ITG Dynamic Daily Risk Model for Europe

Transcription

1 December 2010 Version 1 ITG Dynamic Daily Risk Model for Europe 2010 All righs reserved. No o be reproduced or reransmied wihou permission These maerials are for informaional purposes only, and are no inended o be used for rading or invesmen purposes or as an offer o sell or he soliciaion of an offer o buy any securiy or financial produc. The informaion conained herein has been aken from rade and saisical services and oher sources we deem reliable bu we do no represen ha such informaion is accurae or complee and i should no be relied upon as such. No guaranee or warrany is made as o he reasonableness of he assumpions or he accuracy of he models or marke daa used by ITG or he acual resuls ha may be achieved. These maerials do no provide any form of advice (invesmen, ax or legal). ITG Inc. is no a regisered invesmen adviser and does no provide invesmen advice or recommendaions o buy or sell securiies, o hire any invesmen adviser or o pursue any invesmen or rading sraegy. Do no copy and disribue wihou permission.

2 Table of Conens I. Moivaion.2 II. Facor loadings esimaion a. Using Facor Models o Esimae Covariance Marices..2 b. General Model Formulaion 4 c. Universe...5 d. Inraday bins and reurns 5 III. Realized volailiy esimaion 6 IV. Ou of sample es resuls a. Facor volailiy predicion.8 b. Porfolio consrucion ess 9 V. Deliverables.15 VI. Final remarks.16 VII. References..17 2

3 I. Moivaion Tradiionally, risk models have been used for risk measuremen and decomposiion during he porfolio consrucion sage. Since he ypical holding period of an insiuional porfolio is measured in monhs and even years, earlier risk models were consruced using monhly reurns and several years of hisorical daa. This approach provided much needed sabiliy of he esimaes and allowed he inclusion of risk facors which are beer defined a lower frequency, such as size or growh. Recenly, however, more money is flowing ino he funds ha ake advanage of he shor erm opporuniies which range from a few minues o a few days. The shorened invesmen horizon calls for corresponding changes in he risk conrols wih he goal of measuring and predicing porfolio risk during he ime when porfolio posiions are mainained. Anoher source of demand for a shor erm risk model comes from algorihmic porfolio rading. Since mos rades are compleed wihin a day or a few days, he adequae risk conrol requires models which could respond o daily volailiy shocks almos insananeously. The availabiliy of such risk models allows for a beer risk conrol in buy sell liss which ypically comprise anywhere beween 10 and 500 names. A good risk model for algorihmic porfolio rading or for shor erm buy side sraegies allows for precise porfolio risk measuremen, decomposiion and forecas over fairly shor (e.g. one day ahead) periods of ime. This is ideally achieved via: forward looking forecass of facor covariances and sock specific risk; accurae esimaes of facor loadings on a sock by sock basis; uilizing high frequency informaion which would allow for a quick response ime o volailiy shocks. In addiion, he saisical mehods used o esimae a good risk model have o be simple and robus in order o be suiable for a daily producion process. ITG s Dynamic Daily Risk Model (DDRM) developed for he European region fis all he above requiremens. This whie paper describes he model, he mahemaics behind i and provides ou ofsample es resuls used o evaluae he model s performance. II. The Model Srucure Facor Loadings Esimaion 2a. Using Facor Models o Esimae Covariance Marices. On he mos general level, he risk model consrucion and esimaion falls ino a very wide and rapidly developing field of volailiy modeling. The availabiliy of high frequency daa has caalyzed his area of research generaing ubiquious lieraure on univariae volailiy esimaion. However, he lis becomes much shorer for he mulivariae case, i.e. when boh variances and covariances beween asses are of ineres. Since our focus is on porfolio risk modeling, accurae covariance marix esimaion is essenial. 3

4 High frequency covariance marix esimaion has developed naurally from he concep of realized volailiy. The work by Fleming e al. (2003), Barndorff Nielsen and Shephard (2004), Corsi and Audrino (2008) and ohers provide a good insigh ino his approach. Indeed, using high frequency reurn daa allows o overcome he non inverabiliy of sample covariance since he number of ime periods could be made much greaer han he number of asses. Tradiionally, he singulariy issue wih he sample covariance marix has been resolved by using facor models which have been around since 1970s (e.g. Rosenberg (1974)). Sure enough, i did no ake long unil high frequency reurns have been uilized o esimae a facor model. Bollerslev and Zhang (2003), building on he earlier work by Andersen e al. (2001a), esimaed he risk model wih Fama French facors sampled a 5 minue frequency. In a more recen paper Bannouh e al. (2009) (for breviy we will refer o his paper as BMOD) advocae he mixed frequency facor model, where he facor covariance is esimaed using high frequency inraday reurns, while he facor loadings are esimaed a he convenional daily frequency. This approach aims o exrac he gains resuling from using high frequency daa while avoiding he problems arising from non synchronous rading. I is well known ha when asses have differen rade frequencies, he sample esimaes of he realized covariance can be subjec o a downward bias owards zero. Since facor loadings esimaion is concepually he same as esimaing he covariance, using high frequency reurns can lead o beas which are biased owards zero. Going back o using daily reurns is a possible way o solve he problem. Oher mehods, such as subsampling, suggesed by Zhang, Mykland and Ai Sahalia (2005), aim o correc he bias while sill uilizing high frequency inraday reurns. Cross bin ick maching described in Hayashi and Yoshida (2005) is anoher approach, which expands on Scholes and Williams (1977). Oher correcion mehods are menioned in he comprehensive survey by McAleer and Medeiros (2008). BMOD show ha realized sample covariance esimaes underperform he mixed frequency facor approach boh in Mone Carlo simulaions and in empirical applicaions. In fac, Fan e al. (2008) showed ha when he number of risk facors is small relaive o he number of asses, he inverse of he facor based covariance marix converges faser o he inverse of he rue covariance marix han he inverse of he sample covariance marix. This resul is especially relevan in our conex since he majoriy of pracical porfolio consrucion applicaions require he inverse of he covariance marix raher han he marix iself 1. Ye, he final jury on he relaive meris of sample realized covariance vs. risk facor approaches is sill ou. Fan s (2008) heoreical resul applies o raw sample covariance marices. However, imposing some srucure on a realized sample covariance can improve is performance. In a recen paper Kyj e al. (2009) found ha realized covariance esimaes shrinked oward he hisorical average via an exponenially weighed moving average (EWMA) performed quie well agains high frequency and mixed frequency single facor models. 1 For insance, one of he sandard (and simples) problems is o find he weighs of he global minimum variance porfolio. The soluion is provided by: 1 1 N1 where is N N covariance marix and 1is N 1 column 1 1' 1 vecor of ones. While he real life problems are more complicaed, he inverse of covariance marix is presen in mos soluions. 4

5 In shor, on purely heoreical grounds he comparison beween esimaing porfolio covariance via high frequency reurn sampling and via uilizing facor models is a close call. However, we embrace he risk facor approach for pracical reasons. Specifically, he esimaes of facor loadings and covariances can be performed in advance, and hen combined o calculae he risk of any given porfolio. On he oher hand, compuing he realized covariance marix for a paricular porfolio or a radelis in real ime could be ime consuming and prone o daa errors. Our approach o esimaing DDRM is in he spiri of BMOD wih some noable differences. We use inraday daa boh o esimae he facor covariance marix and facor loadings, while BMOD use daily daa o esimae facor loadings. Kyj e al. (2009), menioned above, offer an indirec comparison of hese approaches. They repor ha a realized one facor risk model slighly ouperforms a mixedfrequency model in a global minimum variance porfolio seing. As a consequence of he difference in esimaion frequency we do no use he Fama French facors and resric ourselves o a marke facor and secor facors. BMOD, on he oher hand, esimae Fama French facors, and include 10 indusry facors. Uilizing inraday daa for facor loading esimaion requires a bias correcion. We describe our approach in he subsecions ha follow. Insead of using an exponenially weighed moving average (EWMA) scheme o provide dynamics o he facor covariance marix, as done in BMOD, we exend he HAR RV approach of Corsi (2004). To he bes of our knowledge, here are no published papers which compare he relaive performances of EWMA and HAR RV models in a mulivariae seing. Our ou of sample resuls indicae ha HAR RV mehod performs very well in providing dynamics o facor and residual sock specific volailiy esimaes. In wha follows, we presen he regression model used o esimae he facor risk loadings, and discuss in more deails he model universe, he inpus and deails of he compuaion. 2b. General Model Formulaion. DDRM is esimaed via a ime series weighed leas squares (WLS) regression on a sock by sock basis uilizing he European marke and European secors as risk facors. While here are 10 secors in Europe, each sock is regressed only agains he secor o which i is assigned by GICS. Therefore, DDRM can be echnically considered a 2 facor model. Equaion (1) provides he mahemaical specificaion of he model: where and R i, i i, mrm, i, srs, i, (1) R i, is he reurn on sock i a ime ; R, and R, are marke and secor reurns a ime ; m i, s are he esimaed loadings on he European marke and secor facors, respecively. Finally, i, is he sock specific risk (i.e. he remainder of he sock s reurn which is no explained by he marke and secor risk facors). Our approach is consisen wih he evidence in he lieraure, which poins ou ha as one moves owards higher frequency daa, he number of relevan risk facors decreases. For example, Connor, Goldberg and Korajczyk (2010) repor he applicaion of he newer Bai Ng (2002) es o daily and monhly US equiy reurns and conclude ha here are hree saisical facors presen a monhly s i, m 5

6 frequency bu only one a daily frequency. Since we esimae (1) using he daa sampled a inraday frequency, we believe ha a wo facor model adequaely describes he covariance srucure of European equiies. We suppor his by providing comprehensive ou of sample ess of he model s performance. Since we esimae facor loadings from inraday daa, we address he poenial bias arising from nonsynchronous rading, as will be shown laer. In order o shoren he risk model s reacion ime, equaion (1) is esimaed over he window of 10 rading days. Considering ha boh facor and individual sock s reurns are measured over 30 minue inerval, his provides us wih up o 180 observaions. We use a form of Bayesian shrinkage in he facor loadings esimaion process. 2c. Universe DDRM for Europe is esimaed for 19 European counries plus Turkey, Israel, Russia and Souh Africa. The European counries included are: Ausria, Belgium, Czech Republic, Denmark, Finland, France, Germany, Grea Briain, Greece, Hungary, Ireland, Ialy, Neherlands, Norway, Poland, Porugal, Spain, Sweden, and Swizerland. There are approximaely 9,000 securiies for which DDRM repors facor loadings. The majoriy of hese counries share he same rading hours: from 8:00 ill 16:30 London ime, which corresponds o he UK marke rading. Therefore, he European marke reurn is available only during hose hours. As a resul, he coefficiens in equaion (1) for counries wih rading hours differen from UK marke are esimaed using only inraday bins which conain boh sock s reurn and European marke reurn. For insance, Czech Republic s marke hours are 8:30 15:00; hence, he inraday bins used for he regression are resriced o be in he period beween 8:30 and 15:00. 2d. Inraday Bins and Reurns All reurns in (1) are compued over a 30 minue inerval uilizing midquoes defined as: r P ln( P mdq, mdq, 1 ) (2) where P mdq, and mdq, 1 P are he las midquoe prices in 30 minueinervals (bins) and 1. For he firs bin of he day he compuaion uses he open price for he day P open in place of P mdq, 1. Therefore, counries wih 8:00 16:30 rading hours have 18 inraday bins. A 30 minue bin is eligible for reurn compuaion only if i conains a leas one rade or midquoe change. Overnigh reurns are compued and adjused for corporae acions as follows: Popen, d r ovn log( ) (3) P close, d 1 where P close, d1 and P open, d are he closing price on day d 1 and he open price on day d, respecively, as repored by Bloomberg. The raio of he old and he new number of shares peraining o a corporae acion is represened by. For example, a 2:1 spli resuls in =2. Examples of corporae 6

7 acions which resul in non zero are splis, sock and cash dividends and spinoffs. Noe ha overnigh reurns are no used o esimae facor loadings. They are only used o compue he realized volailiies of he risk facors and realized sock specific volailiies. European marke and secor reurns are compued as marke cap weighed averages of reurns for all eligible securiies on each rading dae. The securiy s reurn is eligible for inclusion ino he European marke calculaion if: he securiy corresponds o a primary lising for a given company (i.e. no ADRs or oher ypes of inerlised securiies are eligible); i is a common sock (no preferred shares, ETFs or funds); is marke capializaion is available on he day he marke reurn is compued; he number of acive bins (bins wih a leas one rade or midquoe change) is greaer or equal o 2/3 of he oal number of 30 minue bins on he day he marke reurn is compued. The crieria for inclusion ino he European secor reurn calculaions are he same as above plus he GICS secor code should be available on he day he secor reurn is compued. III. The Model Srucure Volailiy Esimaion Forward looking volailiy esimaion is an inheren feaure of DDRM and i grealy conribues o forecasing performance of he model. The risk forecas for dae for an arbirary porfolio wih he weigh vecor ω 1 is given by: risk pred ˆ ' 1 B 1F B' 1 1 ˆ (4) where B 1 is N K marix of facor loadings esimaed in (1), Fˆ is he esimaed K K facor covariance marix and ˆ is a N N diagonal marix of he prediced sock specific volailiy for dae. In secion II we described he esimaion process for he facor loading marix B. Here we describe he process which predics he marices Fˆ and ˆ. We use realized volailiy (RV) as he volailiy measure in he model. Realized volailiy has been shown o be superior o oher popular volailiy models, such as GARCH, in Andersen e al. (2001a, 2001b, 2003) and ohers. While he basic properies of he univariae RV process are well undersood, one needs o resolve wo issues when incorporaing RV based volailiy predicion ino a risk model. Firs, he RV calculaions should be made forward looking and second, one needs o find a way o apply he concep of RV o a mulivariae seing. Below we describe our approach o deal wih hese issues. We adop he Heerogeneous AuoRegressive (HAR) model of Corsi (2004) o provide dynamics o our RV model. The HAR model is given by: RV 0 d RV1 wrv5: 1 mrv21: 1 (5) where 7

8 RV 1/ j1 2 j 2 ovn r r (5a) r j is reurn in bin j uilizing he midquoes as in (2), is he lengh of he bin in fracion of he rading day (i.e. for 30 minue bin 1/ 18 ), and r ovn is overnigh reurn and is compued as in (3). Oher quaniies appearing in (5) are he average realized volailiies weekly (K=5) and monhly (K=21) horizons. RV K: 1 1 K K k 1 RV k defined for The big advanage of he HAR model (5) is is simpliciy as i can be esimaed via OLS regression. Corsi (2004) showed ha he model fied for realized volailiy of USD/CHF ouperformed ou of sample no only simple AR(1) and AR(3) models bu also he ARFIMA model of Andersen e al. (2003). The HAR model in (5) is esimaed for RV of European marke facor and European secors, i.e. he quaniies which appear on he main diagonal of facor covariance marix Fˆ. The off diagonal elemens are esimaed uilizing he sample correlaion beween he risk facors. In addiion o he above, individual HAR processes are also esimaed for sock specific volailiies. For each sock i and each bin we compue he residual reurn as: i, ˆ ˆ ˆ (6) R i, i i, M RM, i, S RS, where R, and M R S, are he EUR marke and secor reurns in bin, respecively, and [ i he esimaed facor loadings and inerceps from (1). residual reurn of sock i in bin. R i, is he reurn of sock i in bin and ˆ, ˆ i, M, i, ˆ S ] are i, is he Once we obain he sock specific residual reurns, we fi he HAR process (5) again, his ime compuing RV as he realized sock specific volailiy: where RV i, j 1/ 2 2 i, i, j i, ovn j1 (7) is sock specific residual reurn in bin j as defined in (6). Overnigh sock specific residual reurns are also compued as in (8) using he corresponding overnigh facor and sock reurns. We use rading days o esimae he HAR processes for facor and sock specific realized volailiy. If he sock has more han 300 available rading days we use all of hem bu no more han 600. If he sock has less han 300 rading days we do no esimae HAR process and use oday s RV as a forecas for omorrow s one. The lack of sufficien number of days for HAR esimaion is no a problem for facor and secor realized volailiy esimaion. 8

9 IV. Ou of Sample Performance of he Model In his secion we presen he resuls of our ou of sample ess o deermine he DDRM performance in several ypical applicaions. Firs, we repor he resuls on facor volailiy esimaion. Then we repor he model performance in porfolio/radelis opimizaion applicaions. 4a. Facor Volailiy Predicion While he forecas of realized volailiy of he risk facors is no a final oupu of he DDRM, i is, neverheless, an imporan ingredien of he overall risk model performance. In Figure 1 below we overlay he forecas of nex day s RV for he European marke facor and he average implied volailiy obained from opions wrien on FTSE100 and CAC40 indexes. The unis are annualized risk in % for implied volailiy and daily risk for RV forecas. Technically, he 1 day ahead RV predicion should no necessarily be equal o he opions marke s percepion of index volailiy during he opion s life. This is eviden from he char as he RV line is much more volaile han he implied volailiy line. Neverheless, our forecass of realized volailiy of European marke facor rack he implied volailiy of he wo European indexes prey well, wih a very quick reacion ime o spikes in Sepember Ocober Figure 1. Realized Volailiy and Implied Volailiy for European Regional Index 9

10 More formally, we calculae he mean absolue error for he forecass of realized volailiy of European marke facor: MAE 1 N N 1 RV RV pred (8) pred where RV is RV predicion from each model for day based on daa up o dae 1 and RV is he realized volailiy on day. Resuls are presened in Table 1 below. Table 1. MAE for European Marke Facor Volailiy Forecas Model/Period Number of rading days HAR GARCH (1,1) RV saic (no HAR) day sample s dev We divided he sample period ino four sub periods which correspond o he differen levels of overall marke volailiy: low ( ), elevaed ( ) and high ( ). The compeiive models are GARCH (1, 1) which uses daily daa, saic RV model (i.e. no HAR predicion) which uses inraday daa, and a 60 day sample sandard deviaion of he facor compued wih he daily daa. Across all marke volailiy regimes he HAR model ouperforms he res. Noe ha he simple RV model (wihou HAR process) performs a par wih GARCH (1, 1) across all hree periods. 4b. DDRM Performance in Porfolio Consrucion Ulimaely he risk model s performance is judged upon how well i performs in porfolio risk measuremen and decomposiion. Boh are inheren pars of porfolio consrucion effor. Consrucing he global minimum variance (GMV) porfolio, in order o conduc he ou of sample performance ess of compeing covariance marices esimaes, has been a paricularly popular approach in he lieraure. I appears in Chan e al. (1999), Jagannaahn and Ma (2003), Fleming e al. (2003) and ohers. As Figure 2 illusraes, due o is locaion on he efficien fronier he GMV porfolio does no require esimaes of expeced reurns. As a consequence, he forecas of porfolio risk plays a crucial role: he bes risk model will resul in he GMV porfolio ha has he lowes ou of sample risk. Shor erm risk models, like DDRM, are used by rading algorihms o conrol he risk exposure of buysell radeliss, which can be viewed as dynamically changing long shor porfolios. Our ou of sample model performance ess reflec ha. Following he large body of he lieraure menioned above, we consruc he global minimum variance (GMV) porfolio allowing for boh long and shor posiions. 10

11 Figure 2. Illusraion of he Global Minimum Variance (GMV) Porfolio The mahemaical problem ha we consider is defined as: min s. BFB N / 2 j1 N / 2 j1 0 L, j S, j 0.1 S, j 1; 1; L, j j j (9) where he noaion is he same as in (4) (wih ime subscrips dropped for clariy). The firs wo consrains fix he leverage raio a 2:1, and he las wo impose he weigh limis for he long and he shor sides of he porfolio. The oal number of socks in he porfolio is N, and he subscrips L and S on he weighs denoe he long and shor sides. Technically, we proceed as follows. On each day saring from June 2005, we randomly selec 30 or 100 socks across all European counries covered in DDRM. We load he corresponding facor loadings and he covariance marix for each of he risk models ha we consider. We opimize he porfolio according o (9), and use he opimal weighs o measure he nex day s inraday reurns of his porfolio. Using inraday reurns we compue he porfolio s nex day s RV as in (5a). We advance one day and repea he seps above. A he end of our es period (currenly i is ) we have he ime series of daily 11

12 RVs for porfolios opimized wih differen risk models. We measure he average of hese RVs and selec he one wih he lowes value. The corresponding risk model is declared he winner. We perform he esing procedure described above for DDRM and for wo oher risk models ha serve as benchmarks: ITG s Global Risk Model (GRM). I s a monhly model, so i s illusraive o look a is performance for forecasing one day ahead RV; a 5 facor PCA model (PCA) using daily reurns. This model is he closes o a convenional daily RM, i.e. he one using daily reurns 2 ; In addiion, we include he equally weighed dollar neural (i.e. un opimized) porfolio as a benchmark. Table 2 presens he resuls of he long shor GMV porfolio opimizaion. Repored are annualized average daily RV of he GMV opimal porfolios, wih sandard errors in parenhesis. The es porfolios conain eiher 30 or 100 socks, and we also vary he upper weigh limi for a single sock. The bes model is highlighed in green. The evidence in Table 2 indicaes ha he DDRM model comforably beas he res for all combinaions of porfolio sizes and weigh limis. In paricular, he opimal porfolio corresponding o a 5 facor PCA model has an average RV which is ~70 130bp higher han ha of DDRM porfolio. Table 2. Annualized RV of GMV Opimal Porfolios # of socks upper weigh limi ±10% ±20% ±5% ±10% DDRM PCA5 GRM EW (0.0018) (0.0018) (0.0010) (0.0010) (0.0018) (0.0018) (0.0011) (0.0010) (0.0021) (0.0023) (0.0012) (0.0013) (0.0020) (0.0020) (0.0011) (0.0011) No surprisingly, he monhly GRM performs he wors bu i is sill superior o he equally weighed dollar neural porfolio consruced wihou running opimizaion. The paern of he ou of sample average RVs across porfolio sizes and weigh limis meris a few words. Larger porfolios have lower ou of sample risk, which is expeced. However, ighening he weigh limi reduces he risk only for he 100 sock porfolios. For he 30 sock porfolios he risk is he lowes wih he 20% weigh limi, raher han wih he 10% limi. I appears ha wih only 15 socks per side, shifing more weigh ino a few socks wih very low risk and/or negaive correlaion wih oher socks beas he usual diversificaion effec. The GMV porfolio es for he long only case demonsraes an even wider ouperformance of DDRM over he benchmark models. However, considering ha he majoriy of he rade liss are buy sell, we 2 We also fi a 1 facor PCA model and found is performance o be similar o a 5 facor PCA model. We only repor resuls for he 5 facor PCA model. 12

13 don repor deails for he sake of breviy 3. Noe ha we include some of our long only porfolio resuls in Figure 3 below. An alernaive way o look a how well DDRM performs relaive o oher models is o look a he fracion of days i comes on op when used o consruc he GMV porfolio. In essence, we adop a more qualiaive (vs. quaniaive) approach o he es described above and simply measure he percenage of days on which he RV of he opimized GMV porfolio was he lowes for each model. We organize he resuls by porfolio ype (long or long shor) and by porfolio size (30 or 100 socks). In order o make he informaion digesible, we average over he weigh limis for each combinaion of porfolio ype and size. Figure 3 presens he resuls. Figure 3. Percenage of Time Model Is he Bes, The porfolio opimized wih DDRM has he lowes RV beween 57 and 72% of he days, depending on he porfolio ype and size. This percenage is higher for long only porfolios and for larger porfolios. The daily PCA 5 facor model is he bes on 22 31% of days and he monhly GRM on 5 15% of days. The percenages repored in Figure 3 downplay he performance of he DDRM relaive o he benchmarks. When we recalculae he percenages and resric ourselves o he universe of days on which he difference beween he differen models forecass is larger han normal, he numbers for DDRM go up by roughly 8% while he numbers for a 5 facor PCA and he monhly GRM models drop by 3% and 5%, respecively 4. Ofen he goals of a porfolio ransiion include conrolling he racking error relaive o he benchmark. We simulae his siuaion by repeaing opimizaion es (9) while replacing he raw reurns wih he reurns in excess of he benchmark porfolio. We consruc our proxy for he benchmark as he equally 3 These resuls are available upon reques. 4 Deails available upon reques. 13

14 weighed porfolio of 50 European socks which are among he larges by marke capializaion 5. The risk of he benchmark porfolio consruced in his manner would consis mosly of he marke risk. Provided he socks in he racking porfolio have marke beas which are no aypical, he marginal imporance of he remaining facors increases. In oher words, o do well in he racking error es, he risk model should adequaely model he oal risk beyond he par conribued by he marke facor. Once he reurns are calculaed in excess of he benchmark, he mahemaical formulaion of he es becomes very similar o (9). There are wo noable differences. Firs, we impose he long only consrain, since i is ypical for index rackers. Second, we narrow down our es universe so i conains only liquid socks wih above average marke capializaion. We do i in order o make sure he socks wih which we aemp o rack he benchmark acually are drawn from he universe similar o he one from which he socks in he benchmark are drawn (oherwise we migh find ourselves rying o rack 50 larges socks wih small and micro cap socks). Table 3 presens he resuls. The noaion is idenical o he one already used. Table 3. Annualized RV of Minimum Tracking Error Porfolios # of socks upper weigh limi ±10% ±20% ±5% ±10% DDRM PCA5 GRM EW (0.0020) (0.0020) (0.0017) (0.0017) (0.0021) (0.0021) (0.0019) (0.0019) (0.0023) (0.0023) (0.0021) (0.0021) (0.0024) (0.0024) (0.0023) (0.0023) The resuls for he minimum racking error porfolio coninue o favor DDRM over he benchmark models. While he magniude of DDRM s ouperformance over a 5 facor PCA model somewha decreased, i is sill in he 40 50bp range. The paired es for equaliy of he means indicaes ha he difference is saisically significan a 1% confidence level. A relaed applicaion of he risk model o porfolio (buy sell lis) consrucion is hedging. Ofen he buysell lis is no symmeric, i.e. here are more names on he long or on he shor side 6. Ye he rader prefers (or is required) o mainain he marke neuraliy of he radelis during he execuion process. We illusrae he usage of DDRM o hedge ou he marke risk from a long shor porfolio (or from a buy sell radelis). The es design is as follows. Every day saring from July 2003, we form a long shor porfolio wih 40 names on he long side and 10 names on he shor side. All socks are picked randomly from he DDRM 5 The lis of socks used o consruc he benchmark proxy is provided in he Appendix. 6 The siuaion described in his secion can arise even wih he equal number of names on he buy and sell sides of he radelis. 14

15 universe. The iniial porfolio is fully invesed, i.e. he sum of iniial porfolio weighs is one 7. We se he equally weighed porfolio of 50 large cap socks used for a minimum racking error es o be our proxy for a European marke ETF. We hen calculae he hedge raio as he iniial porfolio and ETF p ETF, where p is he marke bea of is he marke bea of our ETF proxy. A he nex sep we form he hedged p porfolio by selling shor of our synheic ETF for every dollar invesed in he original porfolio and ETF by keeping he original porfolio inac. As a benchmark we also form he dollar neural porfolio consising of he same names (iniial plus 50 names in synheic ETF), and require ha he sum of weighs on he buy and sell sides is equal o zero. This dollar neural benchmark could be viewed as a naïve aemp o hedge ou he marke risk wihou using a risk model. We measure he nex day s RV of all hree porfolios (iniial, hedged and dollar neural) and hen move forward one day, a which poin we repea he same seps again. Table 4 presens annualized average RV for all hree porfolios calculaed by year. Table 4. Annualized RV of a Hedged Porfolio Year/Porfolio Iniial Marke Hedged Dollar-neural The resuls show ha every year he long shor porfolio wih he European marke hedged ou has lower risk han eiher he iniial or he dollar neural porfolio. The advanages of bea hedging are especially apparen during he high volailiy periods ( ). In he dollar neural hedging also helps o reduce porfolio risk albei no o he same exen as he bea neural hedging. We would like o noe ha he bea hedging es described above relies enirely on he esimaes of he marke facor loadings. The hedging resuls presened in Table 4 aes o he qualiy of hese esimaes and o he effeciveness of our shrinkage procedure designed o overcome he poenial bias from nonsynchronous rading. In addiion o proper risk decomposiion which is necessary for consrucing long shor porfolios (or buy sell liss) a good risk model should help o accuraely measure he level of porfolio risk. We slighly modify he previous es o gauge he models capaciy o do ha. Every day, we form he equallyweighed porfolio of 100 randomly picked socks, load he marices of facor loadings, covariances and sock specific risk and predic he nex day porfolio risk. We hen measure he porfolio s nex day realized volailiy and calculae he predicion error. We repea over all days in our ou of sample period and calculae he mean of absolue errors as in (8) wih RV denoing acual porfolio RV on day pred and RV denoing is predicion. In oher words, we perform no opimizaion, jus predic and measure he RV of a random porfolio. Table 5 conains he resuls organized by year. 7 We also consruced iniial porfolio wih a differen base and obained resuls which are qualiaively similar. 15

16 Table 5. MAE for Porfolio Risk Forecas Year/Model DDRM PCA5 GRM As in he GMV and minimum racking error ess, he DDRM model comforably ouperforms he 5 facor PCA model in forecasing porfolio risk. No surprisingly, ITG s monhly model performs he wors. V. Deliverables There are hree oupu files being generaed by he model on a daily basis and delivered o fp locaion by 1:00AM EST (or 6:00AM London ime): Facor covariance file This file conains facor covariance marix predicion for he nex rading day. There are 11 facors which include he regional marke and 10 regional secor facors. The filename is facorcov_loc_d_eur_1_yyyymmdd.da The file header provides he imesamp of model creaion, model version, he daa fields and few oher model parameers: #<Timesamp> :28:27 #<Time Period> #<Base Currency>Loc #<Delimier> #<Horizon>Daily #<Version>1.0 #<Facors1>"Region1(EUR)","Region1_Secor10","Region1_Secor15","Region1_Secor20","Re gion1_secor25","region1_secor30","region1_secor35","region1_secor40","region1_sec or45","region1_secor50","region1_secor55" Loading file This file conains facor loadings, specific risk, igid, CUSIP, company name and secor code for he universe covered in 23 European counries. The filename is loadings_loc_d_eur_1_yyyymmdd.da The file header provides basic model informaion. <Facors1> liss he model facors, while <Column Name> conains he oupu daa fields: #<Timesamp> :36:54 #<Time period> #<File Type>Loadings #<Delimier> #<Horizon>Daily #<Version>1.0 #<Facors1>"Region1(EUR)","Region1_Secor10","Region1_Secor15","Region1_Secor20","Re gion1_secor25","region1_secor30","region1_secor35","region1_secor40","region1_sec or45","region1_secor50","region1_secor55" 16

17 #<Column Name>"UniqueID","Cusip/Sedol","Ticker","Name","EUR RegionLoading","Regional Secor Loading","Specific Risk","Regional Secor Code" Diagnosic repor file This file provides basic diagnosic informaion on DDRM oupu. I includes he sock coverage informaion, summary of he loadings disribuion (perceniles and average) and he average R 2. We also repor risk decomposiion, risk predicion as well as realized risk (RV) for a sample porfolio corresponding o MSCI EURO index. VI. Final Remarks The proliferaion of algorihmic porfolio rading calls for a risk model which quickly reacs o volailiy shocks, and allows o analyze and forecas he risk of a radelis. The ou of sample ess presened in he previous secion demonsrae ha ITG s laes DDRM performs quie well a one day ahead porfolio covariance forecasing. The ess were designed o mimic ypical applicaions of he model, such as hedging ou marke facor exposure or minimizing he risk of a radelis. The model is parsimonious and easy o esimae on a daily basis. We feel ha he mehodology adoped for DDRM esimaion allows for a sraighforward exension o oher counries and regions. For example, a model for Asian counries and for US and Canada could prove useful. An ineresing direcion for fuure research could include comparison beween forecasing performances of HAR and he componen muliplicaive error model (CMEM) of Brownlees a el. (2010). The laer is used by ITG Financial Engineering o predic volume and volailiy in he nex 15 minue inerval for algorihmic applicaions. A comparison wih EWMA smooher is also warraned. 17

18 VII. References Andersen T., Bollerslev T., Diebold F. and Ebens H. The Disribuion of Sock Reurn Volailiy, Journal of Financial Economics, vol. 61 (2001a), pp Andersen, T., Bollerslev, T., Diebold, F. and Labys, P. The Disribuion of Realized Exchange Rae Volailiy, Journal of he American Saisical Associaion, vol. 96 (2001b), pp Andersen, T., Bollerslev, T., Diebold, F. and Labys, P. Modeling and Forecasing Realized Volailiy, Economerica, vol. 71 (2003), pp Audrino, F., and Corsi, F. Realized Covariance Tick by Tick in Presence of Rounded Time Samps and General Microsrucure Effecs, 2008, Universiy of S. Gallen Economics Discussion Paper No Available a SSRN: hp://ssrn.com/absrac= Bai, J., and Ng, S., Deermining he Number of Facors in Approximae Facor Models, Economerica, vol. 70 (2002), pp Bannouh, K. Marens, M., Oomen, R., and van Dijk, D. Realized Facor Models for Vas Dimensional Covariance Esimaion, 2009, working paper. Bollerslev, T. and Zhang, B. Measuring and modeling sysemaic risk in facor pricing models using highfrequency daa, Journal of Empirical Finance, vol. 10 (2003), pp Brownlees, C., Cipollini F., and Gallo, G. Inra daily Volume Modeling and Predicion for Algorihmic Trading, Journal of Financial Economerics, (2010), forhcoming. Chan, L., Karceski, J., and Lakonishok, J. On porfolio opimizaion: forecasing covariances and choosing he risk model, Review of Financial Sudies, vol (1999), pp Connor, G., L. Goldberg and Korajczyk, R. Porfolio Risk Analysis, Princeon Universiy Press, Corsi, F. A Simple Long Memory Model of Realized Volailiy, 2004, available a SSRN: hp://ssrn.com/absrac= Fan, J., Fan, Y. and Lv, J. High dimensional covariance marix esimaion using a facor model, Journal of Economerics, vol. 147 (2008), pp Fleming, J., Kirby, C. and Osdiek, B. The Economic Value of Volailiy Timing Using 'Realized' Volailiy, Journal of Financial Economics, vol. 67 (2003), pp Hayashi, T. and Yoshida N. On covariance esimaion of non synchronously observed diffusion Processes, Bernoulli, vol (2005), pp Jagannahan, R. and Ma, T. Risk reducion in large porfolios: why imposing he wrong consrains helps, The Journal of Finance, vol (2003), pp Kyj, L., Osdiek, B. and Ensor, K. Covariance Esimaion in Dynamic Porfolio Opimizaion: A Realized Single Facor Model, AFA 2010 Alana Meeings Paper, (2009). Available a SSRN: hp://ssrn.com/absrac= McAleer, M. and Medeiros, M. Realized Volailiy: A Review, Economeric Reviews, vol (2008), pp

19 Rosenberg, B. Exra Marke Componens of Covariance in Securiy Reurns, Journal of Financial and Quaniaive Analysis, vol. 9 (1974), pp Scholes, M., and Williams J. "Esimaing Beas from Nonsynchronous Daa," Journal of Financial Economics, vol. 5 (1977), pp

20 Appendix The lis of socks used o consruc he European ETF proxy HSBC HOLDINGS NESTLE VODAFONE GROUP BP TOTAL BANCO SANTANDER NOVARTIS ROYAL DUTCH SHELL A ROCHE HOLDING GENUSS TELEFONICA GLAXOSMITHKLINE SIEMENS RIO TINTO PLC ASTRAZENECA BHP BILLITON PLC BRITISH AMERICAN TOBACCO BNP PARIBAS SANOFI-AVENTIS UBS NAMEN BARCLAYS STANDARD CHARTERED ENI E. ON BASF BG GROUP CREDIT SUISSE BAYER ANGLO AMERICAN (GB) TESCO BBVA DAIMLER LLOYDS BANKING GROUP DEUTSCHE BANK NAMEN UNILEVER NV CERT ABB LTD GDF-SUEZ DIAGEO UNICREDIT ORD SAP STAMM FRANCE TELECOM NOVO NORDISK B ING GROEP DEUTSCHE TELEKOM SOCIETE GENERALE ANHEUSER-BUSCH INBEV XSTRATA NOKIA CORP ZURICH FINL SERVICES NORDEA BANK ENEL 20