Can Multivariate GARCH Models Really Improve Value-at-Risk Forecasts?

Transcription

1 2s Inernaonal Congress on Modellng and Smulaon, Gold Coas, Ausrala, 29 ov o 4 Dec Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? C.S. Sa a and F. Chan a a School of Economcs and Fnance, Curn Unversy Emal: chow.sa@cbs.curn.edu.au Absrac: Ths paper evaluaes he performance of mulvarae condonal volaly models n forecasng Value-a-Rsk (VaR). The paper consders he Consan Condonal Correlaon (CCC) model of Bollerslev (990), and models ha allow dynamc condonal correlaon such as he Dynamc Condonal Correlaon (DCC) model of Engle (2002) and he Tme-Varyng Condonal Correlaon (TVC) model of Tse and Tsu (2002). Whle he underlyng assumpons vary beween hese models, her common objecve s o model volaly for mulple asses by capurng her possble neracons. Thus, hey provde more nformaon abou he underlyng asses ha could no be recovered by unvarae models. However, he praccal usefulness of hese models are lmed by her complexy as he number of asse ncreases. The paper ams o examne hs rade-off beween smplcy and exra nformaon by applyng hese models o forecas VaR for a porfolo of he Ausralan dollar wh welve oher currences. Ths provdes some nsgh no he praccal usefulness of he addonal nformaon for purposes of rsk managemen. Keywords: Value-a-Rsk (VaR), Mulvarae GARCH 043

2 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass?. ITRODUCTIO Modellng volaly n fnancal me seres has been an mporan research area n he pas decades. The famly of Auoregressve Condonal Heeroskedascy (ARCH) model was frs nroduced by Engle (982) who lad he foundaon for a new approach o descrbe and forecas condonal varance for fnancal me seres. Subsequenly, numerous varans and exensons of ARCH models have been proposed. See for examples, he Generalzed ARCH (GARCH) model of Bollerslev (986) and s asymmerc exenson by Glosen, Jagannahan, and Runkle (993). In many fnancal applcaons, condonal covarance and correlaons play a drec and mporan role n volaly forecasng. A bank s very lkely o rade wh large and complex porfolos daly. I s unlkely ha he asse reurns n a porfolo would move ndependenly of each oher. Therefore, undersandng her correlaon srucures s essenal n dervng sensble nvesmen sraeges o maxmze reurns whle mnmzng rsk. Mos of he exsng unvarae volaly models focus on he dynamcs of a sngle me seres and hey do no provde any nformaon on he poenal dependency beween asse reurns whn a porfolo. I s worh nong ha he correlaon beween asse reurns may be drven by ndvdual heerogeney as well as any poenal common facors. Ths mples ha he correlaon srucures may be me-varyng. For example, he correlaon beween Sandard & Poor's 500 (S&P 500) and kke 225 s lkely o be dfferen before and afer he Global Fnancal Crss (GFC). The correlaon before he crss may be drven by normal marke condon whereas he GFC forms a sngle facor ha caused sgnfcan changes n he correlaon beween he wo ndces. To capure he condonal covarance and correlaons for he dfferen ype of asses n a porfolo, many researchers expanded he unvarae o mulvarae volaly models. McAleer (2005) poned ou ha one mporan aspec n modellng fnancal volaly s o sudy mulvarae exensons of he condonal volaly models. Bollerslev, Engle, and Wooldrdge (988) proposed he dagonal vecor ARCH (DVEC) model ha s a drec exenson of he unvarae Generalzed ARCH (GARCH) model o mulvarae model. Oher alernave approaches for achevng more parsmonous and emprcally racable mulvarae volaly models are he Consan Condonal Correlaon (CCC) model of Bollerslev (990); Baba, Engle, Kraf and Kroner (BEKK) model descrbed by Engle and Kroner (995); he Dynamc Condonal Correlaon (DCC) model of Engle (2002); he Tme-Varyng Correlaon (TVC) model of Tse and Tsu (2002); he Vecor ARMA-GARCH (VARMA-GARCH) model of Lng and McAleer (2003); and he VARMA-asymmerc GARCH (VARMA-AGARCH) model of McAleer, Ho, and Chan (2009). However, he praccal usefulness of hese models can be affeced by he curse of dmensonaly (see Caporn and McAleer 204). Tha s, he number of parameers ncreases dramacally n hese models as he number of asse ncreases. There are a huge number of sudes ha esmae VaR forecass usng mulvarae GARCH models. Hsu Ku and Wang (2008) examned he performance of mulvarae GARCH models, namely he CCC, DCC and BEKK models, n erms of VaR volaons on a porfolo of foregn exchange raes. The DCC model s consdered o be he bes model ha offers a beer forecasng performance among he oher wo models n esmang VaR. da Vega, Chan, and McAleer (20) used boh CCC and DCC models on he porfolos of Chnese A and Chnese B sock reurns. On one hand, DCC model provdes a lower number of volaons han he CCC model. On he oher hand, CCC model ends o generae a lower amoun of daly capal charges han he DCC model. Consequenly, hey showed ha a more severe penaly srucure s probably desrable o dscourage banks from choosng forecasng models ha underesmae VaR. In parcular, hey proposed a new penaly srucure ha s based on he magnude of volaons nsead of he curren penaly srucure ha s based on he number of volaons. An approprae penaly srucure may encourage banks o mprove her rsk models n forecasng VaR more precsely. Whle, Bauwens and Lauren (2005) proposed a mulvarae skewed- dsrbuon for mulvarae GARCH models on he porfolos of he US sock reurns and foregn exchange raes. They found ha he mulvarae GARCH models under mulvarae skewed- dsrbuon mproves he performance of VaR forecass. everheless, hese sudes showed ha accommodang mevaryng condonal correlaons mproves he forecasng performance of VaR. Ths paper s oulned as follows. The srucural properes of he CCC and DCC models and he marke rsk capal requremens by he Basel Accord are provded n Secon 2. Secon 3 descrbes he daa and dscusses he emprcal resuls for he performance of VaR forecass. Secon 4 concludes he paper. 2. CODITIOAL VOLATILITY MODELS AD VAR FORECASTS Consder he followng model: 044

3 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? Φ ( Lr ) =Θ ( L) ε ε = Dη η ~ MV(0, I) D= dagh (,..., h) () k where r ( r r ) =,..., ' s a k vecor of asse reurns and =,..., T, wh L denoes he lag operaor k such ha for any me seres y, Ly = y. Φ ( L) = I φ L and p Θ ( L) = I + θ L are he lag = = polynomals of order p and q, respecvely. η s a k ndependenly and dencally dsrbued mulvarae random vecor wh zero mean and deny varance-covarance marx. q Followng he model as defned n equaon (), he condonal varance and covarance marx of r s Ω Γ, where ( ηη ') = D D Γ =E denoes k k marx of he condonal correlaons beween he condonal shocks. E and E denoes he uncondonal and condonal expecaon wh respec o he nformaon se a me, respecvely. The CCC model assumes ha he condonal correlaons are consan Γ= s a consan condonal correlaon marx wh ρ j = ρ j. Engle (2002) and Tse and Tsu (2002) proposed he DCC model and he TVC model, respecvely, o allow he condonal over me. Hence, { } ρ j correlaons o be me-varyng, so ha he condonal varance and covarance marx of r s me-varyng. In ha case, he dynamc of volaly depends on he specfcaon of Ω. The parameers n hese models are ypcally esmaed by Quas-Maxmum Lkelhood Esmaor (QMLE), whch s defned o be: T ˆ T θ = arg max log Ω + ε ' Ω ε (2) θ Λ 2 = Where θ denoes he parameers o be esmaed n he condonal log-lkelhood funcon. Ω denoes he deermnan of Ω. See McAleer (2005) and McAleer e al. (2008) for more echncal dscussons on hs class of models, ncludng he suffcen condons for he exsence of momens and he suffcen condons for conssency and asympoc normaly of QMLE. Followng equaon (), he VaR forecas a α = 0.0 for asse a me + can be obaned as: m, + VaR = E ( r ) + q h (3) m, + α, d, + where E ( r ), + s he forecas of he asse s reurn based on he nformaon a me, q α s he crcal,d value based on he sgnfcan level of VaR and he dsrbuon of η. Alhough η s ypcally assumed o be normally dsrbued, a suden- dsrbuon wh δ degrees of freedom can be used an alernave., m h + s he esmaed sandard devaon of E ( r ), + wh m denoes he model used. oed ha he superscrps sd and norm denoes esmaes assumng a normal dsrbued reurn and a -dsrbued reurn. The curren regulaory framework requres banks ha use her own nernal rsk models o calculae he VaR on a daly bass a 99 percen confdence level. Backesng procedures have been used o evaluae he performance of VaR models. As such, he marke rsk capal requremens are deermned as follows (Basel Commee on Bankng Supervson 20):. A bank mus backes s nernal VaR models over he prevous 250 radng days. 2. To monor he frequency of volaons, he number of mes ha he acual losses exceed VaR forecass are calculaed. Subsequenly, he percenage of volaons can also be calculaed. A good model wll have a percenage of volaon ha s very close o one percen and should lead o correc esmaon of marke rsk a every pon n me. A VaR model ha overesmaes marke rsk wll lead o nsuffcen volaons and requres a large amoun of capal. On he oher hand, a VaR model ha underesmaes marke rsk wll be penalzed by he regulaor due o excessve volaons. 045

4 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? 3. The marke rsk capal charge (MRCC) s se eher a he lower VaR of he prevous day or he average VaR of he prevous 60 days radng days, mulpled by a scalng facor of (3+k). The scalng facor calculaes he probably ha a volaon occurs for a gven day over he prevous 250 radng days. I can be wren as: 60 MRCC = mn, (3 ) VaR + k VaR α, α, (4) 60 = where, V 4( Green) 250 k = ( ) V = Yellow (5) 250 V 0( Red) Some sascal ess are also carred ou o valdae VaR forecass. These ess nclude Kupec (995) Tes Unl he Frs Falure (TUFF), followed by Chrsoffersen (998) and Chrsoffersen, Hahn, and Inoue (200) Uncondonal Coverage (UC), Seral Independence (ID) and Condonal Coverage (CC) ess. See da Vega, Chan, and McAleer (20) for furher dscusson on hese sascal ess. 3. RESULTS A daase of daly exchange raes on Ausralan dollar (AUD) wh welve oher currences s used. The exchange raes are US Dollar (USD), Japanese Yen (JPY), Pound Serlng (GBP), ew Zealand Dollar (ZD), Korean Won (KRW), Sngapore Dollar (SGD), Swss Franc (CHF), Chnese Renmnb (CY), Hong Kong Dollar (HKD), Indan Rupee (IDR), Malaysan Rngg (MYR), and ew Tawan Dollar (TWD). These exchange raes are colleced from Thomson Reuers DaaSream Professonal, for he perod of 2 January 984 o 3 December 203. Usng he daa above, an equally-weghed porfolo of welve asses s consruced. The sample sze used for esmaon s from 2 January 984 o 3 December 2002 wh 4,950 observaons and he forecasng perod s from 2 January 2003 o 3 December 203 wh 2,87 observaons. The means of he porfolo reurns for boh esmaon and forecas perods are close o zero. The skewness of he porfolo reurns for boh esmaon and forecas perods are negave. Whle, he porfolo reurns dsplay hgh kuross and faaled. There are four ses of VaR forecass esmaed from he CCC-GARCH(,), CCC-GJR(,), DCC- GARCH(,) and DCC-GJR(,) models for normal dsrbuon. The degrees of freedom se by -densy are esmaed from he sandardzed resduals ha follow GARCH(,) and GJR(,) processes ulzed under normal and suden- dsrbuons. Ths gves egh crcal values ha leads o egh ses of VaR forecass. A oal 2 ses of VaR forecass are presened for comparson purposes. All VaR forecass are consruced a % level. The parameer esmaes n he CCC-GARCH(,), CCC-GJR(,), DCC-GARCH(,) and DCC- GJR(,) models are sascally sgnfcan for all currences. As he second momen condon s sasfed, he log-momen condon s necessarly sasfed, so he QMLE s conssen and asympocally normal. Table. VaR Forecass a % level Model Mean Medan Mnmum Maxmum Sandard Devaon CCCGARCH () () CCCGARCH (2) (2) CCCGARCH (3) (3) D CCGARCH () () D CCGARCH (2) (2) D CCGARCH (3) (3) () VaR forecass are esmaed from equaon (6) based on a normal dsrbuon 046

5 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? (2) VaR forecass are esmaed from equaon (6) based on a normal dsrbuon a he degrees of freedom se by -densy (3) VaR forecass are esmaed from equaon (6) based on a suden- dsrbuon a he degrees of freedom se by -densy Table summarzes he resuls for he 2 ses of VaR forecass esmaed by he CCC and DCC models. The means of VaR forecass for he CCC and DCC models ha ulzed under suden- dsrbuon appear o be lower han he means of VaR forecass for he CCC and DCC models under he normal dsrbuon. Hence, he suden- dsrbuon provdes more conservave VaR forecass han a normal dsrbuon. I can also be seen ha he means of VaR forecass esmaed by he DCC models are mosly lower han he means of VaR D CCGARCH forecass esmaed by he CCC models. In parcular, shows he lowes mean of VaR forecass CCCGARCH a whle shows he hghes mean of VaR forecass a Hence, VaR forecass esmaed by he DCC models are crucal o mprove he performance of VaR forecass. These resuls also jusfy he use of model such as DCC o capure he me-varyng condonal correlaon srucures n porfolo reurns. Table 2. umber and Percenage of Volaons for VaR Forecass a % Level Model o. of Volaon % of Volaon CCCGARCH % % CCCGARCH 4.43% 39.36% CCCGARCH 4.43% 40.39% D CCGARCH % % D CCGARCH 4.43% 39.36% D CCGARCH 35.22% 33.5% Table 2 repors he number and percenage of volaons for VaR forecass. Ideally, a good model would have a percenage of volaon ha s very close o one percen. A model ha underesmaes marke rsk gves a percenage of volaon ha s more han one percen. A model ha overesmaes marke rsk gves a percenage CCCGARCH of volaon ha s less han one percen. Hgh percenages of volaons for and D CCGARCH are observed a 2.47% and 2.40%, respecvely. Smlarly, and presen hgh percenages of volaons a 2.37% and 2.23%, respecvely. A good model s gven by wh a CCCGARCH percenage of volaons a.5. The hghes percenage of volaons s gven by a 2.47%. I s worh nong ha he DCC models are preferred o he CCC models gven ha he DCC models provde he percenages of VaR volaons ha are closer o one percen. Also, he VaR volaons under suden- dsrbuon always gve fewer volaons han VaR volaons under he normal dsrbuon. Table 3. Backesng Resuls for VaR Forecass a % level CCCGARCH CCCGJR CCCGARCH CCCGJR CCCGARCH CCCGJR D CCGARCH D CCGJR D CCGARCH D CCGJR Model TUFF () UC () Ind (2) CC (2)

6 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? D CCGARCH sd VaR () The Uncondonal Coverage (UC) and Tme Unl Frs Falure (TUFF) ess are asympocally dsrbued as χ 2 (). (2) The Seral Independence (Ind) and Condonal Coverage (CC) ess are asympocally dsrbued as χ 2 (2). (3) Enres n bold denoe rejecon of he ess. The resuls from TUFF, UC, Ind and CC ess are gven n Table 3. The TUFF resuls for all models lead o correc accepance of he es a a consan value of I can be seen ha boh CCC and DCC models ulzed under a normal dsrbuon, fal UC, Ind and CC ess. Ths suggess ha he VaR volaons performed D CCGARCH by hese models are serally dependen. On he oher hand,,,, and pass UC, Ind and CC ess. Ths shows ha he VaR volaons are lkely o be ndependen and ha a VaR volaon oday should no provde any nformaon abou wheher or no a VaR volaon wll occur CCCGARCH CCCGARCH D CCGARCH omorrow. Whle,,,,, and, fal he UC and CC ess bu pass Ind es. Table 4. Marke Rsk Capal Charges Model Mean Medan Mnmum Maxmum Sandard Devaon CCCGARCH CCCGARCH CCCGARCH D CCGARCH D CCGARCH D CCGARCH Table 4 shows he marke rsk capal charges from equaons (4) and (2). Berkowz and O'Bren (2002) and Pérgnon, Deng, and Wang (2008) showed ha banks end o repor hgh VaR forecass ha lead o an excessve amoun of capal charges. In any case, here s an opporuny cos of msesmang VaR. Hence, pursung a correc VaR model ha can lead o he precson of deermnng mnmum capal charges s crucal for banks D CCGARCH and he regulaor. In hs case, provdes he lowes mean of marke rsk capal charges a D CCGARCH Whle, he lowes capal charge s gven by a Ths s mosly expeced due o he exreme negave reurns durng he GFC of 2008 where hgher amoun of capal charges are mposed o proec banks o cover from he wors possble radng losses. On he oher hand, he hghes capal charge s presened D CCGARCH by a 2.39, followed by a Ths generally occur durng perods of low volaly n he foregn exchange marke. 4. COCLUSIO Ths paper emphaszes he mporance of accommodang me-varyng condonal correlaons n forecasng VaR. These fndngs are crucal for banks and he regulaor snce a correc VaR model leads o ncrease effcency n measurng marke rsk, hence leadng o deermne mnmum capal requremens. In hs paper, wo mulvarae volaly models, namely CCC and DCC models, are consdered o forecas VaR. The resuls show ha a suden- dsrbuon gves more robus esmaon of VaR forecass han a normal dsrbuon, gven ha he foregn exchange reurns exhb heavy als. The resuls also fnd ha he DCC models provde more conservave VaR forecass han CCC models wh he DCC models have lower numbers and percenages of VaR volaons. Consequenly, s reasonable o sugges ha me-varyng condonal correlaons canno be gnored n forecasng VaR. Also, CCC models delver a hgher amoun of capal charges compared o he DCC models. These resuls are conssen wh he emprcal fndngs by da Vega, Chan, and McAleer (20). 048

7 Sa and Chan, Can Mulvarae GARCH Models Really Improve Value-a-Rsk Forecass? Incorporang mulvarae volaly n VaR models s no sraghforward where here are many oher facors o be consdered. These models rase some dffcules n pracce, where banks rade wh relavely large and complex porfolos ha are unlkely o change daly. Ths mples ha each day, he banks would have o compue a seres of hsorcal daa for he new porfolos o esmae VaR. Ths may creae addonal coss o he banks. Insead of usng hese models, banks appear o be akng less compuaonally demandng alernaves. Banks prefer o use a smple VaR model ha aggregaes all of he rsks of a porfolo no a sngle number, whch s suable for use n he boardroom, reporng o he regulaor and dsclosure n her fnancal repors. oneheless, mulvarae volaly models play a sgnfcan role n he sudy of VaR as hey are very useful o measure and manage marke rsk. REFERECES Basel Commee on Bankng Supervson. (20). Basel III: A global regulaory framework for more reslen banks and bankng sysems - revsed verson. Basel, Swzerland: Bank for Inernaonal Selemens. Bauwens, L., and S. Lauren. (2005). A new class of mulvarae skew denses, wh applcaon o generalzed auoregressve condonal heeroscedascy models. Journal of Busness & Economc Sascs, 23(3), Berkowz, J., and J. O'Bren. (2002). How accurae are value-a-rsk models a commercal banks? Journal of Fnance, 57(3), Bollerslev, T. (986). Generalzed auoregressve condonal heeroscedascy. Journal of Economercs, 3, Bollerslev, T. (990). Modellng he coherence n shor-run nomnal exchange rae: a mulvarae generalzed arch approach. Revew of Economcs & Sascs, 72(3), Bollerslev, T., R. F. Engle, and J. M. Wooldrdge. (988). A capal asse prcng model wh me-varyng covarances. Journal of Polcal Economy, 96(), 6-3. Caporn, M., and M. McAleer. (204). Robus rankng of mulvarae GARCH models by problem dmenson. Compuaonal Sascs & Daa Analyss, 76, Chrsoffersen, P. (998). Evaluang nerval forecass. Inernaonal Economc Revew, 39, Chrsoffersen, P., J. Hahn, and A. Inoue. (200). Tesng and comparng value-a-rsk measures. Journal of Emprcal Fnance, 8(3), da Vega, B., F. Chan, and M. McAleer. (20). I pays o volae: how effecve are he Basel Accord penales n encouragng rsk managemen? Accounng & Fnance, 52(), Engle, R. F. (982). Auoregressve condonal heeroscedascy wh esmaes of he varance of Uned Kngdom nflaon. Economerca, 50, Engle, R. F. (2002). Dynamc condonal correlaon: a smple class of mulvarae generalzed auoregressve condonal heeroskedascy models. Journal of Busness & Economc Sascs, 20(3), Engle, R. F., and K. F. Kroner. (995). Mulvarae smulaneous generalzed ARCH. Economerc Theory, (), Glosen, L. R., R. Jagannahan, and D. E. Runkle. (993). On he relaon beween he expeced value and he volaly of he nomnal excess reurn on socks. Journal of Fnance, 48(5), Hsu Ku, Y. H., and J. J. Wang. (2008). Esmang porfolo value-a-rsk va dynamc condonal correlaon MGARCH model - an emprcal sudy on foregn exchange raes. Appled Economcs Leers, 5(7), Kupec, P. (995). Technques for verfyng he accuracy of rsk measuremen models. Journal of Dervaves, 3, Lng, S., and M. McAleer. (2003). Asympoc heory for a vecor ARMA-GARCH model. Economerc Theory, 9(2), McAleer, M. (2005). Auomaed nference and learnng n modelng fnancal volaly. Economerc Theory, 2(0), McAleer, M., F. Chan, S. Ho, and O. Leberman. (2008). Generalzed auoregressve condonal correlaon. Economerc Theory, 24(06), McAleer, M., S. Ho, and F. Chan. (2009). Srucure and asympoc heory for mulvarae asymmerc condonal volaly. Economerc Revews, 28(5), Pérgnon, C., Z. Y. Deng, and Z. J. Wang. (2008). Do banks oversae her value-a-rsk? Journal of Bankng & Fnance, 32(5), Tse, Y. K., and A. K. C. Tsu. (2002). A mulvarae generalzed auoregressve condonal heeroscedascy model wh me-varyng correlaons. Journal of Busness & Economc Sascs, 20(3),