Crisis, Value at Risk and Conditional Extreme Value Theory via the NIG + Jump Model

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1 Journal of Mahemaical Finance, 202, 2, hp://dx.doi.org/0.4236/jmf Published Online Augus 202 (hp:// Crisis, Value a Risk and Condiional Exreme Value Theory via he NIG + Jump Model Samuel Y. M. Ze-To Deparmen of Finance and Decision Sciences, Hong Kong Bapis Universiy, Hong Kong, China symzeo@hkbu.edu.hk Received May 2, 202; revised June 23, 202; acceped July 2, 202 ABSTRACT This sudy develops a new condiional exreme value heory-based model (EVT) combined wih he NIG + Jump model o forecas exreme risks. This paper uilizes he NIG + Jump model o asymmerically feedback he pas realizaion of jump innovaion o he fuure volailiy of he reurn disribuion and uses he EVT o model he ail disribuion of he NIG + Jump-processed residuals. The model is compared o he GARCH- model and NIG + Jump model o evaluae is performance in esimaing exreme losses in hree major marke crashes and crises. The resuls show ha he condiional EVT-NIG + Jump model ouperforms he GARCH and GARCH- models in depicing he non-normaliy and in providing accurae VaR forecass in he in-sample and ou-sample ess. The EVT-NIG + Jump model, which can measure he volailiy of exreme price movemen in capial markes due o unexpeced evens, enhances he EVT-based model for measuring he ail risk. Keywords: Value-a-Risk; Exreme Value Theory; Jump; NIG. Inroducion Along wih he Asian financial crisis of 997 ha caused severe slumps of currencies and he devaluaion of sock markes in Asian counries, he global financial crisis ha occurred in 2007 represens one of he serious financial crises ha riggered exreme volailiy in capial markes of boh developed and developing counries. I led o he collapse of banks, financial insiuions and conglomeraes, bringing serious loss o crediors and invesors. In response o hese evens, regulaors have become more concerned abou he proecion of financial insiuions agains hese caasrophic marke risks. Value-a-Risk (VaR) has become a sandard measure for risk managemen, ye i is well known ha he disribuions of he reurn series in mos financial markes are heavily ailed. Tradiional risk measuring models such as Riskmerics focus on he whole disribuion and fail o provide an accurae measure of exreme price movemens. Addiionally, he radiional VaR mehod has been criicized for violaing he requiremen of subaddiiviy [,2]. To accoun for he heavily ailed disribuion in financial reurns, researchers have exensively derived and modified VaR models. Mos researchers have developed VaR models ha incorporae eiher asymmeric disribuions or he exreme value heory. For insance, Bali and Theodossiou [3] derived a condiional VaR wih a skewed generalized. Longin [4] proposed considering he esimaion of capial requiremens as a problem of exreme value calculaion. Longin also presened an approach for compuing VaR using he EVT model. The parameric exreme value mehod ha focuses on he exreme ails of he disribuion allows for an exension of he curve beyond he range of daa [5]. This allows he EVT model o esimae exreme losses beer han classical mehods ha use normal disribuions [6]. The uncondiional EVT has been applied o measure he downside risk in equiy markes [7]. Bali [8] also inroduced a generalized exreme value approach o financial risk measuremen. Bali [9] esed an asympoic disribuion for exreme changes in US Treasury yields and showed ha he exreme value approach provided a more accurae esimaion of VaR han sandard models. Bali and Nefci [0] developed a condiional approach o derive VaR by specifying he locaion and scale parameers of he generalized Pareo disribuion (GPD) as a funcion of pas informaion, which was found o provide an accurae forecas of he occurrence and size of exreme observaions. The EVT has been commonly applied in boh financial and insurance risk managemen [-3]. Bali [8] proposed a Box-Cox generalized exreme value disribuion model o capure exreme evens in financial markes. Alhough he uncondiional EVT approach provides asympoic resuls in he disribuion of exreme loss for

2 226 S. Y. M. ZE-TO long-erm invesmen decisions, he primary concern for a risk manager is he possibiliy of loss due o adversarial marke movemen during he nex couple of days, where he dynamics of he ime-varying volailiy are imporan. Hence, i seems more appropriae o use he condiional EVT mehod when esimaing day-o-day risk exposures and shor-erm risk managemen [4]. Smih [5] suggesed an approach o deal wih sochasic volailiy via a change-poin model for exreme value parameers. McNeil and Frey [6] aemped o esimae VaR by incorporaing he GARCH model wih hreshold-based EVT ools, while Bysröm [4] modified he approach by creaing a condiional VaR esimae using he block maxima mehod. Their condiional models correced he clusering of exreme evens due o sochasic volailiy. The models were able o capure he dynamics of he curren volailiy beer han he uncondiional model [6]. Bali and Weinbaum [7] inroduced a daily condiional exreme value volailiy esimaor in view of he significan persisence in he parameers for a GEV disribuion of high frequency reurns wihin a fixed ime inerval. Approaches for VaR esimaion ha use GPD and empirical disribuion funcions o model ail evens and o capure normal marke condiions, respecively, have been proposed by some researchers [8]. Bysröm [4] used boh he block maxima approach and he hreshold approach in esimaing condiional VaR and found ha he wo models performed similarly. Hence, his sudy was exended o creae he condiional VaR by firs using he regime swiching model o consruc he condiional volailiy of he disribuion. Then, he EVT model using he hreshold mehod was employed o esimae he disribuion of he residuals. In 982, Engle [9] proposed he auoregressive condiionally heeroskedasic (ARCH) model in esimaing marke volailiy. Subsequenly, he GARCH model was developed by Bollerslev [20] and Taylor [2]. Variaions and enhancemens of he GARCH models have been subsanially generaed by researchers. I is, however, indicaed ha exreme declines in he price in securiies markes due o unexpeced evens like sock marke crashes or financial crises canno be fully explained by hese GARCH models [22]. Researchers such as Maheu and McCurdy [23] demonsraed ha hese unusual evens may be beer capured by jumps. Maheu and McCurdy [23] proposed a GARCH model ha incorporaed a heerogeneous Poisson process wih a condiional inensiy parameer o govern he occurrence of jumps. Maheu and McCurdy [23] explained ha he news process is divided ino wo componens: normal news and unusual news evens. Normal news innovaions cause changes in he condiional variance of reurns. The second componen of he news process, however, leads o sudden jumps in price over a very shor period of ime ha could be beer capured by jumps raher han Brownian moion. Hence, his model capures he excess volailiy resuling from unexpeced changes in he financial ime series. The moivaion of his paper is o propose a model (he EVT-NIG + Jump model) ha furher enhances he performance of he VaR model in capuring exreme losses by incorporaing he NIG + Jump model wih EVT and o compare is performance wih seven oher models in VaR forecasing. This paper focuses on he sudy of negaive ails and examines he fiing and forecasing performance of he models when hey are applied o a series of marke crashes and financial crises. This sudy examines wheher he proposed VaR models can beer forecas he burs of marke bubbles. The sudy focuses on hree major crises: he Asian Financial Crisis (AFC) in 997, he Do-Com Bubble (DCB) burs in 2000 and he Global Financial Crisis (GFS) in The AFC was riggered by a financial overexension of he Thailand economy ha led o he subsequen breakage of he peg of Thai bah wih US dollars and he devaluaion of Asian sock markes. The DCB sared in 998 wih he growh of sock prices in he Inerne secor. The marke expecaion of he fuure earnings growh of hese Inerned-based firms caused NASDAQ o reach is peak in March The GFS ha occurred in July 2007 was driven by a burs of he real esae bubble in he Unied Saes and a loss of marke rus in subprime morgages. The subsequen liquidiy crisis and credi risks caused banks o be relucan and unable o offer loans o companies. A sudden loss of asse values and global marke sock crashes expedie economic recessions. This paper conribues o he lieraure by empirically demonsraing ha he EVT-NIG + Jump model ouperforms oher models in capuring exreme loss in crises. The auoregressive condiional jump inensiy process of he model aids in capuring he clusering of jumps ha normally exis in marke crashes. The model allows for dynamic changes in jump arrival rae, jump size, volaileiy clusering and asymmeric responses o pas reurn innovaions. The model performance in exreme loss esimaion is furher enhanced by adoping he EVT o esimae he disribuion of he residuals. This paper presens five models for compuing he VaR in Secion 2. Secion 3 repors he daa and an analysis of he empirical resuls, and Secion 4 concludes wih he findings and conribuions of he paper. 2. Models This secion sars by presening he EVT-NIG + Jump model and explaining he NIG + Jump process and EVT. Then, he mehodologies of oher VaR esimaions using he Suden- disribuion and GARCH models are dis-

3 S. Y. M. ZE-TO 227 cussed. 2.. EVT-NIG + Jump Model 2... NIG + Jump Process Since he developmen of ARCH models by Engle [9] and he generalizaion of he GARCH model by Bollerslev [20], he GARCH models have been exensively enhanced and used in modeling he volailiy dynamics of financial ime series. On he oher hand, unusual evens like he Asian Crisis or news surprises creae exreme movemens in price ha could be beer capured by jumps han Brownian moion or normal innovaions, as saed by Maheu and Mccurdy [23]. This paper has adoped heir NIG + Jump model o develop he condiional EVT wih a jump model. One of he major feaures of he model is ha he previous innovaions ha are modeled as a serially correlaed condiional Poisson process feed back ino he expeced volailiy hrough he GARCH componen of condiional variance. This allows for condiional conemporaneous leverage effecs and lagged leverage effecs, as described by Maheu and Mccurdy [23]. The NIG disribuion has a densiy funcion expressed as 2 2 ( x m) f x,,, m, exp π xm xm w K w () where, represen he locaion scale pa- 2 rameers and wy Y and K is he modified Bessel funcion of index one in hird order. This paper has adoped heir NIG-Jump model o develop he condiional EVT wih a jump model. One of he major feaures of he model is ha he previous innovaions ha are modeled as a serially correlaed condiional Poisson process feed back ino he expeced volailiy hrough he GARCH componen of condiional variance. This allows for condiional conemporaneous leverage effecs and lagged leverage effecs, as described by Maheu and Mccurdy [23]. The marke reurn is expressed as This parameerizaion incorporaes an auoregressive condiional inensiy governing he likelihood of he arrival of jumps beween and. δ is a ime-varying inensiy residual. I is defined as np X n (5) n0 The jump dynamic is assumed o follow a Poisson disribuion. The condiional densiy of X is expressed as n exp PX n, n0,,2, (6) n! where R, R, R 2 I represens he informaion of he previous reurn. The jump size J,n follows a normal disribuion wih mean J and variance ξ. J n,, NID J (7) Hence, he jump innovaion is expressed as X 2, J, n J (8) n This is he sum of X jumps arriving wihin he ime inerval beween and and is condiionally mean zero. The condiional volailiy of marke reurns is governed by wo condiional variances as follows: 2 Var, Var 2, (9) The firs componen describes he diffusion of pas informaion impacs and is defined as a GARCH model: 2, 0, Var, Var (0) where ι is he oal innovaion of reurn a and is defined as Ψ(.) is expressed as (), 2,, expa ajex D bbjex where aa, j, bb, j is he vecor of parameers. R 0 R, 2, (2) D is when 0 and 0 oherwise. where M 0 R is he condiional mean and, E X is he expeced number of jumps in he is he reurn innovaion a ime, expressed as ime inerval beween and 2. The condiional variance Var, z z~ NID0, 2, is relaed o he heerogeneous in- (3) formaion arrival process ha generaes jumps, and is while 2, is a jump innovaion wih a condiionally expressed as mean zero and is conemporaneously independen of,. 2 2 The condiional jump inensiy λ is expressed as Var 2, J (3) 0 0 (4) where J is he condiional jump size a ime and is ex- (2)

4 228 S. Y. M. ZE-TO pressed as a funcion of he pas reurn. and GR J R G R R 0 2 c c R (4) (5) where G R if R 0 and 0 oherwise. The parameers are esimaed using he maximum likelihood mehod. The densiy funcion of he reurn condiional on he mos recen informaion is expressed as f R X n,, 2π Var 2 R M Jn exp 2 2 n 2 Var, n (6) Inegraing over he number of jumps, he condiional densiy funcion is, f R f R X n P X n (7) n0 The filer is defined as P X R X n, PX n f R f n (8) n 0,, 2,. Insead of solving he infinie summaion, his paper runcaes he summaion a 25 for esimaing he parameers Condiional EVT-NIG + Jump Model This sudy incorporaes he NIG + Jump model wih he EVT o model he ime-varying reurn disribuion. This approach focuses on he enire disribuion raher han he ail disribuion only [4,23] and esimaes VaR via a wo-sage process. The procedure sars wih he NIG + Jump model o esimae he condiional mean and volaileiy of he enire disribuion. Then, in he second sage, he POT mehod of EVT is used o model he disribuion of he residual. The parameer vecor of he NIG + Jump model can be esimaed by maximizing he log likelihood funcion discussed in (6) for N observaions. N log, L f R (9) For he parameer esimaion, N is fixed as 000 (a window of 000 rading days) and y as 00. Wih he se of parameers, he series of N condiional means and sandard deviaions can be esablished. Then, in sage 2, he series of N residual values is calculaed based on he formula, R M (20) where M and σ represens he condiional mean and sandard deviaion. Assume R follows a disribuion F x. Then, based on he approach of exceedances over hresholds [24-26], by fixing a high hreshold τ, he excess disribuion of residual is expressed as F F F y F y P R y R (2) where 0 y R0, and R 0 represens he righ endpoins of F [4]. Balkema and de Haan [27] and Pickands [28] saed ha F (y) ends o a GPD for a large class of disribuion F, expressed as G, y y,if 0 y exp,if 0 (22) where y = R i - and Λ represens he ail index, while is he scaling parameer. The hreshold value affecs he disribuion of exreme values. Bali [9] se he hreshold as wice he sandard deviaion around he sample mean of he asse value. This sudy, however, followed McNeil and Frey s [6] mehod for deermining he hreshold. For every N (number of daily observaions), le k be he number of poins ha exceed he hreshold. I can hen develop a random hreshold k a he (k + )h order saisic. By ordering he residuals as 2 N, he GPD disribuion can hen be fied o he daa series of excess amoun of residual over hreshold k, 2 k, N k. If he hreshold is large enough o reduce he chance of bias and k is IID and follow a GPD disribuion, he parameers Λ and can be esimaed by he maximum likelihood mehod [26]. The log likelihood funcion is expressed as N i k nlg Nlnln i The ail esimaor is hus given as k (23) k l F( ) N (24) I follow he mehod of McNeil and Frey [6] o ake N = 000 and k = 00 and rank he residuals in ascending order. The residual k is aken as a random hreshold o esimae he excess amoun of he hreshold over he

5 S. Y. M. ZE-TO 229 firs 00 residuals y,, y y. The VaR esimae using he condiional EVT via he NIG + Jump model can hen be compued as k k p VaR pgj, k k k N 2.2. VaR Esimaes Using he Suden- Disribuion and he GARCH Model (25) This sudy followed Lin and Shen s [29] model for esimaing VaR using he Suden- disribuion along wih a GARCH model. The densiy funcion of a non-cenral Suden- disribuion is denoed as v v R q f R (26) v v πv 2 where q and are he locaion and dispersion parameers, respecively. denoes he degrees of freedom, and ( ) is he gamma funcion. A p percen of he Suden- disribuion, he VaR esimae is expressed as VaR p, Sd p, v v (27) where p, represens he corresponding criical -value and is he excess kurosis. The daily VaR quaniles via he GARCH and NIG + Jump models are hen deermined by incorporaing he corresponding variance ino Equaion (29). The process was repeaed on a rolling basis for he enire se of daa. 3. Empirical Analysis 3.. Daa Three sock indices, he Indusrial Average Index (), he Bombay Sock Exchange Sensiive Index (Sensex) and he Thailand Sock Exchange Index (SET), were seleced o es he models performance. Daily observaions range from January, 985 o May 9, To examine he performance of he VaR models in financial crisis evens like he Asian financial crisis, his paper used SET daa from Thailand for examinaion. Table summarizes he saisics for he daily reurns of he hree indices. I shows ha Sensex has he highes sandard deviaion (.85 percen) while S&P has he lowes (. percen). The index represens a developed counry while Sensex and SET represen he emerging markes of Asia. Therefore, hey generally exhibi higher volailiy. In addiion, he disribuions of he hree reurn series are heavily ailed. The kurosis and skewness are relaively higher for he Dow Jones. The Jarque-Bera saisics for he hree indices are excepionally high, which evidences he non-normaliy of hese disribuions Parameer Esimaion The parameer esimaes of he models are deermined by he maximum likelihood mehod, wih he resuls presened in Tables 2 and 3. A window of 000 daily observaions is used o esimae he parameers for nex-day esimaion. The resuls show ha he ail indices Λ for he EVT-based models are posiive. Therefore, he limiing disribuions of he hree indices are of he Fréche ype Empirical Tess Three ses of ess were conduced o measure each model s performance relaed o in-sample fiing, ou-ofsample forecasing and backesing by years. In-sample esing measures how good each model is a fiing he daa. The ou-of-sample sudy, acing as a Table. Summary saisics of he sock index reurns. US: India: SENSEX Thailand: SET Mean 0.09% 0.06% 0.00% Maximum. 08% 7.34% 2.02% Minimum 7. 87%. 4% 4. 84% Sandard Deviaion.26%.75%.77% Skewness Excess kurosis Jarque-Bera Saisic The fiing performances of he models are examined using he maxim um log likelihood value and he Jarque-Bera and AIC values. Jarq ue-bera = (Number of 2 2 observaions) skewness 6 kurosis As sugg esed by Akaike (974), AIC = ML func ion K, where K is he number of parameers. As suggesed by Schwarz (978), Schwarz value = ML funcion number of parameers 2 Ln number of observaions. The number of para meers for he NIG + Jump model is 24.

6 230 S. Y. M. ZE-TO Table 2. Esimaion of parameers of he EVT-NIG + Jump model for he sock index reurns. Panel A (Parameers Par: NIG + Jump). US: India: Sensex Thailand: SET α E E E E E E 06 α 4.42 E 0.65 E E E E E 03 ι 3.47 E E 06.93E E E E 07 ι, 3.47 E E 06.93E E E E 07 ι 2, 3.20 E 08.32E E.9E 2.50E 08 2.E 08 λ 5.82 E E 03.2E E E 03.63E 03 β E 03.04E 04.7E 02.42E E E 09 β 6.64 E E E 03.93E E E 05 λ E E 03.20E E E 03.63E 03 J.40E 05.28E 06.93E 03.23E E E 06 ψ E E 03.95E E E E 07 ψ.38e 03.30E E 04.09E E E 04 ψ 2.75E 0 7.7E E E E 04.73E 07 χ 0.58E E 06 7.E 03.2E 04.04E 04.29E 06 Ψ(.) 2.92E E E 0.74E E E 08 χ 8.62E 02.88E 03.55E E E 0.62E 03 δ 6.63E 04.79E E E E E 05 a 2.38E E E E 09.8E E 09 a j 5.94E 05.20E 06.95E E 0.39E E 09 b 5.5E 05.6E 06.50E 04.85E E 05.98E 08 b j 6.0E 05.22E 06.99E E 0.42E E 09 ξ 6.0E 05.22E 06.99E E 0.8E E 08 c 0.22E E E E 05.36E E 0 c 2.0E 02.82E E E E E 0 L A IC Schwarz Panel B (Parameers Par: EVT) US: India: Sensex Thailand: SET k 5.43E E E 0 3.6E E 0.25E 03 Λ k.96e E 03.03E 0. 59E 03.02E E 03.04E+00.68E E E 03.34E+00.2E 03 k

7 S. Y. M. ZE-TO 23 Table 3. Comparison of he performance of he five models in one-day VaR esimaion in erms of he number of exceedences (in-sample performance es). Panel A Sock Index Expeced GARCH D. NIG + Jump D. GARCH + D. 99.5% level Sensex SET MAD % level Sensex SET MAD % leve l Sensex SET MAD % level Sensex SET MAD Panel B Sock Index Expeced NI G + Jump + D. Cond. EV T- NIG + Jump D. 99.5% level Sensex SET MAD % level Sensex SET MAD % leve l Sensex SET MAD % level Sensex SET MAD 32 62

8 232 S. Y. M. ZE-TO back-es, compares he acual re urn wih he daily VaR forecass hroughou he sample period o evaluae each model s performance in forecasin g VaR esimaes. The backes is analyzed on an annual basis o examine each model s abiliy o capure he dynamics of condiional volailiy of he indices. The five models for comparison are he GARCH, NIG + Jump, GARCH-, NIG + Jump- and EVT-NIG + Jump models In-Sample Performance Each model s parameers were esimaed by he maximum likelihood mehod using he in-sample daa. The VaR quaniles for he corresponding confidence level for each dae are calculaed, and VaR forecass of he five models are hen compared agains he acual daily reurn for each index. Table 3 indicaes he relaive in-sample performance of he five models in one-day VaR esimaion. Since he bes model for VaR measuremen should give he exac number of expeced exceedences, he EVT-NIG + Jump model ouperforms he oher five models by giving he smalles deviaion of is number of exceedences from he expeced figures a boh he 99 percen (one percen of he number of daily observaions in he sample period) and 99.5 percen level for all hree sock indices. The EVT-NIG + Jump model is beer han he condiional EVT model for boh he index and he wo indices of emerging markes. I seems ha he jump process is beer a capuring even losses due o unusual evens. The auoregressive condiional inensiy governing he jump process allows he news feedback on variance from jumps o vary when he previous news is good or bad. This helps he jump-based models o beer capure he non-normaliy when esimaing VaR. I is shown ha he NIG + Jump-based models, such as he NIG + Jump- model, are beer han GARCH wihou he jump model in VaR forecasing. The performances for he condiional EVT-based models are generally beer han hose of he GARCH-based models. The EVT-NIG + Jump model has he lowes MAD among he five models. Comparing he GARCH ype models, i is found ha he NIG + Jump- model yields he bes predicion of VaR for all hree indices. While exhibiing a higher MAD in a lower confidence level, is predicion improves subsanially a higher confidence levels. I has he lowes MAD among he GARCH ype models a boh he 99 and 99.5 percen level. A he 99.5 percen level, he ranking of performance saring from he bes is EVT-NIG + Jump, NIG + Jump-, GARCH-, NIG + Jump, and GARCH. A he 99 percen level, he EVT-NIG + Jump model sill ouperforms he oher models in providing he lowes absolue deviaion from he expeced figures. The MAD is only four. The NIG + Jump- model ouperforms he oher GARCH ype mod- els. From he 98.5 percen o 95 percen level, he EVT- NIG + Jump model sill ranks he b es. I gives he lowes number of absolue deviaions from he expeced figures. The performance of he NIG + Jump model, however, improves a lower confidence levels. I has he lowes MAD in he 98.5 o 97.5 percen level, as compared wih he oher GARCH ype models. I seems ha he combinaion of NIG + Jump and he disribuion leads o an over-predicion of VaR and generaes a higher MAD. I is inferred ha he performance of he EVT-NIG + Jump model improves as he confidence level increases. The condiional EVT series models generally perform beer wih he Index han wih he Sensex and SET, especially a higher confidence levels, due o he larger volailiy clusering and kurosis ha exiss in he Dow Jones. The EVT-NIG + Jump model does a good job in capuring hese non-normaliies Backesing (Ou-of-Sample) Performance The five models were backesed o examine how well he models predic exreme losses in he fuure. This is paricularly imporan for shor-erm marke risk managemen, by evaluaing each model s performance in VaR forecasing. Backesing sars wih a window of 000 previous daily observaions in he sample o esimae he parameers of each model. The esimaes are hen used o derive he one-day VaR forecas for he nex day, and he forecass are compared wih he acual reurn of ha day. The procedure is repeaed for he res of he daily observaions in he sample. The exceedences are couned whenever he acual reurn is lower han he VaR forecass. The resuls are summarized in Table 4. Compared wih he resuls in Table 3, Table 4 generally has higher numbers of exceedences for all models in he ou-of-sample es. A he 99.5 percen level, he order of ranking in erms of he number of exceedences remains he same as ha in Table 3. EVT-NIG + Jump is sill he bes model in VaR esimaion. In addiion, he NIG + Jump- model ouperforms all of he oher GARCH ype models. A he 99 percen level, he EVT-NIG + Jump model sill produces he fewes ouliers in he backesing. From he 98.5 percen o 95 percen level, he EVT-NIG + Jump model remains he bes performer. The ranking is he same as ha in he in-sample es, alhough he number of ouliners increases wih decreasing confidence levels. In addiion, i is shown ha he MAD for boh he NIG + Jump- and GARCH- models increases wih decreasing confidence level. This may aribue o he poorer performance of he Suden- funcion in fiing he acual financial ime series disribuion as he confidence level decreases. The NIG + Jump model can beer forecas unusual news evens or earnings surprises wih

9 S. Y. M. ZE-TO 233 Table 4. Comparison of he performance of he five models in one-day VaR forecass in erms of he number of exceedences (ou-of-sample performance es). Panel A Sock Index Expeced GARCH D. NIG + Jump D. GARCH + D. 99.5% level Sensex SET MAD % level Sensex SET MAD % lev el Sensex SET MAD % lev el Sensex SET MAD Panel B Sock Index Expeced NIG + Jump + D. Cond. EVT- NIG + Jump D. 99.5% level Sensex SET MAD % level Sensex SET MAD % level Sensex SET MAD % level Sensex SET MAD he inclusion of a jump process. The GARCH model, how ever, which ha s linear volailiy sei ngs, is less sensiive o changes in reurn volailiy [30] Uncondiional and Condiional Coverage Tess The uncondiional coverage es and condiional co ver- The uncondiional cove rage es age es were conduced. measures he performance of he VaR models based on he proporion of failures in he sample, while he condiional coverage es ess boh he uncondiional coverage and serial independence. The derivaions of boh he uncondiional and condiional coverage ess are given in he Appendix. Table 5 illusraes he LR saisics of boh he uncondiional and condiional coverage for he alernaive VaR models. For he uncondiional coverage es, he LR saisics for he EVT-NIG + Jump model in all hree indices are significanly less han he criical values (chi-squared wih one degree of freedom) of 5.92 (.5 percen), 6.63 ( percen ) and 7.88 (0.5 percen ), and his model ouperforms he oher five models. Regarding he condiional coverage es, he corresponding criical values of chisquared wih wo degrees of freedom were 8.40 (.5 percen), 9.2 ( percen) and 0.59 (0.5 percen). The EVT-NIG + Jump model achieves low LR saisics. The resuls indicae ha, in general, he exceedences occurring in he condiional EVT-models are independen and idenically disribued Yearly Backesing Bysröm s [4] mehod s adoped o compare he performance of condiional and uncondiional models by dividing he sample period ino year-long sub-periods. Table 6 presens he comparaive performance of VaR forecass of he five models on an annual basis a he 99 and 99.5 percen levels. The resuls show ha boh he NIG-Jump- and EVT- NIG-Jump models generally give a lower MAAD (Mean Average Absolue Deviaion). The EVT-NIG + Jump model generaes accurae VaR esimaes, as expeced by he confidence level. The same siuaion applies o he hree sock indices a he 99 and 99.5 percen levels. The daily VaR esimaes produced by he condiional EVT models vary closely wih changes in volailiy. In paricular, he EVT-NIG + Jump model capures he exreme losses during he period well for boh he Sensex and SET indices. The VaR forecass by he GARCH model, however, are less responsive o changing volailiy. The GARCH model ends o underesimae he risk during urbulen periods. In accessing he performance of he VaR forecass of he five models in he hree crises, his paper measures he MAAD of he models for he periods of ,

10 234 S. Y. M. ZE-TO Table 5. Uncondiional coverage and condiio nal coverage ess for alernaive VaR models. Panel A Condiional EVT NIG + Jump NIG + Jump- GARCH- Sock Index LR_UC LR_CC LR_UC LR_CC LR_UC LR_CC Level: 0.5% Sensex SET Level: % Sensex SET Level:.5% Sensex SET Panel B NIG + Jump GARCH Sock Index LR_UC LR_CC LR_UC LR_CC Level: 0.5% Sensex SET Level: % Sensex SET Level:.5% Sensex SET and These periods include he devaluaion o f sock marke values in Asian markes in 997, he echnical bubble burs in 2000 and he collapse of invesmen banks in I is shown ha boh he NIG-Jump- and EVT-NIG + Jump model perform significanly well in capuring exreme losses. In paricular, he EVT-NIG + Jump model gives he lowes average MAAD in he Global Financial Crisis a 99 percen level. I is inferred ha he EVT model incorporaed wih he jump process is more responsive o sudden jumps in sock price and is b eer a capuring exreme price movemens during a crash period. 4. Conclusions This paper proposes a condiional EVT-based model ha incorporaes he NIG + Jump process for VaR esimaion. I explores he possibiliy of improving he EVT-based model in esimaing and forecasing VaR for exreme

11 S. Y. M. ZE-TO 235 Table 6. Annual backesing performance in one-day VaR forecass of he five models a 99.5% and 99% levels (ouof-sample es). Panel A: Level 99.5% GARCH- GARCH NIG + Jump- NIG + Jump EVT-NIG + Jump Yr 3 Mks Avg. 3 Mks Avg. 3 Mks Avg. 3 Mks Avg. 3 Mks Avg Till 22/5/ MAAD Avg Panel B: Level 99% GARCH- GARCH NIG + Ju mp- NIG + Jump EVT-NI G + Jump Yr 3 Mks Avg. 3 Mks Avg. 3 Mks Avg. 3 Mks Avg. 3 Mks Avg Till 22/5/ MAAD Average

12 236 S. Y. M. ZE-TO evens. The EVT-NIG + Jump model includes an auoregressive jump componen in valuing he condiional variance o feedback previous jump innovaions ino he expeced volailiy. This feaure helps he model perform beer around crisis periods. The findings indicae ha he EVT-NIG + Jump model perform s well especially a high confidence levels in boh he in-sample and ou-of-sample ess. In he insample es, he EVT-NIG + Jump model ouperforms all of he alernaive models in scapuring he heavy ail and skewness of he reurn disribuion of he hree indices sudied. The resuls show ha he VaR model developed in he EVT-NIG + Jump framework is more robus in racking he occurrences of exreme losses in emerging sock markes, such as hose in India and Thailand. In he ou-of-sample es, he EVT-NIG + Jump model again performs well for one-day VaR forecass. The improvemen became more significan when a higher confidence level VaR forecas (99 percen or above) was used. The NIG + Jump seing incorporaed in he proposed model helps o beer capure he skewness and heavy ail resuling from an unexpeced exreme price jump. Addiionally, he performance of he condiional NIG + Jump model in VaR forecass improves as he confidence level increases. The model produces excepionally accurae VaR forecasing a he 99.5 percen level. Regarding he year-by-year backesing, he condiional EVT-NIG + Jump model performs well and ouperforms he ohers in capuring he dynamics of he marke condiion. REFERENCES [] P. Arzner, F. Delbaen, J. Eber and D. Heah, Thinking Coherenly, Risk, Vol. 0, No., 997, pp [2] P. Arzner, F. Delbaen, J. Eber and D. Heah, Coheren Measures of Risk, Mahemaical Finance, Vol. 9, No. 3, 999, pp doi:0./ [3] T. G. Bali and P. Theodossiou, A Condiional-SGT-VaR Approach wih Alernaive GARCH Models, Annals of Operaions Research, Forhcoming, Vol. 5, No., 2005, pp [4] F. M. Longin, Opimal Margins Level in Fuure Markes: A Parameric Exreme-Based Mehod, Journal of Fu- of ures Markes, Vol. 9, No. 2, 999, pp [5] S. C. Coles, An Inroducion o Saisical Modeling Exreme Values, Springer, London, New York, 200. [6] F. M. Longin, From Value a Risk o Sress Tesing: The Exreme Value Approach, Journal of Banking & Finance, Vol. 24, No. 7, 24, 2000, pp doi:0.06/s (99) [7] J. Coer, Downside Risk for European Equiy Markes, Applied Financial Economics, Vol. 4, No. 0, 2004, pp doi:0.080/ [8] T. G. Bali, A Generalized Exreme Value Approach o Financial Risk Measuremen, Journal of Money, Credi and Banking, Vol. 39, No. 7, 2006, pp [9] T. G. Bali, An Exreme Value Approach o Esimaing Volailiy and Value-a-Risk, Journal of Business, Vol. 76, No., 2003, pp doi:0.086/ [0] T. G. Bali and S. N. Nefci, Disurbing Exremal Behavior of Spo Rae Dynamics, Journal of Empirical Finance, Vol. 0, No. 4, 2003, pp doi:0.06 /S (02) [] J. Beirlan, J. Teugels and P. Vynckier, Pracical Analy- Veriag, New York, 997. sis of Exreme Values, Leuven Universiy Press, Leuven, 996. [2] P. Embrechs, C. Klüppelberg and T. Mikosch, Modelling Exremal Evens for Insurance and Finance, Springer- [3] R. Reiss and M. Thomas, Saisical Analysis of Exreme Values: From Insurance, Finance, Hydrology, and Oher Fields, Birkhäuser Verlag, Basel, Boson, 200. [4] H. N. E. Bysröm, Managing Exreme Risks in Tranquil and Volaile Markes Using Condiional Exreme Value Theory, Inernaional Review of Financial Analysis, Vol. 3, No. 2, 2004, pp doi:0.06/j.irfa [5] R. Smih, Measuring Risk wih Exreme Value Theory, Exremes and Inegraed Risk Managemen, Vol. 2, 996, pp [6] A. J. McNeil and R. Frey, Esimaion of Tail-Relaed Risk Measures for Heeroscedasic Financial Time Series: An Exreme Value Approach, Journal of Empirical Finance, Vol. 7, No. 3-4, 2000, pp doi:0.06/s (00) [7] T. G. Bali and D. Weinbaum, A Condiional Exreme Value Volailiy Esimaor Based on High-Frequency Reurns, Journal of Economic Dynamics and Conrol, Vol. 3, No. 2, 2007, pp [8] C. Brooks, A. D. Clare, J. W. D. Molle and G. Persand, A Comparison of Exreme Value Theory Approaches for Deermining Value a Risk, Journal of Empirical Finance, Vol. 2, No. 2, 2005, pp doi:0.06/j.jempfin [9] R. F. Engle, Auoregressive Condiional Heeroscedasiciy wih Esimaes of he Variance of UK, Economerica, Vol. 50, No. 4, 982, pp doi:0.2307/92773 [20] T. P. Bollerslev, Generalized Auoregressive Condiional Heeroscedasiciy, Journal of Economerics, Vol. 3, No. 3, 986, pp doi:0.06/ (86) [2] S. Taylor, Modelling Financial Time Series, In: H. Akaike, Ed., A New Look a he Saisical Model Idenificaion, John Wiley & Sons, New York, 986. [22] R. Susmel, Swiching Volailiy in Privae Inernaional Equiy Markes, Inernaional Journal of Finance & Economics, Vol. 5, No. 4, 2000, pp doi:0.002/099-58(20000)5:4<265::aid-ijfe32> 3.0.CO;2-H [23] J. M. Mahe, and T. H. McCurdy, News Arrival, Jump

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