MAPE) Similrly to the men bsolute error, the bsolute vlue of the error is used but ech error is weighted by the current vlue of the bond price. Its expression is NX MAPE = 1 jpe t j N t=1 13

15 5. Root Men Squred Percentge Error (RMSPE). This mesure is similr to the root men squred error but, similrly to the MAPE, the dily errors re weighted by the ctul bond prices. It is given by vu u RMSP E = t 1 NX (PE t ) 2 N t=1 These ve descriptive mesures, for both models, re computed for (within smple period), for 1995 (out-of-smple period) nd for dierent subperiods. The within nd out-of-smple results re reported in Tbles IV- VI nd Tbles VII-X, respectively. The performnce of both models, for the within-smple period, is included in Tble IV. For this period, the Vsicek-Vsicek model underprices the shortnd medium-term bonds nd overprices the bonds whose mturity is beyond one yer. On the other hnd, the Vsicek-CIR model overprices the bonds with mturities up to three months s well s the 5-yer bonds. All the sttistics included in Tble IV reect tht both models provide very big ccurcy to the observed bond prices for ll the mturities. It cn be seen tht the pricing errors re incresing with the mturities (the longer the mturity, the lrger the error price) but the MAPE for the Vsicek-Vsicek nd the Vsicek-CIR models is smller thn :26% nd :2%, respectively. Moreover, in the Vsicek-CIR (Vsicek-Vsicek) model, this sttistic is lwys smller thn :1% (:8%) except for the 5-yer bond price. Compring both models, the Vsicek-CIR model outperforms the Vsicek- Vsicek model for ll mturities nd for ll the sttistics. In short-term bonds, with mturities smller thn one month, both models provide negligible errors nd the improvement obtined with the Vsicek-CIR model over the other one is not very lrge. On the other hnd, focusing on the mturities beyond one month, the Vsicek-CIR model provides huge improvement: the errors from the Vsicek- Vsicek model re decresed in more thn 88% for ll these mturities. The biggest improvement in ccurcy is chieved in the medium-term mturities (six month nd one yer) nd in 5-yer bonds, mturity in which the error mesures from the Vsicek-CIR model re bout 6% of the error mesures provided by the Vsicek-Vsicek model. This conclusion is obtined for ll the sttistics. 14

16 Tble V includes the results obtined for the yer Similrly to the period , the Vsicek-Vsicek model underprices the shortest mturities (up to six months). In contrst to wht hppened in the whole within-smple period, this underpricing cn lso be seen in the Vsicek-CIR model which, on the other hnd, overprices the bonds tht mture beyond one yer. In this yer, both models t the observed dt specilly well. Thus, the error mesures for both models re decresed in more thn hlf for most of the mturities with respect to the whole period. Compring both models, the error mesures of the Vsicek-CIR model, s in the whitin-smple period, re bout 1% of the sttistics provided by the Vsicek-Vsicek model for ll the mturities longer thn 15 dys. Therefore, the min conclusion for this yer is the sme thn the obtined for the period : for the shortest mturities, both models t specilly well to the dt but, for most of the remining mturities, the Vsicek-CIR model provides remrkble lrge improvement in ccurcy. Severl subperiods hve been nlyzed nd, bsiclly, the sme conclusions re reched. For illustrtive purposes, Tble VI includes the whitinsmple results obtined for 1-yer bonds for ech semester of the period Looking t every sttistic, it is shown tht the Vsicek-CIR model ts better thn its competing model in ll the semesters nd it works specilly well in 1991 nd 1994 while the Vsicek-Vsicek model obtins its best performnce in the rst semester of 1992 nd in the second one of Bsed on men bsolute error (MAE or MAPE) criterion, the superiority of the Vsicek-CIR model implies n improvement of bout 9% in ll the semesters but the lst one in which the errors from the Vsicek-Vsicek model re decresed in 'just' 77%. For ll the semesters, the men bsolute percentge error of the Vsicek- CIR model is round :3% tht is bout fteen times smller thn the obtined for the Vsicek-Vsicek model. This superiority is specilly remrkble in the second semester of 1991 nd in the rst one of 1994 when the bsolute vlue of the errors re decresed in more thn 96%. The forecsting power of both models is nlyzed by computtion of onend ve-step hed forecsts 1 of bond prices for every mturity nd for every 1 Ten-step-hed forecsts were lso computed. Results re vilble upon request. 15

17 dy of the yer Mesures of the forecsting pricing errors re included in Tbles VII-X. Tbles VII-VIII include the mesures for one-step-hed forecsts. Thus, Tble VII provides the one-step-hed mesures for the whole out-of-smple period. It cn be seen tht (1) both models forecst quite well, (2) the forecsting power decreses with the time to mturity, nd (3) for both models, the MAPE (RMSPE) is lwys smller thn :36% (:48%). It cn lso be seen tht both models perform similrly for the shortest mturities nd, in fct, the Vsicek-Vsicek model performs slightly better thn the Vsicek- CIR model. For mturities beyond one month, the Vsicek-CIR model forecsts better thn the Vsicek-Vsicek model showing tht the modeling of the voltility in the long-term rte process helps to predict the movements in the mediumnd long-term interest rtes. This superior forecsting performnce is not monotonic in the time to mturity. Thus, the lrgest improvement is obtined in the 1-yer bond prices when the error mesures from the Vsicek-Vsicek model re decresed in more thn 21% (15%) when working on (root) men bsolute criterion. In the remining bonds, the improvement in the reltive forecsting power is much smller (2 7%) nd never exceeds 12%, vlue tht is obtined when forecsting the 5-yer bonds. Tble VIII provides the error mesures obtined, for every month of 1995, from 1-yer bonds in which the Vsicek-CIR model chieves its best reltive performnce. Both models perform better in the second semester (with MAPE (RMSPE) smller thn :6% (:8%)) thn in the rst one, when the MAPE (RMSPE) reches :1% (:12%). It cn lso be seen similr behvior in both models from Jnury to April (with smll superiority of the Vsicek-Vsicek model in this period) nd in the two lst months of In the period My-October, the Vsicek-CIR model chieves n improvement of the out-of-smple performnce tht rnges between 2% (in My) nd 65% (in July). Finlly, Tbles IX-X show the results obtined when the forecsting horizon is ve dys. Thus, Tble IX includes the mesures obtined with the out-of-smple errors for ll the mturities in Although these mesures re bigger thn in the shorter forecsts, they re resonbly smll s reected in the MAPE or the RMSPE tht re, for both models, smller thn :8% nd 1%, respectively. 16

18 The results re nlogous to the obtined with the previous predictions: (1) the forecsting power decreses with time to mturity, (2) the Vsicek- Vsicek model outperforms its rivl model in the mturities smller thn six months, nd (3) the Vsicek-CIR forecsts better thn the Vsicek-Vsicek model in the mturities beyond six months. However, this improvement is usully quite smll (between 1% nd 3%) nd only increses until 5% when forecsting 1-yer bond prices. The lst tble provides the qurterly results obtined with ve-step-hed forecsts for 1- nd 5-yer bonds. In both mturities, it cn be seen better forecsting behvior in the second hlf of 1995 thn in the rst one. For 1-yer bonds, both models show MAPE smller thn :2% in ll the qurters nd the Vsicek-CIR model outperforms the Vsicek-Vsicek model whose error mesures re decresed between 5% nd 15% in the period April-September. Focusing on the forecsting errors for 5-yer bond prices, the MAPE sttistic rnges between :6% nd 1%. In this cse, the Vsicek-CIR model improves the forecsts from its competing model in 1% from July to September nd, in the remining qurters, its improvement is much smller (12%). This pper hs presented two-fctor (Vsicek-CIR) model in continuous time for the nlysis of the terms structure of interest rtes nd its empiricl behvior hs been nlyzed with respect to second lterntive model. The min common chrcteristic of these two models is tht both employ the sme fctors (stte vribles) to explin the unexpected chnges tht interest rtes my show in the future. These fctors re the long-term interest rte nd the spred, the dierence between the long- nd the short-term interest rte. The Vsicek-CIR model ssumes tht the spred follows Vsicek process while the long-rte is modeled s CIR-type process. On the other hnd, the Vsicek-Vsicek model hs ssumed tht both vribles follow Vsicek process. This second model ws previously presented Moreno (1996) which lso developed its pricing properties, the implictions on the term structure of interest rtes nd nlyzed its empiricl properties with smple of dily interest rtes nd bond prices tht cover the period The min objective of this pper is nlyze nd compre the empiricl 17

19 performnce of both models in the period Therefore, we cn determine if modeling the voltility of the long-term interest rte my help us to explin the future movements of interest rtes. As strting point, we hve derived bond pricing eqution whose solution indictes the price of zero-coupon bond under certin ssumptions, nmely, this price depends solely on the current vlues of two stte vribles (mentioned bove) nd the time to mturity of the bond. After this solution is obtined, it is used to nlyze the tting nd forecsting properties of this model. These properties re, in posterior stge, compred with the ones derived from the Vsicek-Vsicek model. The prmeters of our competing models hve been estimted in two steps. In the rst one, the prmeters of the diusion processes hve been estimted by the Generlized Method of Moments by Hnsen (1982). Once these vlues re obtined, the remining prmeters re estimted by using cross-section technique. As result of this combintion of estimtion methods, we hve been ble to obtin the dily mrket prices of risk corresponding to both stte vribles for both models. Thus, it hs been shown tht, for the Vsicek-CIR model, these prices re highly signicnt nd hve constnt sign during ll the period On the other hnd, under the Vsicek-Vsicek model, it cn be seen tht the mrket prices of risk re signicntly dierent from zero nd positive for lthough they show chnging sign when nlyzing yerly this period. Finlly, we hve nlyzed the tting nd forecsting power of both models. The within- nd out-of-smple periods re nd 1995, respectively. After computing the within- nd out-of-smple forecsts, the pricing errors of both models (nd severl ccurcy mesures) for dierent subperiods hve been obtined to compre one ech other. All these sttistics show the following fcts: (1) both models provide very big ccurcy to the observed bond prices for ll the mturities, (2) the pricing errors re incresing with the mturities, (3) the MAPE for the Vsicek-Vsicek nd the Vsicek-CIR models is smller thn :26% nd :2%, nd (4) in the Vsicek-CIR (Vsicek-Vsicek) model, this sttistic is lwys smller thn :1% (:8%) except for the 5-yer bond price. Compring both models, it cn be seen tht (1) the Vsicek-CIR model outperforms the Vsicek-Vsicek model for ll mturities nd for ll the sttistics, (2) in mturities smller thn one month, both models provide 18

20 negligible errors nd the reltive improvement obtined with the Vsicek- CIR model is not very lrge, (3) deling with mturities greter thn one month, the errors from the Vsicek-Vsicek model re decresed in more thn 88% for ll these mturities, (4) the biggest improvement (bout 9 94%) in ccurcy is chieved in the medium-term mturities nd in 5-yer bonds. Severl subperiods hve been nlyzed nd the sme conclusions re reched. The forecsting power of both models hs been nlyzed by one- nd vestep hed forecsts for every mturity nd for every dy of the yer Looking t one-step-hed forecsts, it hs been shown tht - similrly to the within-smple period - both models perform quite well, the forecsting power decreses with the time to mturity nd, for the shortest mturities, both models perform similrly. However, for mturities beyond one month, the Vsicek-CIR model forecsts better thn the Vsicek-Vsicek model lthough this superior forecsting behvior is not monotonic in the time to mturity. Thus, the best reltive performnce is obtined in the 1-yer bond prices when the error mesures from the Vsicek-Vsicek model re decresed in more thn 21%. In the remining bonds, the improvement is much smller rnging between 2% nd 12%. Finlly, deling with ve-step-hed forecsts, ll the sttistics reect worse performnce thn in the previous (shorter) forecsts lthough similr results re shown: (1) the forecsting power decreses with time to mturity, (2) the Vsicek-Vsicek model outperforms its rivl model in the shortest mturities nd (3) the Vsicek-CIR forecsts better thn the Vsicek-Vsicek model in the mturities beyond six months. In this cse, this improvement is usully quite smll, between 1% nd 5%. Therefore, the min conclusion is tht the modeling of the voltility in the long-term rte process cn help (in lrge mount) to t the observed dt cn improve - in resonble quntity - the prediction of the future movements in the medium- nd long-term interest rtes. However, for shorter mturities, it hs been shown tht the pricing errors re, bsiclly, negligible nd it is not so cler which is the best model to be used. 19

21 Appendix: Proofs Proof of Proposition 1 The method of the seprtion of vribles llows us to write the solution of the eqution (5) subject to (6) s where X(s; t; T ) solves the eqution P (s; L; t; T )=X(s; t; T ) Z(L; t; T ) (17) X ss + q1(^1 s)x s + X t sx = (18) subject to the terminl condition nd Z(L; t; T ) is the solution of the eqution X(s; T; T )=1; 8s (19) LZ LL + q2(^2 L)Z L + Z t LZ = (2) with terminl condition Z(L; T; T )=1; 8L (21) To solve eqution (18), we posit solution of the type X(s; t; T )=X(s; ) =A1()e B()s (22) Hence, the eqution (18) becomes " # 1 A B2 () q1(^1 s)b() 1 () A1() B ()s s = (23) where, from (19), the terminl conditions re given by A1() = 1; B() = (24) 2

22 Eqution (23) is liner in the vrible s nd, therefore, it becomes null when the corresponding coecients re equl to zero. Hence, this eqution is equivlent to the following system of rst-order dierentil equtions q1b()+b ()1 = (25) B2 () q1^1b() A 1 () A1() = (26) subject to the terminl conditions (24). We rst solve (25) with terminl condition B() =. Including this solution in (26), integrting this eqution, nd using A1() = 1, we obtin B() = 1 eq 1 A1() = exp q1 ( 2 1 4q1 B2 ()+s 3 (B()) ) (27) where s 3 =^1 1 2 =(2q2 1) Replcing (27) into (22), we obtin the nl expression for X(s; t; T ). In similr wy, to solve eqution (2), we posit solution of the type Z(L; t; T )=Z(L; ) =A2()e C()L (28) Hence, the eqution (2) becomes " # 1 A LC2 () q2(^2 L)C() 2 () A2() C ()L L = (29) where, from (21), the terminl conditions re given by A2()=1; C() = (3) As eqution (29) is liner in the vrible L, this eqution is equivlent to the following system of rst-order dierentil equtions C2 ()+q2c()+c ()1 = (31) q2^2c()+ A 2() A2() = (32) 21

23 subject to the terminl conditions (3). We rst solve (31). It is strightforwrd to show tht this eqution cn be rewritten s 2 dc() 2 (C() c1)(c() c2) = d where c1 = q > ; c2 = q q 2 2 < ; = q2 +22 Integrting this eqution nd using the terminl condition C() =, it is obtined tht 1 ln C()! c 2 = + 1 c C() c1 ln 2 c1 nd little lgebr leds to 2 (expfg1) C() = (33) (q2 + ) expfg +(q2) Once we know C(), we cn solve the eqution (32) or, equivlently k22c()+ A 2() A2() = Integrting, we hve Z ln[a2()] = k22 C()d + k A (34) Let y = expfg. Then, more lgebr gives Z C()d = 2 2 ln((q2 + ) expfg +(q2)) (q2 + ) 2 Replcing this expression in (34) nd pplying the condition A2() = 1, the nl expression for A2() is given by A2() = 2 4 n 2exp (q2 + ) 2 (q2 + ) expfg +(q2) o 3 5 2k 2 2 = 2 2 (35) Including (33) nd (35) into (28), we obtin the nl expression for Z(L; t; T ). This expression, jointly with (22), gives the closed-form formul for the defult-free discount bond prices for ll mturities. 2 22

24 [1] At-Shli, Y. (1996). Testing Continuous-Time Models of the Spot Interest Rte. Review of Finncil Studies, 9, 2, 385{426. [2] Ayres, H.R. nd J.Y. Brry (198). A Theory of the U.S. Tresury Mrket Equilibrium. Mngement Science, 26, 6, 539{569. [3] Boudoukh, J., M. Richrdson, R. Stnton nd R.F. Whitelw (1999). A Multifctor, Nonliner, Continuous-Time Model of Interest Rte Voltility, mimeo, New York University. [4] Brennn, M.J. nd E.S. Schwrtz (1979). A Continuous Time Approch to the Pricing of Bonds. Journl of Bnking nd Finnce, 133{155. [5] Chn, K. C., G.A. Krolyi, F.A. Longst nd A.B. Snders (1992). An Empiricl Comprison of Alterntive Models of the Short-Term Interest Rte. Journl of Finnce, 47, 3, 129{1227. [6] Chen, L. (1996). Interest Rte Dynmics, Derivtives Pricing, nd Risk Mngement. Springer-Verlg, Berlin. [7] Conley, T.G., L.P. Hnsen, E.G. Luttmer, nd J. Scheinkmn (1997). Short-Term Interest Rtes s Subordinted Diusions. Review of Finncil Studies, 1, 3, 525{577. [8] Cox, J.C., J.E. Ingersoll nd S.A. Ross (1985). A Theory of the Term Structure of Interest Rtes. Econometric, 53, 385{48. [9] Di, Q. nd K. Singleton (1998). Speciction Anlysis of Ane Term Structure Models, mimeo, NYU nd Stnford University. [1] Due, D. nd R. Kn (1996). A Yield-Fctor Model of Interest Rtes. Mthemticl Finnce, 6, 4, 379{46. [11] Longst, F.A. nd E.S. Schwrtz (1992). Interest Rte Voltility nd The Term Structure: A Two-Fctor Generl Equilibrium Model. Journl of Finnce, 47, 4, 1259{

25 [12] Nelson, J. nd S.M. Schefer (1983). The Dynmics of the Term Structure nd Alterntive Portfolio Immuniztion Strtegies. In Innovtions in Bond Portfolio Mngement: Durtion Anlysis nd Immuniztion, eds. G.O. Bierwg, G.G. Kufmn, nd A. Toevs, Greenwich, CT: JAI Press. [13] Nu~nez, S. (1995). Estimcion de l Estructur Temporl de los Tipos de Interes en Esp~n: Eleccion entre Metodos Alterntivos. Documento de Trbjo Bnco de Esp~n. [14] Richrd, S.F. (1978). An Arbitrge Model of the Term Structure of Interest Rtes. Journl of Finncil Economics, 6, 33{57. [15] Schefer, S. nd E.S. Schwrtz (1984). A Two-Fctor Model of the Term Structure: An Approximte Anlyticl Solution. Journl of Finncil nd Quntittive Anlysis, 19, 4, 413{424. [16] Vsicek, O. (1977). An Equilibrium Chrcteriztion of the Term Structure. Journl of Finncil Economics, 5, 177{

26 Tble I. Estimtes of the Diusion Prmeters This tble provides the prmeter estimtes (with t-vlues in prentheses) of the processes followed by ech stte vrible. The smple period is from Jnury 1991 to December The prmeters re estimted by mens of the Generlized Method of Moments pplied to the following equtions s t s t1 = 1 + b 1 s t1 + " s t ; "s t IID (; 2 1) L t L t1 = 2 +b 2 L t1 +" L t ; "L t IID (; 2 2 r t1) L t L t1 = 3 +b 3 L t1 +" L t ; "L t IID (; 2 3) Vrible b k Spred (-.21) (-3.756) (3.756) (-.21) Long-Term Rte (CIR process) (3.5991) ( ) (3.6285) ( ) Long-Term Rte (Vsicek process) (2.2968) ( ) (2.3988) (2.881) 25

27 Tble II. Averges of Pure Cross-Sectionl Regressions This tble contins the estimtion results, for ech dy of the period , of the prmeters (q i ; i=1;2; s 3 ) in the closed-form pricing eqution for the Vsicek-CIR model where P (s; L; t; T )=P(s; L; )=A()e B()sC()L A( ) = A 1 () A 2 () 8 " 9 A 1 () = exp 2 1 2exp B 2 ()+s 3 (q 2 + ) 2 (B()) ; A 2 ()= 4q 1 (q 2 + ) expfg +(q 2 ) B() = 1 eq 1 2(expfg 1) ; C()= (q 2 + ) expfg +(q 2 ) with q 1 q 1 = k 1 + b 1 ; ^ 1 =(k )=q 1 ; s 3 =^ =(2q2 1) q 2 = k 2 + d 2 ; ^ 2 = k 2 2 =q 2 ; = p q Numbers in prentheses represent the verge of the t-sttistics of cross-sectionl regressions. The numbers in squre brckets [:] represent the stndrd devition of the time series of prmeter estimtes q1 (12.66) (99.92) (116.21) (7.97) (154.12) (162.88) [.5376] [.7612] [.2526] [.765] [.1749] [.1515] q2 (87.88) (92.79) (77.4) (117.71) (14.31) (46.68) [.643] [.7786] [.1967] [.7156] [.4736] [.1431] s 3 (784.1) ( ) (572.7) (945.38) (58.79) (254.74) [.17] [.13] [.15] [.81] [.174] [.199] # 2k22=

28 Tble III. Averges of Mrket Prices of Risk This tble contins the estimtion results, for ech dy of the period , of the mrket prices of risk ( i ; i=1;2) relted to ech stte vrible in both two-fctor models. Numbers in prentheses represent the verge of the t-sttistics of these estimtes. The numbers in squre brckets [:] represent the stndrd devition of the time series of mrket prices of risk estimtes. Pnel A: Vsicek-CIR Model (Spred) (-89.24) (-89.14) (-76.7) (-86.83) ( ) (-75.26) [13.414] [ ] [7.3795] [11.543] [6.1945] [5.9674] (Long rte) (85.99) (91.24) (74.87) (116.41) (12.51) (44.) [2.9495] [3.639] [.9426] [3.4294] [1.934] [.6534] Pnel B: Vsicek-Vsicek Model (Spred) (5.9) (46.88) (34.1) (1.19) (-2.7) (-54.31) [ ] [1.1866] [5.6779] [ ] [18.933] [7.54] (Long rte) (15.22) (-49.31) (-3.41) (3.75) (-29.76) (1.75) [ ] [32.872] [ ] [ ] [44.95] [2.7386] 27

29 Tble IV. Within-Smple Pricing Error Mesures This tble contins the within-smple pricing error mesures of both two-fctor models for the period We consider zero-coupon bonds with fce vlue of $1 nd with mturities rnging from 1 dy to 1 yers. We hve computed ve dierent error mesures: the men error (ME), the men bsolute error (MAE), the root men squred error (RMSE), the men bsolute percentge error (MAPE) nd the root men squred percentge error (RMSPE). Vsicek-CIR Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer Vsicek-Vsicek Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer

30 Tble V. Within-Smple Pricing Error Mesures This tble contins the within-smple pricing error mesures of both two-fctor models for the yer We consider zero-coupon bonds with fce vlue of $1 nd with mturities rnging from 1 dy to 1 yers. We hve computed ve dierent error mesures: the men error (ME), the men bsolute error (MAE), the root men squred error (RMSE), the men bsolute percentge error (MAPE) nd the root men squred percentge error (RMSPE). Vsicek-CIR Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer Vsicek-Vsicek Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer

31 Tble VI. Within-Smple Pricing Error Mesures for 1-yer Bonds This tble contins the within-smple pricing error mesures of both two-fctor models for ech semester of the period We consider zero-coupon bonds with fce vlue of $1 nd with mturity of 1 yer. We hve computed ve dierent error mesures: the men error (ME), the men bsolute error (MAE), the root men squred error (RMSE), the men bsolute percentge error (MAPE) nd the root men squred percentge error (RMSPE). Vsicek-CIR Model Period ME MAE RMSE MAPE RMSPE 1991:I :II :I :II :I :II :I :II Vsicek-Vsicek Model Period ME MAE RMSE MAPE RMSPE 1991:I :II :I :II :I :II :I :II

32 Tble VII. Comprison of One-Step-Ahed Forecsts This tble contins the out-of-smple pricing error mesures of both two-fctor models for the yer We compute one-step-hed forecsts for prices of zero-coupon bonds with fce vlue of $1 nd with mturities rnging from 1 dy to 1 yers. We report ve dierent error mesures: the men error (ME), the men bsolute error (MAE), the root men squred error (RMSE), the men bsolute percentge error (MAPE) nd the root men squred percentge error (RMSPE). Vsicek-CIR Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer Vsicek-Vsicek Model Mturity ME MAE RMSE MAPE RMSPE 7-dy dy month month month yer yer yer

33 Tble VIII. Comprison of One-Step-Ahed Forecsts. 1-yer Bonds This tble contins the out-of-smple pricing error mesures of both two-fctor models for ech month of the yer We compute one-step-hed forecsts for prices of zerocoupon bonds with fce vlue of $1 nd with mturity of 1 yer. We report ve dierent error mesures: the men error (ME), the men bsolute error (MAE), the root men squred error (RMSE), the men bsolute percentge error (MAPE) nd the root men squred percentge error (RMSPE). Vsicek-CIR Model Period ME MAE RMSE MAPE RMSPE 1995:I :II :III :IV :V :VI :VII :VIII :IX :X :XI :XII Vsicek-Vsicek Model Period ME MAE RMSE MAPE RMSPE 1995:I :II :III :IV :V :VI :VII :VIII :IX :X :XI :XII