Forecasting Bond Returns Using Jumps in Intraday Prices

Transcription

1 Forecasing Bond Reurns Using Jumps in Inraday Prices Auhor: Siawash Safavi Nic Suden Number: Maser Program: QFAS, Tilburg Universiy Supervisor Tilburg Universiy: Prof. Dr. Bas Werker Supervisors Robeco: Drs. Johan Duyveseyn, CFA Dr. Ir. Marin Marens

2 Table of Conens Preface Absrac Relaed Lieraure Classics Recen lieraure Wrigh and Zhou (29) Daa Source Qualiy and availabiliy Mehod Theoreical example jump saisic Realized volailiy and Bi-power variaion Tes saisics Example jump saisics using real daa Variable definiions Replicaion Wrigh and Zhou Bond reurn regression models Graphs and summary saisics forecas variables Main resuls replicaion Wrigh and Zhou (29) Refinemens Definiion variables Resuls regression Robusness checks Changing he significance level and rolling window Three- monhs holding period Insigh in Jumps Momen of Jumps Jumps and macro-economic news announcemens Inuiion Even Plos Even Plo 7:3 jumps Even Plo 9: jumps Long Term Invesmen Sraegies Invesmen sraegies for he in-sample period Ou-of-sample model Recommendaion for furher research Conclusion Appendix Disribuion of Jumps Long/Shor invesmen sraegies References

3 Preface Wih wriing his Maser hesis comes an end o my life as a lazy suden. A he same ime i marks he sar of a new life. Sudying has been someimes exiing and enlighening, someimes boring. My graduaion inernship has been fun all he way. This hesis can be seen (a leas he way I like o see i) as he crown on my work so far. My special hanks go ou o boh my supervisors a Robeco Asse Managemen for giving me he once in a life ime opporuniy o become a super quan. Firs of all I would like o hank Dr. Ir. Marin Marens, co-head Quaniaive Sraegies, for his valuable ideas and insighs. I would like o hank Drs. Johan Duyveseyn, CFA. I could no have wrien his hesis wihou his daily feedback and sharp eye on my work. He always kep me on rack whenever I was in danger of loosing conrol. In addiion I would like o hank my supervisor a he universiy, Prof. Dr. Bas Werker, for his criical view on my work. The monhly meeings always encouraged me o work harder. Of course I would like o hank my parens for heir encouragemen and suppor in (almos) every decision I made during my life so far. My final hanks go ou o my friends and colleagues for giving me he needed disracion during he las half year. Siawash Safavi Nic Roerdam, March 29 h 21 3

4 1. Absrac In 25 Cochrane and Piazzesi published heir paper Bond Risk Premia in he American Economic Review. In his paper he auhors show ha one o five year forward raes are able o forecas US bond reurns wih R 2 s up o 36%. This paper has generaed a number of oher papers including he paper Bond Risk Premia and Realized Jump Risk of Wrigh and Zhou (29). They show ha augmening he regression of excess bond reurns on he erm srucure of forward raes (Cochrane and Piazzesi (25)) wih an esimae of he mean realized jump size almos doubles he R 2 o 6%. This variable is consruced by esimaing jumps in inraday prices using high frequency daa and subsequenly aking he rolling average of all he jumps over he las wo years. In his paper we replicae he findings of Wrigh and Zhou (29) and presen a refinemen of heir jump mean variable, namely reurn jump mean. We show ha his variable performs beer in he regression models: An R 2 of 67% in he regression ogeher wih he forward raes compared o 48% for he jump mean variable of Wrigh and Zhou (29). The reurn jump mean variable also performs well in invesmen sraegies. We show ha using a revised version of his variable we achieve an informaion raio of.64 for our in-sample period and.52 for he ou-of-sample period. We show ha he jump mean and reurn jump mean have a low correlaion wih mean reversion, indicaing ha hese variables are relaed bu no he same. This indicaes ha here is added value in hese variables above mean reversion. In our final regression model we achieve an R 2 of 72% using he forward raes, reurn jump mean and mean reversion as explanaory variables. In addiion we sudy he relaion beween macro-economic news announcemens and jumps in bond fuure prices and show ha announcemens wih he larges impac on bond fuure prices are relaed o employmen, inflaion and confidence announcemens. This confirms he findings of Balduzzi (1997). Abou 44% of all he jumps are caused by macro-economic announcemens. 4

5 2. Relaed Lieraure In recen years much academic research has been done in he field of bond forecasing lieraure. The auhors have found significan forecasable variaion in US excess bond reurns. Firs we refer o some key research (ha forms he basis of new resuls), and hen we will coninue wih some new ideas presened in recen lieraure Classics Fama and Bliss (1987) find ha he spread beween he n-year forward rae and he one-year yield predics (one year) excess reurns of he n-year bond (n=2,3,4 and 5). The R 2 repored is 18%. Campbell and Shiller (1991) use yield spreads o forecas excess reurns. Ilmanen (1995) uses a small se of global insrumens o predic bond excess reurns in six differen counries. Cochrane and Piazessi (25) recenly showed a large amoun of predicabiliy of US excess bond reurns (wih an R 2 of 35%) wih mauriies ranging from wo o five years using a linear combinaion of forward raes. The Cochrane and Piazessi (CP) paper has become a benchmark in he lieraure. I is cied o by a number of oher papers. Robeco has already analysed hese papers and uses he mos relevan ideas from hese in quaniaive invesmen sraegies. For his inernship we focus more on he recen research Recen lieraure Ludvigson and Ng (29) invesigae he predicable power of macro facors o forecas one-year excess reurns. They find predicable variaion in one year excess bond reurns ha is associaed wih macroeconomic aciviy, even afer conrolling for he CP forecasing facor, wih R 2 s up o 45%. Greenwood and Vayanos (28) find ha a larger proporion of long erm debs predic posiive bond excess reurns, on op of he CP facor and he slope of he yield curve. The proporion of long erm deb is posiively relaed o he seepness of he yield curve. A seeper curve is known o predic posiive bond reurns because of ime varying risk premia. This is known as a carry sraegy where you borrow agains a low rae and inves a a high rae in he longer mauriy bonds. Faus and Wrigh (29) find ha using he CP facors is mos significan when esimaion is based on 15 minues around 1 imporan macro-economic news announcemens. They show a sraegy based on esimaed coefficiens for hese 15 minues applied o monhly posiions improves over using he more noisy coefficiens obained from monhly reurn daa, wih a Sharpe raio of.52 versus -.18 for he 1-year fuure. Almeida e al (28) use arbirage-free affine erm srucure models o explain bond and ineres rae opions (caps) simulaneously. The oupu conains facors wih predicive power for bond markes. They argue ha ineres rae opions may conain informaion abou he risk premium because heir prices are sensiive o he volailiy and marke prices of he risk facors ha drive ineres raes Wrigh and Zhou (29) Wrigh and Zhou (29) augmen he regression model of CP wih measures of Implied Volailiy, Realized Volailiy, Jump Inensiy, Jump Mean and Jump Volailiy. The auhors use high-frequency Treasury bond fuures daa o consruc realized volailiy and jump risk measures. Assuming ha he price of an asse (a bond in his paper) follows a jump-diffusion process, he auhors ake a direc approach o idenify realized jumps based on Barndorff-Nielsen and Shephard (24, 26). This approach uses high-frequency daa o decompose realized volailiy ino separae coninuous and jump componens (see Huang and Tauchen (25) and Andersen, Bollerslev, and Diebold (27) as well) and hence o deec days on which jumps occur and o esimae he magniude of hese jumps. The jump mean variable in his paper has power o forecas fuure US bond reurns wih an R 2 of 15%. Including he forward rae variables of Cochrane and Piazzesi (25) ino he model, he R 2 increases o 6%. 5

6 We choose his paper for furher research. I shows promising resuls and uses a new source of daa: high frequency (Inraday). This is no used wihin Robeco ye. A plus is ha he daa is easy o updae from Bloomberg. Inraday daa are available for he las 5 days for differen counries. Afer a meeing wih he porfolio managers we have agreed o furher invesigae he idea in his paper. Firs we wan o replicae and check he resuls in he paper. Furhermore we wan o ge more inuiion ino he daa and he resuls. Our second preference for furher research was he paper of Ludvigson and Ng (25), Macro Facors in Bond Risk Premia. This paper shows good empirical resuls, however he echnique used in he paper (summarize macro-economic daa using principal componen analysis) is already analyzed wihin Robeco (wih promising resuls) and shows herefore overlap wih previous research done on macro daa. The same holds for he paper of Faus and Wrigh (29), Risk Premia in he 8:3 Economy. The idea presened in he paper is ineresing, bu i leans owards Cochrane and Piazzesi (25) which is already exensively sudied wihin Robeco. The resuls of Greenwood and Vayanos (28) are less promising. The deb mauriy is a very slow variable (high auocorrelaion) and he resuls are no robus over subinervals. The auhors spli he sample in wo and find predicabiliy only in he second par of he sample. Also he inuiion is no clear. The model of Almeida e al (28) is complex and no ransparen. Moreover, he resul is no consisen wih he disappoining resuls for implied volailiy of Wrigh and Zhou (29). In heir paper he implied volailiy variable is highly insignifican wih an R 2 of less han 1%. 6

7 3. Daa 3.1. Source For consrucing excess reurns, yields and forward raes, we used end-of-monh daa on zero-coupon bonds from he CRSP Fama-Bliss daase. The CRSP daa consiss of end-of-monh prices of one o five-year US zero coupon bonds. The daa are available for sudens and employees of Tilburg Universiy a he WRDS.com (Wharon Research Daa Services). We also use bond daa from Daasream and JP Morgan (marke sandard for bonds) o compare i o he CRSP daa. The laer is mosly used in academic research. We obained he inraday daa (US 1-year fuures) from Tickdaa.com. Complee fuures & index daa by symbol coss 65 USD. Wrigh and Zhou (29) consider 3-year fuures, bu we choose for he 1-year fuure, since he laer is more liquid (higher volume, lower bid-ask spread). Inraday 1-year fuures daa is available from January 1983 ill Ocober 29, bu before 1989 daa is available only from 8: AM Chicago ime. For our research however i is crucial o also consider daa around 7:3 Chicago ime, since around his ime here are ofen macro-economic daa releases in he US, which usually cause exreme price movemens. Therefore we sar our analysis from January Wih each order Tickdaa includes a daabase managemen sofware, TickWrie, which enables users o creae ASCII files cusom-ailored o heir needs. TickWrie can consruc ime inervals of any granulariy, generae coninuous fuures files, and adjus for rollover gaps. We look a five-minue frequency from January 1989 ill Ocober 29. The five-minue frequency is he sandard in he lieraure for high frequency daa. Daa a higher frequencies are subjec o marke microsrucure noises like he bid and ask bounce. For each day we look a he period from 7:2 ill 14: Chicago ime. This means we have a oal of 8 observaions for each day. Our in-sample se is from January 1989 ill December 22 and our ou-of-sample analysis is based on he period January 23-Ocober 29. By explicily choosing an ou-ofsample period, we ry o avoid daa mining Qualiy and availabiliy The daa availabiliy (1 year US fuure from Tickdaa) is beer for he second par of he sample han he firs par as follows from he figure below. Figure 1: # of observaions each day Availabiliy Daa # observaions each day /3/1989 1/3/199 1/3/1991 1/3/1992 1/3/1993 1/3/1994 1/3/1995 1/3/1996 1/3/1997 1/3/1998 1/3/1999 1/3/2 1/3/21 1/3/22 1/3/23 1/3/24 1/3/25 1/3/26 1/3/27 1/3/28 1/3/29 We (ideally) should have 8 observaions for each day, bu for he firs par of he sample here are a lo of holes in he daa as follows from Figure 1. Missing observaions are because of no rade wihin ha five-minue inerval or because of shorer rading days like 24 and 31 December. Dae 7

8 We have in oal for 41% of all rading days 8 observaions. In case of no daa for a specific ime, we fill he empy inerval wih he price of he previous observaion. Therefore, for our analysis we do have 8 daa poins for each day. The same paern follows from he able below. For he firs 7 years of our sample only for 2% of he days we have a oal of 8 observaions. For he periods and he percenage increases o respecively 27% and 94%. The 2% for he firs 7 years is no much, bu he average number of observaions for each day for he same period is 65, which is sill considerable. For he periods and he average number of observaions increases o 73 and 79. Table 1: Daa availabiliy for differen sub periods period # days wih 8 obs. # oal days % days wih 8 obs. average # obs % % % 79 Anoher ineresing feaure is o look a he daa availabiliy for differen ime poins. As follows from he figure below a 7:2 AM here is always daa available. The availabiliy declines ill noon and increases afer ha. Lower availabiliy coincides wih lower volume. This paern is common wihin high frequency daa. Around lunch ime here is less daa available and also he raded volume is less. Figure 2: % available observaions for each ime poin Availabiliy Daa % available observaions :25 7:4 7:55 8:1 8:25 8:4 8:55 9:1 9:25 9:4 9:55 1:1 1:25 1:4 1:55 11:1 11:25 11:4 11:55 12:1 12:25 12:4 12:55 13:1 13:25 13:4 13:55 Time (Chicago) Inraday daa is also available from Bloomberg, bu only for he las 5 days. To check he qualiy of our source we compared daa from Tickdaa o Bloomberg. See he figure below for 16 Ocober 29. The wo lines are very close (we see similar paerns for oher days) and when here is a difference i is eiher 1/64 or 1/32. This could be caused by rounding errors, since fuure prices (in he US) are always noed in fracions insead of decimals or iming and source differences. In general however he qualiy of he daa looks good. 8

9 Figure 3: Comparison inraday daa from Bloomberg and Tickdaa 1y US fuure (5min frequency) Price TD BB :2: 7:4: 8:: 8:2: 8:4: 9:: 9:2: 9:4: 1:: 1:2: 1:4: 11:: 11:2: 11:4: 12:: 12:2: 12:4: 13:: 13:2: 13:4: Time (1/16/29, Chicago) We use Bloomberg also as our source for updaing he daa. 9

10 4. Mehod In his secion we give an ouline of he echnique used o deermine jumps in he ime series of fuure (or more generally asse) prices. A jump is informally defined as an exreme change in he price in a shor ime window. We illusrae he mehod wih a heoreical example. Nex we coninue wih he formal procedure and afer ha we give an example using real daa. A he end of his chaper we presen he definiions of he variables used in he forecas regressions Theoreical example jump saisic In his secion we give an example o illusrae he mehod presened in his paper o deec jumps in inraday prices. In he nex secion we give he formal definiions of he variables. The mehod is based on he difference beween realized volailiy (RV) and bi-power variaion (BV). Assuming ha he volailiy of an asse consiss of wo pars, a coninuous and a jump componen, he RV measures boh pars, while BV measures only he coninuous par. The difference beween he wo denoes he jump. Based on his idea you can derive various es saisics o derive wheher he difference is significan or no. In he figure below he inraday log reurns of a ficional day is shown. The leers denoe absolue five-minue reurns. Figure 4: Inraday absolue log reurns of an asse heoreical example abs log reurns (in %) a b c d e f g h i j reurn bars The figure above graphs he (heoreical) inraday absolue log reurns of an asse. The reurn corresponding o he h bar is much larger compared o he res. Looking a he graph one would expec a jump, since one of he reurns (h) dominaes he oher ones in size. The RV and BV for his day are equal o: RV = a + b j 1 BV = π ( ab + bc ij ) 2 9 RV measures he oal price variance (boh coninuous par and he jump componen) during a rading day, while he BV measures only he coninuous par of he variance. A large difference beween hese wo indicaes a 2 jump. The RV here ges dominaed by he h par. In he BV he h -reurn appears wice: gh and hi. These 2 wo producs are much smaller han h and his is wha causes he RV o be larger han he BV. The example provided here gives he essence of he mehod o deermine jumps in asse prices. To formally es for a jump, we use he es saisic presened in he nex secion. 1

11 4.2. Realized volailiy and Bi-power variaion To idenify jumps we use high-frequency Treasury bond fuures daa o consruc realized volailiy and jump risk measures. Assuming ha he price of an asse (a bond fuure in his paper) follows a jump-diffusion process, we ake (following Wrigh and Zhou (29)) a direc approach o idenify realized jumps based on Barndorff-Nielsen and Shephard (24, 26). This approach uses high-frequency daa o decompose realized volailiy ino separae coninuous and jump componens (see Huang and Tauchen (25) and Andersen, Bollerslev, and Diebold (27) as well) and hence o deec days on which jumps occur and o esimae he magniude of hese jumps. We use inraday (high frequency) 1-year Treasury bond fuure daa o consruc realized variance and jump measures. We follow he same approach and noaion as Wrigh and Zhou (29). Le s denoe he logarihmic asse price a ime. I is assumed ha his quaniy evolves as a coninuous-ime jump diffusion process. I is expressed in sochasic differenial equaion as: ds = µ d + σ dw + J dq where process and µ and σ are he drif and he sricly posiive (sochasic) volailiy funcions. W is a sandard Wiener q is a couning process wih inensiy JI (possibly ime varying), and J is he corresponding 2 (log) jump size wih mean JM and variance JV. Also he jump mean and variance are allowed o be ime varying in a compleely unresriced way. Time is measured in daily unis and he inraday reurns are defined as: r s, j s, j s,( j 1) Here, s r, j denoes he j-h inraday reurn on day and is he sampling frequency wihin a day. This means ha here are a oal of 1/ observaions every rading day. In he lieraure i is common o use five-minue inervals raher han more frequen observaions. This is because more frequen observaions may be subjec o disorion from marke microsrucure noise like he bid-ask bounce (See Ai-sahalia e-al. (25) and Bandi and Russell (26)). s The quadraic variaion process over day of he cumulaive reurn process, r, is hen: 2 ( r, r) = σ du 1 u + 1 J 2 u dq u (1) where he second par of he summaion in he RHS of (1) is equal o he squared jumps occurred a day. Of course, in he absence of jumps, i is equal o zero and he quadraic variaion process becomes he inegraed volailiy of he coninuous sample pah componen. Barndorff-Nielsen and Shephard (24) propose in heir paper wo general measures for he quadraic variaion process, namely realized variance (RV) and realized bi-power variaion (BV). These wo measures converge (uniformly as he sampling frequency or equivalenly he number of observaions m = 1/ ) o differen funcionals of he underlying jump-diffusion process: RV m j = 1 1 s r σ du + J dq = ( r, r) (2), j π m BV 2 m 1 m j = 2 r s, j u r s, j 1 u u u σ du As follows from (2) and (3) realized variance and bi-power variaion coincide (in he limi) in case of no jump and he difference is sricly posiive when here is a jump. This is he basis of he mehod o idenify jumps. (3) 11

12 4.3. Tes saisics A number of jump deecion echniques are proposed by Barndorff-Nielsen and Shephard (24). Wrigh and Zhou (29) use in heir paper he raio saisic favored by Huang and Tauchen (25) and Andersen e al. (27). We apply he same measure which is defined as: RJ RV BV RV RJ is a measure of he relaive jump. The es saisic based on i is: ZJ π [( ) 2 RV BV π 5] m TP d N(,1) (4) This es saisic 1 has in he limi (as he number of observaions m ) under null a sandard normal disribuion. However Huang and Tauchen (25) repor based on heir simulaions ha ZJ in (4) ends o over-rejec he null hypohesis of no jump. Huang and Tauchen (25) find ha he raio-saisic, ZJ RJ π 2 1 TP [( ) + π 5] max(1, ) 2 2 m BV d N(,1) (5) is very closely approximaed under he null by a sandard normal disribuion. Moreover, hey find ha his es saisic in (5) has grea power (small ype II error). We follow Andersen e al. (27) and Wrigh and Zhou (29) and use (5) as our es saisic. Huang and Tauchen (25) showed ha his es is quie accurae in deecing jumps in Mone Carlo simulaions. Following Tauchen and Zhou (21) and Wrigh and Zhou (29) we assume ha here is a mos one jump each day and ha when a jump in he fuure price occurs he jump dominaes he reurn on ha day. Using hese assumpions, we can filer ou he realized jumps in fuure prices as: J = sign (6) s ( r ) * ( RV BV )* I 1 ( ZJ Φα ) The Φ is (6) is he cumulaive disribuion funcion (CDF) of a sandard normal random variable. I is 1 ( ZJ Φ α ) he indicaor funcion which akes he value 1 if he es saisic exceeds he criical value and α is he corresponding significance level. Here we choose α equal o.1 (corresponding o a criical value of 3.719). The small significance level is jusified since we are ineresed only in exreme marke movemens. Andersen e 1 TP is he ri-power quariciy, and as show by Barndorff-Nielsen and Shephard (24) is holds: TP m µ 3 4 / 3 wih µ 2 k k / 2 m m 2 m j = 3 r s, j 2 Γ(( k + 1) / 2) / 4 / 3 r s 4 / 3, j 1 r s, j Γ(1/ 2) for k > 4 / 3 1 σ 4 u du 12

13 al. (27) documen ha for heir sample ( year T-bond) aking a significance level of.1 produces jumps a 7.6% of he days. Wrigh and Zhou (29) do no menion heir significance level, bu hey also find jumps a 8% of he days for heir sample ( year T-bond) Example jump saisics using real daa Below in Figure 5 wo graphs are shown. The firs one denoes a coninuous price movemen, while he second shows a jump. This example is mean o show ha he mehod used o filer jumps from inraday daa is able o discriminae beween large coninuous price movemens and jump componens. Figure 5: Two examples of large inraday price changes June Price :5: 8:2: 8:35: 8:5: 9:5: 9:2: 9:35: 9:5: 1:5: 1:2: 1:35: 1:5: 11:5: 11:2: 11:35: 11:5: 12:5: 12:2: 12:35: 12:5: 13:5: 13:2: 13:35: 13:5: Time (Chicago) April Price :25: 7:4: 7:55: 8:1: 8:25: 8:4: 8:55: 9:1: 9:25: 9:4: 9:55: 1:15: 1:3: 1:45: 11:5: Time (Chicago) The wo figures above graph he inraday price movemens of he 1 year US bond fuure a five minues frequency. The firs figure graphs a day wih large reurn bu coninuous movemens and he second figure denoes a jump in he inraday price movemens. 11:2: 11:35: 11:5: 12:5: 12:2: 12:35: 12:55: 13:1: 13:25: 13:4: 13:55: As explained in he ex he RV measures he oal price variance (boh coninuous par and he jump componen) during a rading day, while he BV measures only he coninuous par of he variance. To look for jumps, deermine wheher he difference beween he wo is significan. Since boh RV and BV are variance measures and herefore no in he same dimension as he price and he reurns we will denoe here he square roo of he wo o have more feeling for he numbers. The firs graph in Figure 5 follows from he price movemen a June The square roo of he RV and he BV saisics are for his day equal o: RV = 1.6% BV = 1.62% The square roo difference beween RV and BV (if significan) denoes he jump size, bu he wo values are close o each oher and he difference is no significan (herefore no jump a his day) as follows formally from he es: ZJ = -.28 < 3.72 (= criical value corresponding o α =.1). As you can see, he BV is slighly larger han he RV (his corresponds o a negaive value of ZJ). This migh seems srange since RV measures he oal price variance and BV measures only he coninuous par of he variance, 13

14 bu in he absence of jump he wo saisics measure he same parameer and in a finie sample i can happen ha BV becomes (slighly) larger. The second graph in Figure 5 follows from he price movemen a April From he figure you can see an exreme price movemen around 7:3 (Chicago ime). Around his ime here ofen are macro-economic daa releases in he US (for example unemploymen, CPI, PPI). One would expec a jump from a formal es. The RV and BV saisics are for his day equal o: RV =.91% BV =.4% Unlike he previous example here is a big difference here beween RV and BV. The quesion is, is i significan? The null hypohesis of no jump is rejeced wih grea confidence: ZJ = 9.21 > 3.72 (= criical value corresponding o α =.1). The (log) jump size equals he square roo difference beween RV and BV: Sqr(.91%^2 -.4%^2 ) =.82%. Looking a he graph, you can see ha here is a jump beween 7:3 AM and 7:35 AM. The closing prices a 7:3 and 7:35 were respecively The log difference of he wo prices is: log(16.44) log(15.53) =.86%, which is very close o he esimaed jump size. The same paern is visible if you look a he five-minue reurns a he same days. The firs graph in Figure 6 follows from he reurns a June , he second from April Figure 6: Inraday five-minue absolue log reurns June absolue (5min) log reurns (in %) :1 8:35 9: 9:25 9:5 1:15 1:4 11:5 11:3 11:55 12:2 12:45 13:1 13:35 14: Time (Chicago) April absolue (5min) log reurns (in %) :3 7:55 8:2 8:45 9:1 9:35 1: 1:25 1:5 11:15 11:4 12:5 12:3 12:55 13:2 13:45 Time (Chicago) The wo figures above graph he inraday five-minue absolue log reurns of he 1 year US bond fuure for he same days as in Figure 5. Again he firs figure graphs a day wih large reurn bu coninuous movemens and he second figure denoes a jump in he inraday price movemens. For June here are differen large reurn bars corresponding o large bu coninuous price movemens, while for April here is clearly one bar exremely larger han he res, corresponding o a jump in he fuure price. 14

15 4.5. Variable definiions Afer idenifying he jumps we can hen, like Wrigh and Zhou (29), esimae he jump mean, volailiy and inensiy as: JI = Number of realized Jump Days Number of Toal Trading Days JM = Mean of realized Jumps JV = S. Deviaion of realized Jumps The nex sep is o invesigae he forecas power of hese jump measures for bond excess reurns. Wrigh and Zhou (29) already showed ha an esimae of he jump mean has grea power o forecas one year bond excess reurns. We wan o replicae he resuls of Wrigh and Zhou (29) and check wheher we can ge similar resuls. Our esimaes of bond fuure realized volailiy, jump mean, jump volailiy and jump inensiy are based on 1- year Treasury bond fuures. We look a daa a five-minue frequency from January 1989 ill Ocober 29, obained from ickdaa.com. For each day we look a he period from 7:2 ill 14: Chicago ime (corresponding o 8:2 ill 15: New York ime). This means we have a oal of 8 observaions for each day. We use he daa o calculae (coninuously compounded) reurns as he log difference in he fuure quoes, as explained in he secion mehod. We coninue o calculae realized volailiy and bi-power variaion a daily frequency, es for jumps on each day, and esimae he magniude of he jumps on hose days when jumps are deeced. Again using he echnique explained in he secion mehod. DAILY Le D denoe he dummy variable, which akes he value 1 when a jump is deeced a day and is j oherwise, and J denoes he corresponding jump size a day. For our empirical work, le he h-monh rolling average realized volailiy, jump mean, jump volailiy and jump inensiy be defined as (in he formulas below 22 refers o he # of rading days each monh): RV h = h 1 *22 h * 22 1 j = RV j JV JM h * 22 1 J h j = = h * 22 1 h * 22 1 ( J j = j j D h j = = h * 22 1 j = D D DAILY j JM DAILY j h ) DAILY j 2 D DAILY j JI h = h 1 *22 h * 22 1 DAILY D j j = The jump mean and volailiies are calculaed only over days where a jump is deeced. Wrigh and Zhou (29) ake one-monh rolling window for realized volailiy, since i can be esimaed accuraely wih a fixed span of 15

16 sufficienly high-frequency daa. Also for realized volailiy you have one observaion for every day. This is cerainly no he case for jumps; given he presumpion ha jumps in asse prices are boh rare and large. Wrigh and Zhou (29) herefore argue ha you need much longer rolling window for calculaing he jump mean, jump volailiy and jump inensiy. They ake he parameer h equal o 24 monhs or 12 monhs. The radeoff in choosing h is ha a shorer window gives a noisier bu also imelier measure of agens percepions of jump risk. 16

17 5. Replicaion Wrigh and Zhou In his secion we replicae he resuls of Wrigh and Zhou (29). The firs objecive is o see wheher we ge similar resuls. Noice ha we can no exacly replicae heir findings, since we have a differen sample period and our analysis is based on he 1-year US fuure while Wrigh and Zhou (29) use he 3 year fuure Bond reurn regression models Wrigh and Zhou (29) use in heir paper realized volailiy and jump risk measures (as consruced in he previous secion) o forecas excess bond reurns. We use end-of-monh daa on zero-coupon bonds from he CRSP Fama-Bliss daa o consruc excess reurns. We use he following noaion for log bond prices: (n) p = log price of he n-year discoun bond a ime. The log yield is denoed by: y = p. n ( n) 1 ( n) Following Cochrane and Piazzesi (25) we wrie he log forward rae a ime for loans beween ime + n - 1 and + n as: f ( n) ( n 1) ( n) = p p. We denoe he log reurn from buying an n-year bond a ime and selling i as an n-1 year bond a ime + 1 as: r ( n) ( n 1) ( n) 1 = p 1 p. + + We wrie one-year log excess reurns by: ex ( n) ( n) (1) + 1 = r + 1 y. We use over bars o denoe averages across mauriy: ex 1 5 ( n) + 1 = ex n = 2 All he regressions for excess bond reurns ha we consider in his secion are nesed wihin he specificaion: ex h h h = + β1 f + + β 2 f + β 3 f + β 4 RV + β 5 JM + β 6 JI + β 7 JV ε + 1 β ( 7) In his secion we copy he seings of Wrigh and Zhou (29), using heir variables and also a one-year holding period. The nex secions will include oher (our own) variables and we will also consider shorer holding periods (3 monhs). Using jus he forward raes in he regression you ge Cochrane and Piazzesi (25), excep ha, following Wrigh and Zhou (29) we use hree forward raes insead of five, o minimize he near-perfec collineariy problem. Also he R 2 of he CP regression wih hree forward raes is nearly he same as he regression using al he five forward raes (he difference is less han 1 percenage poin). 17

18 5.2. Graphs and summary saisics forecas variables The figure below graphs he realized jumps for our in-sample period ( ). For he firs par of he analysis we will only look a excess reurns unil end of 22 (jump variables unil end of 21 because of oneyear holding period). We do his o minimize he risks of daa mining. In he nex secions we will perform (again for our in-sample period) some robusness checks and afer choosing our opimal model, we perform an ou-ofsample analysis unil end of 29. Figure 7: Esimaed daily jumps for he 1-year US bond fuure (in basis poins), consruced as described in he ex. Realized Jumps (basis poins) /1/89 7/1/89 1/1/9 7/1/9 1/1/91 7/1/91 1/1/92 7/1/92 1/1/93 7/1/93 1/1/94 7/1/94 1/1/95 7/1/95 1/1/96 7/1/96 1/1/97 7/1/97 1/1/98 7/1/98 1/1/99 7/1/99 1/1/ 7/1/ 1/1/1 7/1/1 Dae Using he realized jumps graphed in Figure 7 we can consruc our jump variables described previously. For our complee sample jumps occur on abou 8% of he days, which is similar o Wrigh and Zhou (29) and Andersen e all (27). We use 24-monh rolling windows for he consrucion of he jump inensiy, jump mean and jump volailiy. This seems a long rolling window, bu since jumps are rare (on average 2 jumps a monh), we need a long rolling window o have enough observaions o consruc our variables. In he robusness secion we will also look a a one-year rolling window. For Realized Volailiy however we ake a rolling window of onemonh, since in conras wih he jump variables we have for he RV an observaion for each day and we do no need a long horizon o ge an accurae esimae of he mean. The figure below plos he one-monh rolling window of he realized volailiy: Figure 8: Realized volailiy for he 1-year US bond fuure (in annualized percenage poins) using a one-monh rolling window, consruced as described in he ex. Realized Volailiy Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 18

19 Below you can see he graphs of he jump variables. The RV is less smooh (because of he shorer rolling window) compared o he jump mean, jump inensiy and he jump volailiy. The jump mean will appear o be a significan forecas variable of he one-year bond excess reurns. Figure 9: Realized jump risk measures. 24-monh rolling esimaes of he jump mean, jump inensiy and jump volailiy, consruced as described in he ex..16 Jump Inensiy Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 Jump Volailiy Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 Jump Mean Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 19

20 Some summary saisics for he key variables are given in Table 2. The averages of he jump inensiy, jump mean and jump volailiy for his sample are respecively 11%, 2 basis poins and 28 basis poins. The average of he jump mean does no seem o be large, bu remember ha we ake he average over posiive and negaive jumps. Turning o he correlaion marix, you can see ha he excess reurns are highly correlaed. Also he forward raes are posiively correlaed. The jump mean variable is fairly negaively correlaed wih he excess reurns. ex 2 ex 3 ex 4 ex 5 f1 f3 f5 RV JI JM JV mean sd corr marix ex 2 ex 3 ex 4 ex 5 f1 f3 f5 RV JI JM JV ex 2 1% 98% 96% 93% 32% 57% 45% % 7% -38% 22% ex 3 1% 99% 97% 29% 57% 46% 3% 4% -39% 22% ex 4 1% 99% 27% 6% 52% 5% 7% -38% 23% ex 5 1% 27% 62% 54% 6% 5% -38% 25% f1 1% 62% 31% -12% 14% -44% 41% f3 1% 9% 8% 33% -25% 32% f5 1% 7% 38% 2% 13% RV 1% -28% 8% -3% JI 1% 1% 2% JM 1% -78% JV 1% Table 2: Summary saisics and correlaion marix for excess reurns, forward raes, and jump measures for he sample All he variables are in percenages excep jump inensiy (raw value). All he variables in Table 2 are in percenages expec he jump inensiy. The average excess reurns increase wih mauriy: 1.5% for he wo year bond versus 2.67% for he five year bond. This higher reurn comes wih greaer risk: 1.46% sandard deviaion for he wo year bond versus 4.91% for he five year bond. Since he 2, 3, 4 and 5 year US bond excess reurns are srongly posiively correlaed we ake, following Wrigh and Zhou (29), he average of hese variables in he regression analysis. So he y-variable is equal o: 5 1 ( n) ex + 1 = ex n = 2 2

21 5.3. Main resuls replicaion Wrigh and Zhou (29) In his secion we presen he regression model esimaes of he jump variables as presened in Wrigh and Zhou (29). Table 3 shows he coefficien esimaes, he corresponding -values and R 2 s for differen specificaions of (7). The seings are as follows: one-year holding period, wo-year rolling window for he jump variables (onemonh for RV), he y-variable in he regression is he excess reurns on a bond (average 2 o 5 year) over hose on a one-year bond. In his able he forward raes are no included in he regressions. The jump mean variable is on is own a significan negaive predicor of bond excess reurns as follows from he -value and he R 2. This implies ha posiive jumps are followed by negaive excess reurns and visa versa. The R 2 of he jump mean is comparable wih using one single forward rae wih a maching mauriy, Fama-Bliss (1987). This is in conras wih he oher variables. For he jump volailiy you ge an R 2 of 6%, however he variable is no significan. The RV and JI show no predicive power a all. These resuls coincide wih he findings of Wrigh and Zhou (29). We look here only a our in-sample period: We use inraday daa of he 1-year US fuure saring from 1989, bu we use wo-year rolling window and herefore our firs esimae of he jump variables is from he beginning of We use a one-year holding period, herefore he las esimae of he jump variables is a December 21 (for he excess reurns i is December 22). From Table 2 we observe ha he sandard deviaion of boh jump mean and jump inensiy is 4 basis poins. So an increase of one sandard deviaion of he jump mean variable means ceeris paribus a decease of 1.2% in he average excess reurn (-3.2 *.4). An increase of one sandard deviaion of he jump volailiy variable means (again ceeris paribus) an increase of.76% in he average excess reurn (19.2 *.4). However one has o be careful wih he inerpreaion of his las example, since he jump volailiy is no significan. Inercep -value variable -value R2 JI % JM % JV % RV % sample 1991:22 Table 3: This able repors coefficien esimaes in a regression of he excess reurns on a bond (average 2 o 5 year) over hose on a oneyear bond, wih a holding period of 12 monhs, an one-monh rolling window of realized volailiy, and 24-monh rolling windows of he bond jump mean, jump inensiy and jump volailiy. Observaions are a he monhly frequency (end-of-monh). The -values are based on Newey-Wes sandard errors. Table 4 shows he resuls from oher specificaions of (7), bu now wih he forward raes included. The coefficiens of he forward raes show more or less he well-known en paern. However he paern is no as srong as in Cochrane and Piazzesi (25), bu his is also an oher sample. Running he regression of he excess reurns using he forward raes alone gives an R 2 of 41%, which is a lo and similar o he findings in Cochrane and Piazzesi (25). Including he jump mean variable ino he regression he R 2 increases o 48%, however he jump mean looses a bi of is significance (significan a 5%, bu no a 2.5%) compared o he case wih no forward raes included in he regression. This is a bi differen han he resuls of Wrigh and Zhou (29). For heir sample ( ) he jump mean remains srongly significan wih -values and R 2 s around respecively -6 and 6%. However in he nex chaper we will presen a refinemen of he jump mean variable, which will perform much beer han he jump mean even afer including he forward raes in he regression. Meanwhile, he realized volailiy, jump volailiy and jump inensiy show no predicive power, also afer including he CP variables in he regression. 21

22 The coefficien of he jump mean variable in he mulivariae regression of Table 4 is no much changed compared o he univariae regression, suggesing ha he forward raes and he jump mean variable measure differen componens of he bond risk premium. This also suggess ha he inerpreaion of he jump mean variable as presened in his secion does no change much. Inercep -value f1 -value f3 -value f5 -value variable -value R2 CP % CP + JI % CP + JM % CP + JV % CP + RV % sample 1991:22 Table 4: This able repors coefficien esimaes in a regression of he excess reurns on a bond (average 2 o 5 year) over hose on a oneyear bond, wih a holding period of 12 monhs, an one-monh rolling window for realized volailiy, and 24-monh rolling window for he bond jump mean, jump inensiy and jump volailiy. Observaions are a he monhly frequency (end-of-monh). In hese regressions also he 1-, 3- and 5-year forward raes (CP variables) are included. The -values are based on Newey-Wes sandard errors. In his secion we replicaed he resuls presened in Wrigh and Zhou (29). In he nex secion, Chaper 6, we presen a refinemen of he jump mean variable and inroduce a mean reversion variable, which we relae o he jump mean. The reason for his is ha he jump mean feels like mean reversion, since he coefficien esimae in he regression model is negaive: Upward jumps forecas negaive excess reurns. In Chaper 6 we show ha hese wo variables are correlaed o some exend, bu hey are definiely no he same. In Chaper 7 we presen differen robusness checks (shorer holding period, shorer rolling window and lower hreshold value for idenifying jumps) and coninue o find imporan role for he jump mean in forecasing excess reurns. 22

23 6. Refinemens In his chaper we presen a refinemen of he jump mean variable which performs beer in he regression model and we also inroduce a mean reversion variable o relae i o he jump mean Definiion variables The jump mean variable is a (wo-year) rolling average of he jumps in 1-year US fuure prices. You can ask he quesion wheher all he jumps are of equal imporance in forecasing excess reurns. For example i can happen ha afer a small jump he fuure price will move back owards is previous level before he jump. This indicaes ha he jump was of lile significance (or no immediaely righly undersood by he marke). On he oher hand, i is possible ha afer a big jump he price will coninue o move owards he same direcion. The jump mean variable is no able o capure such feaures, because we measure he jump size only. I migh be more useful o look a he oal reurn occurred on a jump day, insead a he jump size iself. I will indeed urn ou ha his variable, which we call reurn jump mean (RJM), will perform beer han he jump mean variable. The RJM variable is hus he (wo-year) rolling average of he log reurns occurred a days where a jump is deeced. In formula we ge: Again he h denoes he rolling window, RJM DAILY j h * 22 1 R h j = = h * 22 1 j = j D D DAILY j DAILY j D is he dummy variable which akes he value 1 when a jump is deeced a day and is oherwise and R is he log reurn a day. The inuiion behind he jump mean variable in Wrigh and Zhou (29) is weak. The jump mean variable is srongly significan and he resuls presened in he paper are grea, bu i is no presened wih a good explanaion (we need o improve he inuiion in order o have confidence in he resuls). One of our firs ideas was o compare he jump mean o he mean reversion (MR) variable presened in Fama (26). The jump mean feels like mean reversion since he coefficien in he regression is negaive. Our mean reversion variable is he cumulaive (log) reurn of he pas wo years (same rolling window as he jump mean). We calculae he MR no as he difference beween oday s log fuure price and he price wo years before, since hese denoe differen fuure conracs. Insead we calculae for each day he log reurn and add ha up. In formula you ge: MR h = h * 22 1 R j j = Table 5 presens he correlaion beween he jump mean, reurn jump mean and mean reversion. JM and RJM are as expeced srongly correlaed. You would expec ha a jump would dominae he reurn a ha day and ha indeed follows from he high correlaion beween JM and RJM. JM and MR show a correlaion of 9% bu MR is more correlaed wih RJM. Denoe he difference beween hese wo variables. MR is he cumulaive reurn over every day for he pas wo days, where RJM is he average reurn only over days where a jump is deeced. JM RJM MR mean sd corr marix JM RJM MR JM 1% 8% 9% RJM 1% 35% MR 1% Table 5: Summary saisics and correlaion marix for JM, RJM and MR for he sample JM and RJM are in percenages. 23

24 These correlaions show ha he jump variables JM and RJM are relaed o mean reversion, bu hey are definiely no he same and hus here is added value in he jump mean variable above mean reversion. Figure 1 graphs he JM ogeher wih RJM and MR. JM and RJM show similar paerns as follows from he high correlaion. The sign of boh variables usually coincides excep for he end of he sample (from April 99). The JM remains posiive while RJM falls below zero. Jump Mean and Reurn Jump Mean JM RJM Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 Jump Mean and Mean Reversion JM MR Dec-9 Apr-91 Aug-91 Dec-91 Apr-92 Aug-92 Dec-92 Apr-93 Aug-93 Dec-93 Apr-94 Aug-94 Dec-94 Apr-95 Aug-95 Dec-95 Apr-96 Aug-96 Dec-96 Apr-97 Aug-97 Dec-97 Apr-98 Aug-98 Dec-98 Apr-99 Aug-99 Dec-99 Apr- Aug- Dec- Apr-1 Aug-1 Dec-1 Figure 1: Two year rolling window esimaes of he jump mean ogeher wih reurn jump mean (firs panel) and mean reversion (second panel). The variables are consruced as described in he ex. Looking a he second graph in Figure 1, you can see ha in he firs half of he sample JM and MR show he same paern, bu his is no longer he case from 95 onwards. For he period he MR is posiive while he JM is negaive. This means ha around ha period (remember ha he variables are consruced by aking a rolling window of wo-year) he reurns were posiive while he jumps were mosly downwards. In he nex secion we will esimae regression models including hese new wo variables (RJM and MR) Resuls regression Table 6 shows coefficien esimaes and corresponding -saisics and R 2 s aking he jump mean, reurn jump mean en mean reversion as forecas variables. The dependen variable is he average excess reurn of he wo o five year US bond. The resuls for he jump mean were already shown in Table 3, bu we will also include i here 24

25 for he sake of compleeness. The RJM shows a higher R 2 compared o JM (18% agains 15%) and he coefficien is also esimaed more precisely (he -value is larger in absolue value). The mean reversion variable is also srongly significan. The sign of MR is negaive and herefore i is indeed appropriae o call he variable mean reversion: posiive reurns occurred over he las wo years are a negaive predicor for reurns of he coming year. Look a Table 5 again for he inerpreaion of he coefficiens. The sandard deviaion of RJM and MR is respecively 7 basis poins and 5%. So an increase of one sandard deviaion of he RJM variable means ceeris paribus a decease of 1.3% in he average excess reurn ( *.7), which is very similar o he jump mean variable. An increase of one sandard deviaion of he MR variable means (again ceeris paribus) a decrease of 1.45% in he average excess reurn (-.29 *.5). Inercep -value variable -value R2 JM % RJM % MR % Sample 1991:22 Table 6: This able repors coefficien esimaes in a regression of he excess reurns on a bond (average 2 o 5 year) over hose on a oneyear bond, wih a holding period of 12 monhs and 24-monh rolling window of he bond jump mean, reurn jump mean and mean reversion. Observaions are a he monhly frequency (end-of-monh). The -values are based on Newey-Wes sandard errors. Table 7 shows he resuls of he regression bu now wih he forward raes included. Running he regression of he excess reurns using he forward raes alone gives an R 2 of 41%, as we already noed in he previous secion. Including he jump mean variable ino he regression he R 2 increases o 48%, however he jump mean looses a bi of is significance (significan a 5%, bu no a 2.5%) compared o he case wih no forward raes included in he regression. The RJM performs again beer han he JM. The difference here is huge. The RJM variable is srongly significan (-value of -4.76) and he R 2 increases o 67%. The informaion conained in he RJM variable seems o complemen ha of he forward raes, since he R 2 in he regression of he combined variables (RJM and he forward raes) is larger han he sum of he R 2 s of he separae regressions. I is ineresing o noice ha he hree year forward rae, f3, is always significan, while f1 and f3 are no significan a convenional significance levels. Mos informaion in forecasing excess reurns seems o be hidden in he hree year forward rae. Also he MR variable remains significan afer including he Cochrane-Piazzesi variables. Conrolling for mean reversion does no change he coefficien of he forward raes grealy, suggesing ha he erm srucure of he forward raes and he mean reversion variable are measuring differen componens of he bond risk premia. Inercep -value f1 -value f3 -value f5 -value variable -value R2 CP % CP + JM % CP + RJM % CP + MR % sample 1991:22 Table 7: This able repors coefficien esimaes in a regression of he excess reurns on a bond (average 2 o 5 year) over hose on a oneyear bond, wih a holding period of 12 monhs and 24-monh rolling window of he bond jump mean, reurn jump mean and mean reversion. Observaions are a he monhly frequency (end-of-monh). In hese regressions also he 1-, 3 - and 5-year forward raes (CP variables) are included. The -values are based on Newey-Wes sandard errors. From he fac ha boh MR and RJM remain significan afer including he forward raes in he regression and he fac ha he wo are no highly correlaed, i seems sraighforward o esimae a regression model aking he forward raes, MR and RJM as regressors. I follows ha boh variables are again significan and keep heir sign (boh negaive). The R 2 increases furher o 72%. You can argue ha including variables ino a regression model always increases he R 2. Therefore i is also useful o look a he adjused R 2, where you ge a 25