Risk contributions of trading and non-trading hours: Evidence from commodity futures markets

Transcription

1 Risk cotributios of tradig ad o-tradig hours: Evidece from commodity futures markets Qigfu Liu Istitute for Fiacial Studies Fuda Uiversity, Shaghai, Chia Yubi A 1 Odette School of Busiess Uiversity of Widsor Widsor, Otario, Caada N9B 3P4 Abstract This aer focuses o risk cotributios of tradig ad o-tradig hours i Chiese commodity futures markets. We first examie itegrated risks of Chiese coer, rubber, ad soybea futures markets withi the coula-var (value at risk) ad coula-es (exected shortfall) frameworks, exlicitly accoutig for both the tradig ad o-tradig iformatio. The, we evaluate the comoet VaR ad comoet ES of the tradig ad o-tradig eriods to gauge their resective risk cotributios to the itegrated market risks. We fid that coula-based VaR models ca aroriately measure itegrated risks, as the tyical VaR ad ES based o close-to-close returs uderestimate overall market risks. I additio, we documet that the fiacial iformatio accumulated durig otradig hours cotributes substatially to the overall risk of futures markets, with comoet VaR ad ES weights ragig from more tha 40% to early 60% i these markets. I articular, the iformatio durig o-tradig hours is more imortat tha that i tradig hours i exlaiig the total risk of coer futures i Chia. Moreover, the risk cotributio of o-tradig eriods icreases with their legths, reflectig the fact that iformatio flows costatly over time. Keywords: risk cotributio; value at risk, exected shortfall; futures markets; tradig hours; o-tradig hours JEL classificatio: C32, G15 1 Corresodig author. yubi@uwidsor.ca 1

2 1. Itroductio Chiese fiacial markets trade oly durig the daytime hours from Moday to Friday, ad the total tradig eriod of ay market is geerally less tha seve hours, which is less tha oe half of the o-tradig hours. Cosequetly, the fiacial iformatio accumulated durig o-tradig hours reresets a sigificat source of the itegrated market risk, ad lays a imortat role i rice discovery i fiacial markets. This iformatio arises from ublic aoucemets made durig o-tradig hours i Chia as well as from tradig activities i overseas markets. For risk hedgig uroses, it is crucial ot oly to roerly measure the overall risk i fiacial markets by accoutig for all rice sesitive iformatio but also to locate sources of risk, articularly risks related to tradig ad o-tradig hours. The urose of this aer is to quatify itegrated risks ad risk cotributios of tradig ad o-tradig hours i major Chiese commodity futures markets. To measure the itegrated risk of a futures market, we emloy the value at risk (VaR) ad exected shortfall (ES) cocets i our aalysis, as both rovide a clear iterretatio i moetary terms ad a direct alicatio i risk maagemet. VaR is defied as the maximum loss of a ivestmet at a certai cofidece level over a secified horizo; it became a widely acceted stadard i the risk maagemet idustry after J.P. Morga itroduced their RiskMetrics documet i However, the major drawback of this method is that it is ot sub-additive, which meas that the VaR of a ortfolio ca be larger or smaller tha the sum of the VaRs of its comoets. Moreover, it caot measure the exected loss resultig from extremely ulikely market factor chages. To overcome these drawbacks, Egle ad Magaelli (1999) roose the cocet of exected shortfall (ES), 2

3 which is the mea of losses, give that losses are greater tha the VaR at the cofidece level. Some research (Acerbi ad Tasche, 2002; Frey ad McNeil, 2002; Magaelli ad Egle, 2001) illustrates that ES rovides a better estimate of risk tha VaR whe loss distributios exhibit fat tails or emirical discreteess. I cotrast with VaR, ES is a coheret risk measure i the sese of Artzer et al. (1999), ad satisfies the sub-additivity roerty. For these reasos, ES is cosidered a imortat alterative to VaR for the urose of risk measuremet. Itegrated risk estimates deed critically o the retur distributio assumed for a articular market. I this aer, we utilize coula fuctios based o eaks over threshold (POT) theory to lik both tradig ad o-tradig retur distributios ad to costruct the market retur distributio. Traditioally, the joit distributio of multi-asset returs or risk factors is assumed to be multivariate ormal with a liear correlatio matrix. However, it is a stylized fact that fiacial asset returs tyically exhibit o-ormal roerties ad oliear deedecies. As we kow, the coula aroach is more flexible i modelig multivariate distributios, as it ca searate uivariate margial distributios from the multivariate deedece structure. This allows us to select a rich deedece structure while reservig o-ormality roerties of margial distributios. Give these advatages, the coula aroach has recetly received substatial attetio i the fiace literature. For examle, Ward ad Lee (2002) use a multivariate ormal coula to aggregate differet tyes of risks to create a itegrated risk distributio for a isurace comay. O the other had, Roseberg ad Schuerma (2006) combie market, credit, ad oeratioal risks with coulas to obtai a total risk distributio for a fiacial istitutio. They fid that the coula-based aroach is more accurate tha other methods 3

4 at estimatig the overall risk as measured by VaR ad ES. We follow this lie of thought ad combie the correlated tradig ad o-tradig retur distributios ito a joit risk distributio usig a coula fuctio that better fits our data. Tradig ad o-tradig returs ca exhibit quite distict distributioal characteristics due to differet iformatio flow atters i these eriods. To cature the o-ormality characteristics of tradig ad o-tradig retur distributios, such as skewess ad fat-tails, we aly the extreme value theory (EVT) to form these distributios. The EVT is desiged esecially for tail estimatio, ad it eables us to estimate extreme robabilities ad extreme quatiles without makig assumtios about a ukow aret distributio. Therefore, our aroach ca better measure the itegrated market risk tha the tyical method based solely o close-to-close daily returs, as it exlicitly accouts for iformatio flows durig both tradig ad o-tradig eriods. Aother imortat objective of this aer is to quatify the risk cotributios of tradig ad o-tradig hours i a market. To this ed, we decomose the total risk of the market ito comoet VaRs ad comoet ESs to gauge the imact of the iformatio accumulated durig o-tradig hours o market risk. We also distiguish amog differet tyes of o-tradig eriods to further ivestigate the role of the legth of o-tradig eriods i risk cotributios. The imortace of o-tradig iformatio i rice discovery ad market volatility has bee well documeted i the literature. Tsiakas (2008) fids that the size ad redictive ability of o-tradig iformatio for both Euroea ad US stock markets are substatial. Taylor (2007) also cofirms the sigificat imact of overight iformatio o iformatio flow i the regular S&P 500 futures market. O the other had, Cliff, Cooer, ad Gule (2008) fid that ight returs are higher tha day returs, ad 4

5 coclude that the US equity remium over the last decade is solely due to overight returs. I cotrast, this aer rooses a ew aroach to measurig the risk cotributio of overight returs, ad rimarily focuses o the risk asects of tradig ad o-tradig iformatio i the market. This is of articular iterest to both academics ad ractitioers, as our results hel exlai sources of risk ad rovide imortat imlicatios for risk maagemet. Additioally, it hels us better uderstad how the tradig iformatio i iteratioal develoed markets imacts Chiese markets. Note that Euroea ad US markets are oe to trade durig Chiese o-tradig hours. Usig data from Chiese coer ad rubber futures traded o the Shaghai Futures Exchage (SHFE) ad soybea futures traded o the Dalia Commodity Exchage (DCE), we show that our model ca aroriately measure itegrated risks, ad that the tyical VaR ad ES calculatios based o close-to-close returs sigificatly uderestimate overall market risk. Additioally, the fiacial iformatio accumulated durig otradig hours cotributes substatially to the overall risk of commodity futures markets i Chia. The comoet VaR ad comoet ES weights of o-tradig hours are much higher tha 40% for all markets uder both 95% ad 99% cofidece levels. Secifically, o-tradig hours cotribute more to the itegrated risk tha tradig hours i the coer futures market, idicatig that o-tradig hours cotai more imortat iformatio tha do tradig eriods. Moreover, the risk cotributio of o-tradig hours icreases with the legth of the o-tradig eriod. We also demostrate that both the VaR ad ES are articularly sesitive to the cofidece level whe the level is high. The remaider of this aer is orgaized as follows. Sectio 2 describes the coula methodology. Sectio 3 discusses the data used for the aalysis ad rovides the 5

6 descritive statistics. Sectio 4 reorts the emirical results, while Sectio 5 cocludes this aer. 2. POT-based coula- VaR ad ES models 2.1. Tradig ad o-tradig returs Let c F t deote the daily closig rice of a futures cotract at date t, ad o F t the daily oeig rice. Followig Tsiakas (2008), the close-to-close daily retur r t, the close-tooe o-tradig retur r t, ad the oe-to-close tradig retur r d t are calculated as er the followig defiitios: r t r t r d t c c 100l( Ft / Ft 1), (1) o c 100l( F / F 1), (2) t t c o 100l( F / F ). (3) t t Thus, we ca searate o-tradig returs from tradig returs. The o-tradig retur ad its variability are due to the iformatio released durig o-tradig hours, as this iformatio is reflected i oeig rices. The tyical daily retur is simly a sum of the tradig ad o-tradig returs. Searatig tradig from o-tradig returs allows us to model their distributios differetly to esure their uique distributioal characteristics are catured. Further, o-tradig eriods ca be ormal weekights, weekeds, or holidays. As a geeral ractice i Chia, whe a ublic holiday(s) haes to be o a weekday(s) from Tuesday to Thursday, a short holiday eriod is created by extedig the earest weeked to iclude those dates before or after the holiday(s). As a result, ay o-tradig holiday eriod cotais at least 72 hours. Presumably, a loger o-tradig eriod may accumulate more iformatio, thereby makig a higher cotributio to itegrated risks 6

7 tha a shorter o-tradig eriod. We will ivestigate whether a loger o-tradig eriod ideed cotais more (or more imortat) iformatio about returs tha does a shorter oe Modelig joit distributios with coula fuctios Our focus i this aer is o a bivariate model, where the radom variables are tradig ad o-tradig returs. I geeral, let X, ) be a vector of two radom variables with ( X 1 2 a joit distributio deoted as F ( x, x 1 2), ad the margial distributio fuctios F 1 ad F 2, resectively. Additioally, we assume that each margial distributio fuctio deeds o oe sigle arameter i ( i 1, 2 ). The, Sklar s theorem states that there is a coula fuctio C u, u ; ) such that ( 1 2 F x, x ) C( F ( x ; ), F ( x ; ); ), (4) ( where is the arameter vector of the coula fuctio C. If F 1 ad F 2 are cotiuous, the the coula fuctio is uique; otherwise, it is uiquely determied o Ra( F1 ) Ra( F2 ). Coversely, for ay uivariate risk distributios F 1 ad F 2, Equatio (4) defies a joit distributio F ( x, x 1 2) with margis F 1 ad F 2. Thus, Sklar s theorem imlies that we ca combie ay uivariate distributios ito a joit distributio together with a coula. Differetiatig F x 1, x ) with resect to its variables yields the joit desity fuctio ( 2 f x, x ) c( F ( x ; ), F ( x ; ); ) f ( x ; ) f ( x ; ), (5) ( where coula desity c( u1, u2; ) C( u1, u2; ) / u1u 2, ad f 1( x1; 1 ) as well as f x ; ) are margial desity fuctios. Equatio (5) shows that for ay cotiuous joit 2( 2 2 distributio, the uivariate margis ad the deedece structure ca be searated, where 7

8 the deedece structure ca be comletely determied by a roer coula fuctio. For this reaso, coula fuctios eable us to obtai a joit distributio with a variety of ossible, but ot ecessarily equal, arametric uivariate distributios. Cosequetly, we ca reserve the origial characteristics of margial distributios, while allowig for a wide rage of deedece relatios. For a give set of retur observatios ( x T 1, t, x2, t) t 1, model arameters,, ) ca be joitly estimated by maximizig the followig log-likelihood fuctio T l f1( x1, t; 1 ) l f2( x2, t; 2 T 1, x2) l c( F1 ( x1, t; 1 ), F2 ( x2, t; 2); ) ) t1 t1 L( x ( 1 2 This method may be comutatioally exesive esecially for high dimesios. As we ca see from Equatio (6), the coula arameter is uaffected by the arameters of the margial distributios. As a cosequece, these arameters ca be estimated searately. I this aer, we adot the Iferece Fuctio for Margis (IFM) method, i which arameters are estimated i two stages. More secifically, we first estimate arameters 1 ad 2 of margial distributios. The, we estimate the coula arameter vector coditioal uo the margial distributio estimates i the first ste. For the estimatio at each stage, the maximum likelihood method is used. The IFM method is able to rovide estimators as well as the joit distributio method i terms of the mea square errors (Xu, 1996), but is aaretly comutatioally simler Margial distributios Excess distributios. (6) 8

9 I this aer, we emloy the extreme value theory (EVT) to select ad secify the margial distributios of tradig ad o-tradig returs. As is oited out by Logi (2000), the major advatage of EVT is that it takes ito accout rare evets cotaied i distributio tails, with o articular retur distributio assumtio. There are two mai aroaches to estimatig extreme values withi the EVT cotext: the block maxima model (BMM) ad the eaks over threshold (POT) model. The BMM method cosiders the maximum (or miimum) observatios i successive eriods, which are re-chose as blocks. I cotrast, the POT method focuses o the realizatios that exceed a give high threshold. The latter method aaretly uses the available data more efficietly, ad therefore is adoted i this aalysis. I articular, we cosider the most widely used distributio i modellig excesses: the geeralized Pareto distributio (GPD). To illustrate this method, let F( x) Pr( X x) be the distributio fuctio of a radom variable X. For a give threshold u, the distributio of the excess values is give by F u F( y u) F( u) ( y) Pr( X u y X u), y 0. (7) 1 F( u) I ractical alicatios, we will have to aroximate the coditioal excess distributio for high threshold values, as the aret distributio F (x) is ukow. Balkema ad De Haa (1974) ad Pickads (1975) rove that for a sufficietly high threshold u, the excess distributio coverges to the GPD, which is give as 1/ y 1 1, 0 G ( ) ( ), ( u) y u (8) y ( u) 1 e, 0 9

10 where (u) is a ositive fuctio of u, reresetig a scale arameter, ad is a shae arameter that determies the GPD shae. Whe 0, 0, or 0, the corresodig excess distributio is from the Fréchet, Gumbel, or Weibull families, resectively. The Fréchet family is articularly iterestig, as it is most suitable for modellig fat-tailed retur distributios. Give that the GPD well aroximates tail distributios of actual data, we use it to simulate the uer ad lower tails of tradig ad o-tradig returs. The margial distributio is as follows: L L 1/ L ku L x u 1, L K F ( x) ( x), R R ku R x u 1 1 R K R 1/, x u L x u L x u R u R (9) where R u ad u corresod to the uer ad lower thresholds, resectively. x ) is the L 10 ( L R emirical fuctio of x i [ u, u ]. K is the umber of total observatios. k R u reresets the umber of returs that are larger tha the uer threshold umber of returs that are smaller tha the lower threshold R are arameters to be estimated i the margial distributio selectio R u, while L u. Further, L k u reresets the R, L, L, ad Before estimatig the arameters i Equatio (9), a roer level of threshold u eeds to be selected. If we choose too high a threshold, the there may be isufficiet exceedaces, ad this may result i high variace estimators. O the other had, too low a threshold may ot satisfy well the coditios for covergece i the distributio of

11 threshold exceedaces to the GPD, thereby yieldig biased estimators. Thus, we face a tradeoff betwee bias ad variace i the threshold determiatio. The followig is a review of three major aroaches to threshold choice. The first method is based o the mea excess fuctio (MEF), which is defied as e( u) E( X u X u). (10) It ca be show that the MEF of the GPD is a liear fuctio of threshold u. For this reaso, we cosider the emirical MEF 1 u eˆ ( u) ( X i u), (11) u i1 where X ( i 1, 2,, ) reresets observatios that exceed the threshold, ad lot the i u MEF as a fuctio of u. The threshold we choose is the lowest u such that the emirical MEF is aroximately liear. The secod method selects the otimal threshold accordig to the stability of arameter estimates. If the excess distributio for a iitial threshold u 0 is a GPD with arameters ( u 0 ) ad, the the ew excesses over a higher threshold u u0 are distributed as a GPD with corresodig arameters (u) ad, where u) ( u ) ( u ). Defie the modified scale arameter as ' ( u) u ( 0 u0, the ' is a costat with resect to threshold u. Cosequetly, we lot ' ad versus u together with cofidece itervals for the estimated quatities, ad select the smallest u such that these estimates remai early stable. 11

12 The third aroach is based o Hill s (1975) lot. Secifically, let X X 1 2 X be ideedet ad idetically distributed radom variables. The tail idex statistic is give by k 1 X i H k k l, i1 X k. (12) 1 Defie Hill s lot as the dot set of ( k, H k ),1 k 1. The observatio X k that, corresods to the begiig oit i the tail idex stable zoe is selected as the threshold. Overall, the first ad third methods deed crucially o the umber of exceedaces, ad may yield a large bias i the case of small samles. O the other had, the secod method rovides a relatively objective criterio, as the threshold is usually determied by regressios i this aroach. With this method, differet thresholds ca be selected for differet recisio levels. To make the threshold estimates more accurate, we first decide the rage for the otimal threshold accordig to the first aroach, ad the choose the value as er the secod ad third methods Itegrated VaR ad ES estimatio Give the margial distributios of tradig ad o-tradig returs F ) ad F ) 1 ( x 1 2 ( x 2 for the selected thresholds, we the estimate coula arameters by substitutig them ito various tyes of coula fuctios 1 1 C( u1, u2) F( F1 ( x1 ), F2 ( x2)), (13) where F 1 ( ) is the reverse of the margial fuctio F x ). Next, these coula fuctios i x i are simulated ad comared, ad the oe with the smallest Bayesia Iformatio Criteria (BIC) (or Akaike s Iformatio Criteria (AIC)) is chose as the otimal coula for our i ( i 12

13 aalysis. 2 Fially, the VaR ad ES at a cofidece level ca be comuted usig the Mote Carlo method. Secifically, VaR is solved from the followig defiitio: Pr( X VaR ), (14) where VaR w1 M-VaR1 w2 M-VaR2. Namely, M -VaR1 M -VaR2 c i i) i1 F ( x ), F ( x ) f ( x dx dx The ES is defied as the exected loss that exceeds VaR. Namely, 1 2. (15) ES E X X VaR ) VaR E( X VaR X VaR ), (16) ( where E( X VaR X VaR) is the mea excess fuctio corresodig to the VaR. We kow that the mea excess fuctio for the GPD with arameters 1 ad is ( VaR u) e( VaR ) E( X VaR X VaR ). (17) 1 Equatios (16) ad (17) together imly VaR u ES. (18) Comoet VaR (C_VaR) ad comoet ES (C_ES) I this sectio, we decomose the itegrated risk measures ito the risk cotributios of tradig ad o-tradig hours. For this urose, we adot the cocets of margial ad comoet risks i the risk attributio framework. 2 The coula fuctios cosidered i this aer iclude those from ellise coula fuctio family (Normal coula), Archimedes coula family (Gumbel coula, Joe coula, Frak coula, BB1 coula, BB3 coula, BB6 coula, ad BB7 coula), extreme value coula family (Taw coula), ad Archimax coula family (BB4 coula). 13

14 The margial VaR (M_VaR) of tradig or o-tradig returs reresets the margial imact of a small chage i the weight of tradig or o-tradig returs o the itegrated VaR. I articular, VaR M_ VaR r VaR, (19) i E ri wi where w corresods to the weight assiged to tradig iformatio ( i 1) or o-tradig i iformatio ( i 2 ), ad r is the sum of tradig ad o-tradig returs. Sice VaR is homogeeous of degree oe, 3 we have the followig formula, accordig to Ruler s theorem (Zhag ad Rachev, 2006): VaR VaR VaR w VaR. (20) 1 w2 w1 M_ VaR1 w2 M_ w1 w2 2 Thus, the first term i Equatio (20) measures the risk cotributio of tradig returs, while the secod term measures the risk cotributio of o-tradig returs. The sum of both terms is just equal to the itegrated risk of the market uder cosideratio. For this reaso, the comoet VaR is simly defied as C _ VaR i w M_ VaR, i 1, 2. (21) i i Equatio (21) idicates that if we obtai the estimate of M_VaR, the C_VaR follows immediately. Garma (1996; 1997) derives exressios for the M_VaR ad C_VaR metrics uder the assumtio that returs are multivariate ormally distributed. To avoid the striget ormality assumtio, Hallerbach (2003) derives a distributio-free exressio for the margial cotributio of a istrumet to the diversified ortfolio VaR. Based o this method, Huag ad Yag (2007) roose a ew aroach that ca rovide 3 A risk measure is homogeeous of degree oe if ( kx ) k( X ) for all X ad 0 14 k.

15 more accurate estimates of M_VaRs tha ca Hallerbach s rocedure, esecially whe the observatios are sarse. Therefore, i this aer we utilize this modified aroach to estimate margial VaRs. Huag ad Yag s (2007) modified aroach starts with selectig N differet observatios i r VaR VaR, VaR VaR ] for differet ( 0 1), where [ each observatio is a vector r j, r j, r ). Ituitively, the closer r, j ad VaR are, the ( 1, 2,, j better r i, j ca aroximate i r give that r VaR is true. As a result, i j r, should receive a high weight. The estimate for M_ VaRi is give by N M _ VaRi m jri, j j1, (22) where r, is the jth ( j 1,2,, N ) observatio of returs for the ith ( i 1, 2 ) comoet i j (tradig or o-tradig). for ay j, the m j reresets the corresodig weight o r i, j. If r, j VaR m j N 1/ r 1, j 1/ r VaR, VaR. (23) If there exists some r, k l VaR for l k ( l 1,2,, L ), the 1/ r, j VaR, j k1, k2,, k N L m ( L 1) 1/ r, VaR j (24) 1 1/( L 1), j k1, k2,, kl. N I additio, m 1 ( m 0 ), ad also j1 j j 2 i1 VaR M_ VaR i should be close to 1. 15

16 Give that the tyical daily retur is simly a sum of tradig ad o-tradig returs (i.e., w w 1), the comoet VaR is the same as margial VaR i our aalysis. Based 1 2 o the estimates of M_VaRs ad all the observatios from set r, VaR ), we ca comute the M_ESs as follows i r i, j N j1 16 ( N 1 M_ ES. (25) Accordigly, the ercetage risk cotributio of each comoet is give by M_ VaRi PC _ VaRi 2, (26) M_ VaR i1 i1 i M_ ESi PC _ ESi 2. (27) M_ ES 2.6. Backtestig i Sice the late 1990 s, a variety of backtests have bee roosed for gaugig the adequacy of VaR models. The two well-kow aroaches to backtestig a risk model iclude Kuiec s (1995) ucoditioal coverage test ad Christofferse s (1998) coditioal coverage test. To illustrate the basic idea of backtestig, we deote the actual loss for a give horizo as L t, ad the defie the followig idicator variable as the hit fuctio, 1 if Lt VaRt ( ), It ( ) (28) 0 if Lt VaRt ( ). Aaretly, the hit fuctio series tracks the history of whether or ot a realized loss exceeds the model estimated VaR at the give cofidece level 1. Christofferse (1998) demostrates that uder the ull hyothesis that the VaR model is correct, the resultig hit

17 series I T t ( ) t 1 must satisfy both the so called ucoditioal coverage roerty ad ideedece roerty. The ucoditioal coverage roerty requires that the observed frequecy of violatios, which is defied as losses exceedig the VaR estimates for that eriod, be the same as, the exected frequecy of exceedaces accordig to the model. O the other had, the ideedece roerty states that ay two elemets of the hit series must be ideedet of each other. Kuiec s (1995) ucoditioal test focuses oly o the ucoditioal coverage roerty. Namely, it examies whether the observed frequecy of violatios T 1 I t ( ) T 1 t is statistically sigificatly differet from. Followig the biomial theory, the robability of observig N violatios out of T observatios is T N N ( 1 ). Thus, the likelihood ratio test statistic is give as follows: LR UC T N N T N N 2l[(1 ) ] 2l[(1 N / T ) ( N / T ) ]. (29) Uder the ull hyothesis that the exected violatio frequecy is, this ucoditioal test statistic is distributed as 2 (1). While this method is very ituitive ad straightforward, it suffers from at least two shortcomigs. The first roblem is the low ower of test, as oited out by Kuiec (1995), while the secod is that it fails to detect whether these VaR violatios are ideedet of each other. A VaR model that violates the ideedece roerty may result i clustered violatios, idicatig that the model caot roerly cature the variability of losses uder certai coditios. I cotrast, Christofferse s (1998) coditioal coverage aroach joitly tests both the ucoditioal coverage ad ideedece roerties. I additio, we ca also test the 17

18 sub-hyothesis regardig the frequecy ad ideedece of violatios with this method. This rocedure allows us to searate clusterig effects from the distributioal assumtio. The statistic for Christofferse s test is give by LR CC T N N l[(1 ) ] 2l[(1 ) (1 ) ], (30) where ij ( i, j 0,1) is the umber of times that value i is followed by j i the hit series I T t ) t 1 (, ad ij is the corresodig frequecy, which is defied as /. ij ij 1 j0 ij Uder the ull hyothesis that the model is correct, the distributio of LR CC is asymtotically 2 (2). 3. Data I this aer, we focus o Chiese commodity futures markets. I articular, the data sets cosist of daily oeig ad closig rices for coer ad atural rubber futures traded o the Shaghai Futures Exchage (SHFE), as well as those rices for soybea futures o the Dalia Commodity Exchage (DCE), obtaied from their resective exchages. The samle eriods exted from Setember 15, 1993 to July 20, 2010 for coer futures, from November 3, 1995 to July 20, 2010 for rubber futures, ad from October 18, 1994 to July 20, 2010 for soybea futures, resectively. The SHFE curretly trades futures o alumium, coer, gold, zic, atural rubber, ad fuel oil, while the DCE rimarily trades soybea futures. By 2009, both coer ad atural rubber futures o the SHFE raked first i the world i terms of tradig volume, while the tradig volume of soybea futures o the DCE is 23% of that o the Chicago Mercatile Exchage (CME), the largest soybea futures market i the world, ad 13 times the tradig volume of the third largest market, 18

19 the Tokyo Grais Exchage. 4 Therefore, the futures cosidered i this aalysis are well reresetative of Chiese commodity futures markets. Followig the geeral ractice i the literature, each futures rice series is costructed by rollig over the earby futures cotract o the first tradig day of the ext moth (for coer ad rubber cotracts) or the cotract s exiratio moth (for soybea cotracts). The earby futures cotracts are used, as these are the most liquid ad actively traded cotracts i markets. Based o these rice series, close-to-close daily returs, close-tooe, ad oe-to-close returs are calculated as er Equatios (1)-(3), resectively. The tradig hours of Chiese futures markets are from 9:00 a.m. to 3:00.m. (Moday to Friday). Cosequetly, the o-tradig hours o weekdays are three times as log as the tradig hours. Figure 1 lots the tradig ad o-tradig returs for each futures cotract, whereas Table 1 reorts the descritive statistics of these retur series. From the results, we ca see that there are ideed some big differeces i the distributioal characteristics betwee tradig ad o-tradig returs. O average, tradig returs are substatially lower tha the average o-tradig returs for coer ad soybea markets, while the oosite is true for the rubber market. This is rimarily due to how good/bad the ews released durig o-tradig hours is relative to that released durig tradig hours i a articular market. The returs for all the three futures are egatively or ositively skewed with excess kurtosis, idicatig that they are ot ormally distributed. Further, i terms of stadard deviatios, o-tradig returs are more volatile tha tradig returs for all three futures. As a result, the iformatio accumulated durig o-tradig hours is sigificat i Chiese 4 Sources: ad Shaghai Securities News, August 18,

20 commodity futures markets, give the fact that volatilities are directly related to iformatio flows. 4. Emirical results 4.1. Margial distributio estimatio Figures 2, 3, ad 4 dislay the mea residual life, stabilities of GPD arameters, ad Hill lots, resectively. Based o these lots, uer ad lower thresholds for tradig ad o-tradig returs for these futures are selected, ad are reorted i Table 2. For the selected thresholds, scale arameter ad shae arameter i the GPD are the estimated usig the maximum likelihood method. These estimates are reseted i Table 2 as well. To evaluate the goodess of fit of the data series to the model, Figure 5 deicts the QQ-lot of residuals from GPD fit to the actual loss data over the threshold ad the estimated tail for tradig ad o-tradig returs i each futures market. 5 As we ca see from these lots, all the oits o the grahs lie very early alog the solid lie, idicatig that the estimated GPDs fit the data satisfactorily. For the estimated margial distributios F ˆ ) ad F ˆ ) of tradig ad o-tradig 1 ( x 1 2 ( x returs, we have a coula fuctio such that C u, u ) FF ( xˆ ), F ( ˆ ) ( x2, accordig to Sklar theorem. To idetify the coula fuctio that best describe the biary characteristics of the margial fuctios, we simulate 10 differet coula fuctios ad comute their resective log-likelihood, AIC, ad BIC usig the maximum likelihood method. The results are reorted i Table 3. 5 A QQ-lot (quatile-quatile lot) lots the quatiles of a emirical distributio agaist the quatiles of a hyothesized distributio. It is usually used i statistics to examie whether a samle comes from a secific distributio. 20

21 Aaretly, Taw fuctio geerates the lowest BIC ad AIC, ad the highest loglikelihood amog these coulas cosidered, idicatig that it fits the data the best. 6 Thus, we estimate arameters,, ad i Taw fuctio for various futures markets, which are reseted i Table 4. We ote that these estimates are all sigificat at the 1% level. I additio, Figure 6 lots the cotours of emirical coula ad the Taw fuctio for coer, rubber, ad soybea futures markets. Clearly, the cotour of the emirical coula iosculates with that of the Taw fuctio very well for all three futures, which further demostrates that usig Taw fuctio is aroriate to icororate F ˆ ) ad F ˆ ) 1 ( x 1 2 ( x 2 ito their joit distributio Itegrated VaR ad ES estimatio ad backtestig results The itegrated VaRs ad ESs with cofidece levels of 95% ad 99% for various futures are reseted i Table 5. At the 95% cofidece level, the itegrated VaRs are , , ad for coer, rubber, ad soybea futures, resectively, while the corresodig itegrated ESs are , , ad , resectively. Therefore, measured by the itegrated VaR ad ES, the Chiese rubber futures market exhibits the highest overall risk, followed by the coer market, with the soybea market least risky. As exected, this orderig of market risks is cosistet with that imlied by volatilities 6 Taw coula (Taw, 1988) belogs to the extreme value coula class, which is rereseted i the form lu of C ( u, v) exl( uv) A, where A (z) is called the deedece fuctio. Taw coula has l( uv) A( z) 1 ( ) z (1 z), where 0, 1, ad a deedece fuctio

22 observed i Table 1. Not surrisigly, the results at the 99% cofidece level are higher tha those at the 95% level, but idicate a similar atter. To examie the robustess of the above results, backtestig is coducted, ad the results are reseted i Table 6. We fid that the failure ratios are all lower tha the target violatio rate, ad both ucoditioal ad coditioal backtestig statistics are sigificat at the 1% or 5% level. This further demostrates the adequacy of our model. To evaluate the accuracy of our risk measures relative to the measures comuted by the tyical method, the VaRs ad ESs are comuted based o close-to-close market returs for each futures, ad the results are reorted i Table 7. The backtestig results are also reorted i Table 7, which clearly idicate that the VaR model ca accurately redict both the frequecy ad the size of exected losses. Comarig these VaR ad ESs with those i Tables 5 shows that they are lower tha the corresodig itegrated values obtaied by the coula method. This imlies that risk estimatio based o close-to-close returs uderstates market risk relative to the itegrated measures. This is because the calculatio of close-to-close returs oly uses closig rices, ad it caot fully cature idividual risk comoets i tradig ad o-tradig hours. O the cotrary, the itegrated VaR ad ES emloy both closig ad oeig rices, which cotai more iformatio i a futures market. Cosequetly, these measures exlicitly take ito cosideratio the risk characteristics of tradig ad o-tradig returs ad their deedece structure. Note that the itegrated VaR or ES greatly deeds o the cofidece level. To gauge the sesitivity of our itegrated risk measures to chages i cofidece levels, we simulate VaRs ad ESs for various cofidece levels ragig from 10% to 100% by iteratig 10,000 times of samles. Figure 7 lots these VaRs/ESs agaist for coer, rubber, ad 22

23 soybea markets, resectively. This figure shows that both the itegrated VaR ad ES icrease with the cofidece level, 1, which is exected give the defiitios of these measures. Moreover, the lots become steeer whe the cofidece level is higher, idicatig that the itegrated VaR ad ES are more sesitive to the cofidece level whe the level is higher tha whe it is lower. This fidig is i lie with those i the literature (Fu ad Xig, 2009), ad is true regardless of the futures market uder cosideratio Comoet VaR (C_VaR) ad comoet ES (C_ES) I this sectio, we focus o the risk cotributios of tradig ad o-tradig hours, which are measured by their resective C_VaR ad C_ES. The results i Table 8 show that uder the 95% cofidece level, the C_VaRs of tradig ad o-tradig returs are ad for coer, ad for rubber, ad ad for soybea futures, resectively. Accordigly, the o-tradig hours cotribute 64.55%, 45.94%, ad 42.94% to the overall risk measured by itegrated VaRs for coer, rubber, ad soybea futures, resectively. Uder the 99% cofidece level, the resective PC_VaRs are 58.80%, 47.16%, ad 48.83%. Overall, the o-tradig PC_VaRs are substatial for all markets. I articular, i the case of coer, the risk cotributio of o-tradig hours is eve larger tha that of tradig hours. This is sharly i cotrast with the Frech ad Roll s (1986) fidig that the tradig iformatio is far more imortat tha o-tradig iformatio i stock markets. This result highlights the huge amout of iformatio accumulated durig o-tradig hours i Chiese futures markets ad its imact o the overall market risk. This iformatio icludes ot oly aoucemets made cocerig these commodities i Chia, but also the tradig activity i corresodig iteratioal futures markets. News released durig o-tradig hours ca be more egative relative to that released durig 23

24 tradig hours (Patell ad Wolfso, 1982; Bagoli, Clemet, ad Watts, 2005); it could also be relatively ositive or eutral (Doyle ad Magilke, 2008; Cliff, Coer, ad Gule, 2008). The risk associated with these aoucemets, articularly egative aoucemets made durig o-tradig hours, is catured by the o-tradig PC_VaR, as VaR focuses o the dowward tail of retur distributios. Our results also reflect the fact that Chiese futures markets have become more ad more itegrated with iteratioal futures markets, esecially US ad Euroea markets. Give that these overseas futures markets trade durig the o-tradig hours i Chiese markets, their tradig iformatio aaretly cotributes greatly to the high risk comoet of o-tradig hours i Chia. This is cosistet with fidigs i Liu ad A (2011), who documet the leadig role of US coer ad soybea futures markets i iformatio trasmissio ad the rice discovery rocess betwee Chiese ad US markets. Fially, we believe that the high liquidity risk durig o-tradig hours is aother factor that ca artially exlai our fidigs. The results of the C_ESs for differet markets uder the 95% ad 99% cofidece levels are also reorted i Table 8. Measured by the C_ES uder the 99% level, it seems that the o-tradig returs have a slightly higher cotributio to the overall risk for both coer ad rubber markets. All other major coclusios are re-cofirmed i this case A further aalysis of VaRs ad ESs for o-tradig returs To further uderstad the risk of o-tradig hours, we aalyze how the VaR ad ES are related to the legth of o-tradig eriods. For this urose, we estimate the VaRs ad ESs of weekights, weekeds, ad holidays based o the idividual distributio of each tye of o-tradig returs. The results are reseted i Table 9. We fid that the VaRs of weekights, weekeds, ad holidays uder the 95% cofidece level are , , 24

25 ad for coer, , , ad for rubber, ad , , ad for soybea futures markets, resectively. This suggests that the loger the otradig eriod, the higher the risk of the o-tradig eriod for all three markets. However, i the case of the 99% cofidece level, this effect is less roouced for the rubber market, ad the weeked VaR is eve greater tha the holiday VaR for the soybea market. We observe a similar atter whe ES is cosidered. Nevertheless, weekight VaR/ES is geerally lower tha weeked/holiday VaR/ES, regardless of the market. Ituitively, the iformatio cotiues to accumulate as time goes by, ad therefore, weekeds or holidays cotai more rice sesitive iformatio tha weekights. 5. Coclusios This aer ivestigates the risk cotributios of tradig ad o-tradig hours i Chiese coer, rubber, ad soybea futures markets. Usig the coula method, we icororate the distributios of tradig ad o-tradig returs ito a joit distributio, ad examie the itegrated VaRs ad ESs for these markets to gauge overall market risk. The we decomose these risk measures ito comoet VaRs ad ESs to evaluate the risk cotributios of tradig ad o-tradig returs to overall risk. Backtests are also erformed to assess the adequacy of the models. Our emirical results show that the coula method rovides aroriate measures for itegrated risks, ad the tyical VaR ad ES based o close-to-close returs uderestimate overall market risks. Measured by both itegrated VaRs ad ESs, Chiese rubber futures are most risky, followed by coer futures, ad soybea futures are least risky. Our results also demostrate that the risk cotributios of o-tradig hours are substatial for all futures cosidered, with C_VaR ad C_ES weights beig more tha 40% i ay market. I 25

26 the case of coer futures i articular, o-tradig hours seem to cotribute to the total risk more tha do the tradig hours. This sheds lights o the imortat role of o-tradig iformatio i redictig market returs ad exlaiig market risks. Fially, we fid that weekight VaRs ad ESs are lower tha weeked/holiday VaRs ad ESs for all markets, idicatig that the risk of o-tradig eriods is ositively related to the legth of otradig eriods. 26

27 Refereces Acerbi, C. ad Tasche, D., O the coherece of exected shortfall. Joural of Bakig ad Fiace 26(7), Arzeer, P., Delba, F., Eber, J.M., ad Heath, D., Coheret measures of risk. Mathematical Fiace 9(3), Bagoli, M., Clemet, M., ad Watts, S., Aroud-the-clock media coverage ad timig of earigs aoucemets. Workig aer. Balkema, A. ad De Haa, L., Residual life time at great age. Aals of Probability 2, Christofferse P.F., Evaluatig iterval forecasts. Iteratioal Ecoomic Review 39, Cliff, M., Cooer, M.J., ad Gule, H., Retur differeces betwee tradig ad otradig hours: Like ight ad day. Workig aer, Available at htt://ssr.com/abstract= Doyle, J. ad Magilke, M., The timig of earigs aoucemets: A examiatio of the strategic-disclosure hyothesis. Workig aer. Egle, R.F. ad Magaelli, S., CAViaR: Coditioal value at risk by quatile regressio. NBER, workig aer Frech, K.R. ad Roll, R., Stock retur variaces: The arrival of iformatio ad the reactio of traders. Joural of Fiacial Ecoomics 17, Frey, R. ad McNeil, A.J., VaR ad exected shortfall i ortfolios of deedet credit risks: Cocetual ad ractical isights. Joural of Bakig ad Fiace 26,

28 Fu, Q. ad Xig, L., Calculate the coditioal VaR by extreme value theory ad Coula fuctios (i Chiese). Joural of Systems Egieerig 24(5), Garma, M.B., Imrovig o VaR. Risk 9/5, Garma, M.B., Takig VaR to ieces. Risk 10/10, Hallerbach, W.G., Decomosig ortfolio value-at-risk: A geeral aalysis. Joural of Risk 5(2), Hill, B.M., A simle geeral aroach to iferece about the tail of a distributio. Aals of Statistics 3, Huag, L. ad Yag, Y., Decomosig ortfolio risk uder o-ormal distributios (i Chiese). Joural of Beijig Uiversity of Chemical Techology 34(6), Kuiec, P., Techiques for verifyig the accuracy of risk measuremet models. Joural of Derivatives 3, Liu, Q. ad A, Y., Iformatio trasmissio i iformatioally liked markets: Evidece from US ad Chiese commodity futures markets. Joural of Iteratioal Moey ad Fiace 30, Logi, F.M., From value at risk to stress testig: The extreme value aroach. Joural of Bakig ad Fiace 24, Magaelli, S. ad Egle, R.F., Value at risk models i fiace. ECB workig aer No. 75. Available at SSRN: htt://ssr.com/abstract= Patell, J. ad Wolfso, M., Good ews, bad ews, ad the itraday timig of cororate disclosures. The Accoutig Review 47, Pickads, J., Statistical iferece usig extreme order statistics. The Aals of Statistics 3,

29 Roseberg, J.V. ad Schuerma, T., A geeral aroach to itegrated risk maagemet with skewed, fat-tailed risks. Joural of Fiacial Ecoomics 79, Taw, J.A., Bivariate extreme value theory: Models ad estimatio. Biometrika 75, Taylor, N., A ote o the imortace of overight iformatio i risk maagemet models. Joural of Bakig ad Fiace 31, Tsiakas, I., Overight iformatio ad stochastic volatility: A study of Euroea ad US stock exchages. Joural of Bakig ad Fiace 32, Ward, L.S. ad Lee, D.H., Practical alicatio of the risk-adjusted retur o caital framework. CAS Forum Summer 2002, Dyamic Fiacial Aalysis Discussio Paers, available at htt:// Xu, J.J., Statistical modellig ad iferece for multivariate ad logitudial discrete resose data. PhD thesis, Statistics Deartmet, Uiversity of British Columbia. Zhag, Y. ad Rachev, S., Risk attributios ad ortfolio erformace measuremets. Joural of Alied Fuctioal Aalysis 4(1),

30 Table 1. Descritive statistics for tradig ad o-tradig returs Markets Returs Maximum Miimum Mea Std. Dev. Skewess Kurtosis Tradig returs Coer No-tradig returs Traditioal returs Tradig returs Rubber No-tradig returs Traditioal returs Tradig returs Soybeas No-tradig returs Traditioal returs This table reorts the descritive statistics of various retur series for Chiese coer, rubber, ad soybea futures markets. Tradig, o-tradig, ad traditioal returs are defied i Equatios (3), (2), ad (1), resectively. The samle eriods exted from Setember 15, 1993 to July 20, 2010 for coer futures, from November 3, 1995 to July 20, 2010 for rubber futures, ad from October 18, 1994 to July 20, 2010 for soybea futures, resectively. 30

31 Table 2. Estimated arameters of the margial distributio fuctios Markets Returs R R R u Log-likelihood L L L u Log-likelihood of uer tail of lower tail Tradig Coer returs No-tradig returs Tradig Rubber returs No-tradig returs Tradig Soybeas returs No-tradig returs This table reorts the estimated arameters of the margial distributio fuctios of tradig ad o-tradig returs, as well as the log-likelihood values for coer futures, rubber, ad soybea futures markets. 31

32 Table 3. Testig results for various coula fuctios Coula Gumbel Joe Frak Taw BB1 BB3 BB4 BB6 BB7 Normal fuctios Coer Log-likelihood AIC BIC Rubber Log-likelihood AIC BIC Soybeas Log-likelihood AIC BIC This table reorts the testig results of log-likelihood, BIC ad AIC for 10 differet coula fuctios for coer, rubber, ad soybea futures markets, resectively. 32

33 Table 4. Estimatio results for the Taw fuctio Markets Parameters Value Std. Error t-value Coer α ** β ** γ ** Rubber α ** β ** γ ** Soybeas α ** β ** γ ** This table resets the estimated α, β, ad γ values i the Taw fuctio for coer, rubber, ad soybea futures markets, resectively. ** idicates sigificace at the 1% level. 33

34 Table 5. Itegrated VaR ad ES estimates for differet futures markets Markets Risk measures Cofidece level Values Coer itegrated VaR 95% % itegrated ES 95% % Rubber itegrated VaR 95% % itegrated ES 95% % Soybeas itegrated VaR 95% % itegrated ES 95% % This table reorts the estimated itegrated VaR ad the corresodig itegrated ES uder the 95% ad 99% cofidece levels i coer, rubber, ad soybea futures markets. 34

35 Table 6. Backtestig results Markets Cofidece Failure Failure ratio LR UC LR CC level umber Coer 95% % * ** 99% % * ** Rubber 95% % * * 99% % * ** Soybeas 95% % * ** 99% % * ** This table reorts the backtestig results for itegrated VaRs uder the 95% ad 99% cofidece levels i coer, rubber, ad soybea futures markets. The samle is iterated 10,000 times. * ad ** idicate sigificace at the 5% ad 1% levels, resectively. 35

36 Table 7. Estimates of VaR ad ES, ad backtestig results based o close-to-close returs Markets Cofidece VaR ES Failure Failure level times ratio Coer 95% % * ** LR UC 99% % * ** Rubber 95% % ** ** 99% % ** ** Soybeas 95% % ** ** LR CC 99% % * ** This table resets the VaR ad ES estimates ad corresodig backtestig results uder the 95% ad 99% cofidece levels i coer, rubber, ad soybea futures markets. The samle is iterated 10,000 times. * ad ** idicate sigificace at the 5% ad 1% levels, resectively. 36

37 Table 8. Comoet VaR ad ES, as well as risk cotributios of tradig ad otradig returs Markets Risk measures 95% cofidece level 99% cofidece level Tradig returs No-tradig returs Tradig returs No-tradig returs Coer Comoet VaR Risk cotributio (PC_VaR) 35.45% 64.55% 41.20% 58.80% Comoet ES Risk cotributio (PC_ES) 42.84% 57.16% 44.19% 55.81% Rubber Comoet VaR Risk cotributio (PC_VaR) 54.07% 45.93% 52.84% 47.16% Comoet ES Risk cotributio (PC_ES) 51.91% 48.09% 49.59% 50.41% Soybeas Comoet VaR Risk cotributio (PC_VaR) 57.06% 42.94% 51.17% 48.83% Comoet ES Risk cotributio (PC_ES) 51.97% 48.03% 53.12% 46.88% This table reorts the comoet VaRs ad ESs uder the 95% ad 99% cofidece levels, as well as the risk cotributios of tradig ad o-tradig hours i coer, rubber, ad soybea futures markets. 37

38 Table 9. VaRs ad ESs of weekights, weekeds, ad holidays for differet futures markets Cotracts No-tradig eriods 95% cofidece level 99% cofidece level VaR ES VaR ES Coer weekights weekeds holidays Rubber weekights weekeds holidays Soybeas weekights weekeds holidays This table reorts the estimated VaRs ad ESs uder the 95% ad 99% cofidece levels based o weekight, weeked, ad holiday returs for coer, rubber, ad soybea futures markets. 38

39 Figure 1. Plots of tradig ad o-tradig returs 10 8 Coer 10 8 Coer 6 6 Daytime returs overight returs Daytime returs 13 Rubber Overight returs Rubber Daytime returs 9 Soybea Overight returs Soybea This figure lots the tradig (daytime) ad o-tradig (overight) returs for coer, rubber, ad soybea futures markets (from to to bottom) i Chia. The samle eriods exted from Setember 15, 1993 to July 20, 2010 for coer futures, from November 3, 1995 to July 20, 2010 for rubber futures, ad from October 18, 1994 to July 20, 2010 for soybea futures, resectively. 39