Forecasting Value-at-Risk Using the. Markov-Switching ARCH Model

Transcription

1 Forecasing Value-a-Risk Using he Markov-Swiching ARCH Model Yin-Feng Gau Wei-Ting Tang Deparmen of Inernaional Business Naional Chi Nan Universiy Absrac This paper analyzes he applicaion of he Markov-swiching ARCH model (Hamilon and Susmel, 1994) in improving value-a-risk (VaR) forecas. By considering a mixure of normal disribuions wih varying variances over differen ime and regimes, we find ha he spurious high persisence found in he GARCH model is adjused. Under relaive performance and hypohesis-esing evaluaions, he VaR forecass derived from he Markov-swiching ARCH model are preferred o alernaive parameric and nonparameric VaR models ha only consider ime-varying volailiy. JEL classificaion: C, C5, G8. Keywords: Value-a-Risk, Swiching-regime ARCH models. Corresponding auhor. Address: Deparmen of Inernaional Business, Naional Chi Nan Universiy, 1 Universiy Rd., Puli, Nonou 545, Taiwan. Tel: ex. 491, Fax: , yfgau@ncnu.edu.w.

2 1. Inroducion Wih he increasing flucuaions in asses prices and severe financial urmoil occurred recenly, he issue of risk managemen has received considerable aenions recenly. Since is adopion by he Basel Commiee (Basel Commiee on Banking Supervision of Bank for Inernaional Selemens, 1996), value-a-risk (VaR) has become one of he mos widely used ools for measuring he marke risk by major rading insiuions. VaR is used o quanify he exposure of a porfolio o fuure marke flucuaions. The purpose of his paper is wofold. Firs, we consider approaches ha allow for he lepokurosis in he disribuion of he porfolio reurn. Since assuming normaliy in calculaing VaR will resul in a sysemaic under-esimaion of he riskiness of he porfolio, especially when reurns are heavily fa-ailed. To capure he lepokurosis many researchers use he GARCH model of Bollerslev (1986) o generae volailiy forecas (Duffie and Pan, 1997). 1 However, GARCH forecass are oo high in volaile periods. Hamilon and Susmel (1994) argue ha he problem of spurious persisence can be solved afer considering regime swiches in he volailiy. Using he Markov-swiching ARCH (SWARCH) model proposed by Hamilon and Susmel (1994), we forecas VaR allowing for regime swiches in ime-varying condiional variance of reurns. Second, we evaluae VaR forecass sysemaically hrough relaive performance comparison and hypohesis ess on forecas accuracy. While he concep of VaR is simple and aracive, here is no unique approach wih VaR implemenaions adop. Because a wide variey of alernaive models are used in VaR 1 I can be seen ha a mixure of normal disribuions wih differen variances will lead o an overall series ha is lepokuric (Duffie and Pan, 1998). Cai (1994) also proposed a Markov-ARCH model o incorporae he feaures of boh Hamilon s (1989) swiching-regime model and Engle s (198) ARCH model. Since boh models of Cai (1994) and Hamilon and Susmel (1994) aim o inegrae Markov Swiching model and ARCH model, and he wo Markov-swiching ARCH models are relaed in paramaerizaion (see Cai (1994)), we only esimae he model of Hamilon and Susmel (1994) in his paper wihou loss of generaliy. 1

3 implemenaions, i is essenial o use sysemaic evaluaion crieria in selecing a preferred VaR model. This paper underakes four case sudies in model evaluaion, including he S&P 500 index, Nikkei 5 index, FTSE 100 index and CAC 40 index, a he 95% and 99% levels significance. The empirical resuls show ha he SWARCH model can solve he problem of spurious high perisience found in he GARCH model and yield a beer forecas of VaR. 3 The evaluaion resuls indicae ha SWARCH-based VaR forecass are generally more accurae han hose generaed by models ha only consider ime-variaion in he condiional volailiy, including he EWMA (exponenial weighed smoohing average), hreshold GARCH (TGARCH) mehods and he hisorical simulaion adjused for ime-varying variance. This paper is organized as follows. Secion inroduces he evaluaion framework for VaR forecass. Secion 3 describes he differen models used o derive VaR forecass. Secion 4 compares he resuls of he empirical invesigaion of compeing models on S&P 500, FTSE100, Nikkei5 and CAC40 indices reurns. Secion 5 concludes.. Evaluaion of VaR Forecass.1 Definiion of VaR The concep of VaR is o summarize he wors loss over a arge horizon wih a given level of confidence. VaR is defined as he maximum loss on a porfolio ha can be expeced wih a cerain level of confidence (1-α) over a cerain inerval of ime (T), and can be expressed as: 3 Besides he SWARCH model, here are alernaive mehods o incorporae boh srucural change and ime-varying sochasic volailiy o solve he problem of he excessive GARCH forecass in volaile periods. For example, Gray (1996) and Ang and Bekaer (00) exend he specificaion of SWARCH o he Markov swiching GARCH model. However, for he regime-swiching GARCH specificaion one is unable o compue he muli-period ahead volailiy forecass.

4 ( VaR ) = α Pr (1) r, + T <, + T where r + T, represens he porfolio reurns over he T periods in he fuure, ha is, P P r,, where P is he value of porfolio a ime. T T = + + P. Evaluaion Mehods of VaR Forecass VaR models are only useful when hey predic risk reasonably well. To compare various VaR forecass, we mus check sysemaically he validiy of he evaluaion of VaR models hrough he comparison of prediced and acual loss levels. When a VaR model is perfecly calibraed, he number of realized observaions falling ouside VaR predicion should be in line wih he confidence level. Wih oo many excepions ha exceed he esimaed VaR, i means ha he model underesimaes he risk. This is a major problem because oo lile capial maybe allocaed o risk-aking unis. Wih oo few excepions is also a problem because i leads o excess or inefficien allocaion of capial. Recenly, here is a rapidly growing lieraure on he evaluaion of VaR models. One ype of mehods judges he beer VaR forecass based on he relaive performance derived from some loss funcions. The oher offers a esing framework based on cerain heoreical properies of he VaR measures. A key issue abou evaluaion based on he hypohesis-esing framework is he power of es. If he hypohesis ess exhibi low power, he probabiliy of classifying an inaccurae VaR model as accepably accurae will be high...1 Relaive Performance The cenral concep of hese mehods is o compare among VaR models and selec he mos accurae one. Hendricks (1996) proposed several crieria o examine differen VaR measures. He emphasized ha hese considered performance crieria do no have sraighforward sandard error ha i is no possible o discriminae beween 3

5 mehods using formal saisical hypohesis. Neverheless, hese crieria provide a relaively complee picure of he performance of seleced VaR esimaes. Lopez (1998) proposed a measure of relaive performance ha can be used o monior he performance of VaR esimaes. The general form of a loss funcion is ( i) ( i) f ( r, + T, VaR, + T ) if r, + T < VaR, + T C i, T = () ( i) ( i) g( r, + T, VaR, + T ) if r, + T VaR, + T where C i T, represens he numerical scores generaed for individual VaR model i, and r + T, represens he porfolio reurns over he T days in he fuure. The score for he complee regulaory sample of size h is + C i = C i, T. Once a loss funcion is = s + 1 defined and C i is calculaed, a benchmark can be consruced and used o evaluae he performance of a se of VaR forecass. In his paper, we apply he following five crieria o evaluae he relaive performance of various VaR forecass. (1) Mean Relaive Bias (MRB) MRB examines wheher differen VaR models produce similar forecass. We firs calculae VaR under each VaR models on each sample dae, and hen compue he average VaR over he forecas sample. Given h forecasing periods and N VaR models, he MRB of model i is compued as: s h MRB i 1 = h VaR VaR s + h ( i), + T = s+ 1 VaR, + T, + T N 1 ( i, where VaR, + T = VaR, N i = 1 ) + T (3) () Roo Mean Square Relaive Bias (RMSRB) RMSRB examines he degree o which cerain VaR measure varies from he average risk measure for a given dae. I capures wo effecs: he exen o which he average risk esimae provided by a given model sysemaically differs from he average risk measure, and he variabiliy of each model s risk esimae. The RMSRB is compued as: 4

6 1 ( ),, + i s h VaR + T VaR N + T = 1 RMSRB i, where VaR, + T = h = s+ 1 VaR, + T N i = 1 VaR ( i), + T (4) (3) Correlaion beween Risk Measure and Absolue Value of Oucome A simple efficiency es is o measure he correlaion beween calculaed VaR and he absolue value of realized reurn. I assesses how well he risk measures adjus over ime o underlying changes in risk. This correlaion saisic has wo advanages. Firs, i is no affeced by he scale of he porfolio. Second, he correlaions are relaively easy o inerpre. (4) Binary Loss Funcion The loss funcion implied by he binomial mehod is ( i) 1 if r, + T < VaR, + T C i, T = (5) ( i) 0 if r, + T VaR, + T If a loss exceeding he VaR is observed, his is ermed an excepion. Here, we are simply concerned wih he number of excepions raher han he magniude of hese excepions. If a VaR model is ruly providing he level of coverage defined by is confidence level, he score for he complee regulaory sample + C i = C i, T will = s + s h 1 equal 0.05 h and 0.01 h for he 95 h percenile VaR and he 99 h percenile VaR, respecively. (5) Quadraic Loss Funcion Quadraic loss funcion akes accoun of he magniude of he excepions. Comparing wih a binary loss funcion, an addiional quadraic erm is imposed when an excepion occurs. Lopez (1998) found ha he use of he addiional informaion embodied in he size of he excepion provides a more powerful measure of model accuracy han he binary loss funcion. The loss funcion is defined by: 5

7 ( i) ( r VaR ) ( i) 1+ <, + T, + T if r, + T VaR, + T C i, T = (6) ( i) 0 if r, + T VaR, + T In he same manner, we compue he score of he quadraic loss funcion as s + h = s + 1 C i = C i, T. When he score of he binary loss funcion is similar under differen models, he quadraic loss funcion goes in deph o examine he magniude of hese excepions... Hypohesis-Tesing Framework Evaluaion mehods based on a hypohesis-esing framework allow us o es he null hypohesis ha VaR forecass are accepably accurae. The null hypohesis is ha VaR forecass in quesion exhibi a specified propery or characerisic of accurae VaR forecass (Lopez, 1998). If he null hypohesis is rejeced, he VaR forecass do no exhibi he specified propery, and he underlying VaR model can be said o be inaccurae. If he null hypohesis canno be rejeced, he model is said o be accepably accurae. Kupiec (1995) is he firs one o develop he performance-based verificaion echniques o es he accuracy of VaR forecass. He consruced VaR verificaion ess from he series of Bernoulli rial oucomes generaed by a daily performance comparison. Tha is, rea he loss on rading aciviies less han he VaR esimaed as a success, and beyond he VaR as a failure. According o his assumpion, he derived he TUFF (Time Unil Firs Failure) and PF (Proporion of Failures) ess. In a performance-based verificaion scheme, he iniial monioring saisic of ineres is he number of observaions unil a failure is observed. Kupiec (1995) defined T ~ as a random variable ha denoes he number of days unil he firs failure is recorded. If p is he probabiliy of a failure on any given day, he probabiliy of observing he firs failure in period V is given by: 6

8 ~ Pr( V 1 T = V ) = p(1 p) (7) where T ~ has a geomeric disribuion wih an expeced value of (1/p). For example, when p = 0. 01, he average ime unil he firs failure is 100. Given a realizaion for T ~, he likelihood raio (LR) saisic for esing he null hypohesis LR(V, p { * * V 1 } { V p (1 p ) + Log (1/ V)(1 1/ V 1 } * ) Log ) * p = p is given : = (8) * Under he null hypohesis, LR(V, p ) has a chi-square disribuion wih 1 degree of freedom. According o Kupiec (1995), when esing p * = 0. 01, he TUFF(0.05) criical values for V are 6 and 439. Tha is, if he firs failure occurs before he sevenh rading day, i can be concluded ha p > If he firs failure occurs afer he 438 h rading day, i can be concluded ha p < Ye i has been suggesed ha he TUFF saisics has poor abiliy o disinguish reliably beween alernaive underlying values for he ail probabiliy associaed wih a VaR forecas. The PF es is used o compare he oal number of failures observed o he oal accumulaed sample size. The PF es is based on he proporion of failures in he sample. When he TUFF es canno rejec he null hypohesis, coninued monioring beyond an observed failure will clearly add informaion ha can be used o verify poenial loss esimaes. The probabiliy of observing x failures in he sample of size h is: h x x Pr( h, x) = (1 p) p ~ binomial( h, x) (9) where p is he probabiliy of a failure on any one of he independen rails. The LR saisic is given by: { h x x } { [ ] h p *) ( p*) + log (1 x / h ) x ( x / h) x } log (1 (10) Under he null hypohesis, p = p *, he PF es has a chi-square disribuion wih 1 degree of freedom. In a daily monioring scheme, he PF es is used o compare he oal number of failures observed o he oal accumulaed sample size. Like he TUFF 7

9 es, he PF es has poor power in small samples. Kupiec (1995) concluded ha sample performance-based VaR verificaion ess require large samples o produce a reliable accuracy assessmen. The Basel rules for backesing he inernal models approach are derived direcly from his failure rae es. The Basel Commiee has decided ha up o four excepions are accepable, which defines a green zone for he bank. If he number of excepions is more, he bank falls ino a yellow or red zone and incurs a progressive penaly. 3. Calculaion of VaR Forecass To calculae VaR, we need furher informaion regarding he disribuion of fuure reurn r + T,. Since VaR is equal o he appropriae quanile of he disribuion of fuure porfolio reurns, he ask of VaR calculaion is o esimae he quanile. By focusing on he quanile or exreme value direcly, several approaches have employed he quanile regression or he exreme value analysis o calculae VaR direcly, including Engle and Manganelli (1999), and Longin (000) among ohers. 4 On he oher hand, several approaches esimae he full disribuion of porfolio reurns and hen calculae he corresponding quanile as VaR. Depending on he parameerizaion, approaches o calculaing VaR via he whole disribuion can be characerized as parameric and nonparameric mehods. The parameric, or namely he variance-covariance or facor approach, involves specifying a parameric disribuion and esimaing he parameers wih hisorical daa. Based on he esimaed disribuion, ofen assuming normaliy, one can calculae he appropriae quanile eiher 4 For example, Engle and Manganelli (1999) proposed he CAViaR (Condiional Auoregressive Value-a-Risk) model o sudy he evoluion of he quanile over ime. They specified a special ype of auoregressive process for he condiional quanile. One disadvanage of his model is ha i requires he specificaion of a dynamic equaion for he condiional quanile and is validness is subjec o misspecificaion errors. Insead of underaking he approach of quanile regression or exreme value analysis, we consider an appropriae model o model he condiional volailiy of reurns, allowing for jumps or regime changes and ime-varying volailiy a he same ime. 8

10 analyically or numerically. On he oher hand, he nonparameric or porfolio approach involves consrucing or simulaing he disribuion of porfolio reurns ha mimic he pas performance of he porfolio. 3.1 Parameric Models Parameric models are he mos popular models for calculaing VaR, and he normaliy of reurns is usually assumed. Under he assumpion of normaliy of daily porfolio reurns, ~ N ( µ ( p), σ ( )), where µ ( p) and σ ( p) are he mean and r p, p variance of r, respecively, he value of VaR can be calculaed by a muliple of he p sandard deviaion of he porfolio reurns. Tha is, VaR +, + T = Cα σ ( p), T (11) where C α is he consan ha gives he appropriae one-ailed confidence inerval, a he ( 1 α ) confidence level, for he sandard normal disribuion, while σ ( ), T is he sandard deviaion of porfolio reurns over he chosen ime horizon, T Time-Varying Volailiy p + When implemening he parameric mehods o obain VaR forecass, we need o forecas σ ( a firs. While he assumpion of normaliy simplifies he p ), + T calculaion of VaR, i may lead o an inaccurae VaR. If porfolio reurns are lepokuric, 5 he normal disribuion will significanly underesimae he likelihood of exreme reurns, and so he esimaed VaR of he porfolio will generally be oo low. One cause of lepokurosis in he uncondiional disribuion of reurns is volailiy clusering or ime-varying volailiy. Duffie and Pan (1997) idenified he 5 The daily changes in many variables exhibi significan amouns of posiive excess kurosis (Hull and Whie, 1998). Duffie and Pan (1997) found ha S&P 500 daily reurns for 1986 o 1996 have an exremely high sample kurosis of 111, while he kurosis of a normal disribued shock is 3. These fa ails are paricularly worrisome precisely because VaR aemps o capure he behavior of he porfolio reurn in he lef ail. In his siuaion, a model based on a normal disribuion would underesimae he proporion of ouliers and he rue VaR. 9

11 empirical volailiy of hisorical daa is changing over ime in some persisen manner. As Engle (198) suggesed, if reurns are normally disribued wih ime-varying condiional variance, hen he uncondiional disribuion of reurns will have ails ha are faer han hose of he normal disribuion. To allow for ime-varying volailiy, he parameric approach is ypically modified wih a model for he condiional variance of reurns 6, such as an exponenially weighed moving average (EWMA) or generalized condiional heeroskedasic (GARCH) model (Bollerslev, 1986). Boh models specify he curren variance of reurns as a funcion of he lagged variance and lagged squared reurns. The RiskMerics model (J.P. Morgan/Reuers (1996)) proposes he exponenially weighed moving average (EWMA) model o esimae σ ( : p ), + T σ = (1) ( 1 λ) r 1 + λσ 1 where λ is he decay facor which is chosen arbirary by used and is usually aken he value of 0.94 for daily daa. The developmen of volailiy models for measuring and forecasing volailiy dynamics began wih he ARCH model proposed by Engle (198). The ARCH model is useful o esimae he variance of r condiional on Ω 1, he informaion se available a ime -1. The ARCH (q) model is wrien as: q σ = w + α ε (13) i = 1 i i where ε = r E(r Ω 1), ε Ω 1 ~ N(0, σ ), and E ( r Ω 1) is he condiional mean of r. 6 An alernaive approach o obaining volailiy forecass is he implied volailiy approach. The implied volailiy is derived from maching rading prices of opions and an opion pricing formula, for example, Black and Scholes (1976). The implied volailiy reflecs he marke opinion on he volailiy of asse reurns. However, his approach requires more inpus han he hisory of reurns. Therefore, we only discuss volailiy models ha only require pas reurn. 10

12 Bollerslev (1986) exended he ARCH model o he GARCH model. The GARCH model assumes ha he condiional variance depends on he laes innovaion bu also on previous condiional variance. A GARCH model is he more general form for esimaing volailiy. The represenaion of he GARCH (p,q) model is: q p = + + w αiε i i = 1 i = 1 σ β σ (14) i i where ε = r E(r Ω 1), ε Ω 1 ~ N(0, σ ). The condiional variance equaion is a funcion of hree erms: he mean c, news abou he volailiy from he previous period, measured as lagged squared residual from he mean equaion ε i, and pas condiional variance σ i. To ensure he posiive variance and saionariy, i requires ha w > 0, αi 0, i = 1,, K, q, βi 0, i = 1,,..., p, and α i + β i < 1 q p i = 1 i= 1. For he model specificaion, (r Ω ) is he condiional mean and could be modeled as an E 1 AR or MA process when reurns are auocorrelaed. Alexander and Leigh (1997) examined he performance of hree volailiy models: he equally weighed moving average of squared reurns, he exponenially weighed moving average, and GARCH models. They concluded ha GARCH models give more conservaive risk capial esimaes, which can more accuraely reflec a 1% value a risk measuremen. However, for equiies, i is ofen observed ha downward movemens in he marke are followed by higher volailiies han upward movemens of he same magniude. To accoun for his phenomenon, he TGARCH (hreshold GARCH) models allows for asymmeric impacs of shocks on curren volailiy. The specificaion of he TGARCH(1,1) model, suggesed by Glosen, Jagannahan, and Runkle (1993), is wrien as: 11

13 σ w αε β σ δ d ε (15) = where ε = r E(r Ω 1), ε Ω 1 ~ N (0, σ ), d 1 = 1 if ε 1 < 0, and d 1 = 0 oherwise. In his model, good news (when ε 1 0) and bad news (when ε 1 < 0 ) have differen effecs on curren condiional variance. By he definiion, he impac of good news is α, while he impac of bad news is ( α + δ ). If δ > 0, we say ha a leverage effec exiss in ha bad news increases volailiy. The relaion δ = 0 implies ha he news impac on he curren condiional variance is symmeric. On he oher hand, as noed in Duffie and Pan (1997), one possible source of fa ails is jumps, or significan unexpeced disconinuous changes in prices. The jump diffusion model has been reaed as a recipe for fa-ailed disribuions. The major implicaion of he jump diffusion model for exreme loss shows up much farher ou in he ail. To consider boh he ime-varying volailiy and possibiliy of jumps in he volailiy process, we esimae he class of Markov-swiching models for he ime-varying volailiy, namely Markov-swiching ARCH (SWARCH) model, proposed by Hamilon and Susmel (1994), o allow for boh ime-variaion and regime swiches in he condiional volailiy Time-Varying Volailiy and Regime Swiches GARCH forecass are usually oo high, especially in periods of high volailiy. This is due o he high degree of persisence implied from he GARCH model. The problem of spuriously high persisence resuls in he weak forecasing performance, since he impacs of shocks usually do no las for such a long period. As poined by Hamilon and Susmel (1994), he spuriously high persisence migh be relaed o srucural changes in he variance process. The volailiy forecas will be less persisen if we model changes in parameers hrough a Markov-swiching process, as shown in Hamilon and Susmel (1994) and Cai (1994) among ohers. 1

14 The SWARCH models proposed by Hamilon and Susmel (1994) allow he volailiy dynamics o change under differen saes or regimes. Tha is, parameers in he ARCH(q) process are allowed o be changed in differen saes. Sae variable s indicaes he sae ha he process is in a ime and i is assumed o follow a Markov chain. Tha means he probabiliy of sae s = j will be affeced by only he realized sae in he las period: Pr( s = j Ω 1 ) = Pr( s = j s 1 = i) = p ij We denoe σ ~ SWARCH(K,q) if and only if σ follows a K saes, q-h order Markov-swiching ARCH process. The model can be wrien as σ = σ~ (16) gs q ~ σ = w + α ε (17) i = 1 i i where ε = r E(r Ω 1), ε Ω 1 ~ N (0, σ ) as before. Tha is, σ ~ ARCH(q). g s is he muliplicaive facor ha depends on he sae ~ s. Under his model, ~σ is muliplied by he consan g 1 when he process is in he sae 1 or s = 1, muliplied by g when s =, and so on. An exension of he SWARCH model is he SWARCH-L model ha capures he leverage effec as he specificaion of a hreshold ARCH model. The process of condiional volailiy becomes σ = σ~, gs q ~ = w + αiε i + δ d 1 ε 1 i = 1 σ (18) where ε = r E(r Ω 1), ε Ω 1 ~ N (0, σ ), d 1 = 1 if ε 1 < 0, and d 1 =0 oherwise. For parameers esimaion and forecass calculaion, please refer o 13

15 Hamilon and Susmel (1994) for more deails. 3. Nonparameric Models Nonparameric models are independen from he parameerized disribuion of asses reurns or marke facors reurns. One of which is he hisorical simulaion mehod. 7 The mehod was proposed iniially by Efron (1979) as a nonparameric randomizaion echnique ha consrucs he empirical disribuion by drawing from he observed disribuion of he daa. I simply requires relaively few assumpions abou he saisical disribuions of he underlying marke facors because i assumes ha marke prices innovaions in he fuure are drawn from he same empirical disribuion as hose marke price innovaions generaed hisorically. Insead of esimaing parameers, such as he sandard deviaion, he mehod of hisorical simulaion simply uses he acual perceniles of he observaion period as VaR measures. This mehod involves creaing a daabase consising of he daily movemens in all marke variables over a period of ime. If we assume ha he reurns in he nex day are simply associaed wih he period of hisorical observaions, we could direcly rank he observed hisorical reurns, and apply hese ranked hisorical reurns o consruc he disribuion of reurn in he nex period. 8 The mehod of hisorical simulaion requires no parameer in esimaing he empirical disribuion. However, if he fuure disribuion of marke facors differs 7 Sress esing is anoher kind of nonparameric models. The goal of sress esing is o idenify unusual scenarios ha would no occur under sandard VaR models. In some sense, sress esing can be viewed as an exension of he hisorical simulaion mehod a increasingly higher confidence level. (Jorion, 000) On he oher hand, he Mone Carlo simulaion mehod is an alernaive approach of parameric models. The mehod is used o simulae a variey of differen scenarios for he porfolio value on he arge dae by generaing random draws for he risk facors from a predeermined disribuion. In a Mone Carlo simulaion, one chooses a saisical disribuion ha is believed o adequaely approximae he possible changes in he marke facors. Then, a pseudo-random number generaor is used o generae housands (or perhaps ens of housands) of hypoheical changes in he marke facors. These hypoheical changes are used o consruc housands of hypoheical porfolio reurns on he curren porfolio and he disribuion of reurns. Finally, he VaR is compued from his disribuion. 8 For example, suppose ha 1,000 days of daa are used and he 1 percenile of he disribuion is required. VaR would be esimaed as he enh wors change in he porfolio value. 14

16 subsanially from he hisorical disribuion, compued resuls can be misleading. Hull and Whie (1998) modified he mehod of hisorical simulaion using an adjusmen on he variance. Insead of using he acual hisorical percenage changes in marke variables o calculae VaR, hey used hisorical changes ha have been adjused o reflec he raio of he curren daily volailiy o he daily volailiy a he ime of he observaion. Le r be he hisorical percenage change in he price on day, a period covered by he hisorical sample N (ha is, < N ); σ be he hisorical esimae of he variance of reurn for day. Then he mos recen esimae of he daily variance is σ N, he variance esimae made a he end of day N-1. Assuming he process of r / σ is saionary, hen he adjused r, * r, is given by: r r * = σn, where σ σ can be he esimaed volailiy from he hisorical daa. In his paper, we use he TGARCH (hreshold GARCH) model o esimae he volailiy for adjusing he hisorical observaions. 4. Empirical Resuls The daa sudied in his paper are reurns o major sock indices, including S&P 500, FTSE 100, NIKKEI 5, and CAC 40 indices. The daily daa are colleced from he Daasream and cover he period from January 1990 hrough December 00. We calculae daily log reurns by aking he difference of log prices for each index. Table 1 repors he descripive saisics of hese sock index reurns. I shows ha values of sample mean are close o 0. Values of sample kurosis lie beween and The kurosis is higher han 3, he kurosis of a normal disribuion, which shows ha disribuions of index reurns exhibi fa ails. By he Jarque-Bera saisic, 15

17 he null hypohesis of normal disribuions is also rejeced for all four reurns. We also deec weak firs-order auocorrelaion in reurns o four indices. Values of he Ljung-Box Q saisic sugges he exisence of significan serial correlaion in reurns and squared reurns from he four indices. By he phenomenon of auocorrelaed squared reurns, we see ha he daa exhibi he characerisic of volailiy clusering. The number of observaions for each index reurn in his sudy is We use he las 500 observaions for ou-of-sample forecasing. The esimaion procedure ha we apply is as follows. For each model, 891 observaions of daily daa are used in esimaion, and used o form a VaR forecas for day 89. Afer his, daa from day unil 89 is used in esimaion o obain a VaR forecas for day 893. For each of he VaR models compeing in his paper, 500 ou-of-sample forecass are generaed recursively by moving he esimaion-window forward hrough ime. Assuming ha he condiional disribuion of reurns is normal, we can obain VaR via he formula: VaR, + T = C σ r, r + T α, where C α is a muliplicaive facor ha depends on he confidence level ( 1 α ) of a normal disribuion, and σ, + T is he sandard deviaion of reurns over T periods. For he simples case, we se T=1. Since VaR α, we calculae VaR wih V ˆaR, + 1 = Cα ˆ σ + 1, where, + 1 = C σ, + 1 = Cα σ + 1 ˆ σ and ˆ σ 1 is he forecas of Var ( r 1 ) condiional on informaion + 1 = ˆ σ available a dae. In he EWMA model, ˆ σ = (1 λ) r λσ, we se λ o 0.94 as J.P. Morgan suggess. Afer forecasing he variance, daily VaR is compued as σ ˆ for he 95%, and σ ˆ for he 99% VaR. However, he Ljung-Box Q saisics for he squared sandardized reurns show ha he squared sandardized reurns are sill 16

18 auocorrelaed when we use he EWMA o esimae he ime-varying volailiy. This indicaes ha he EWMA does no capure he ime-varying volailiy well enough. On he conrary, we find GARCH (1,1) is sufficien o capure he volailiy clusering, and he leverage effec is significan. To capure he leverage effec in index reurn volailiy, we use he TGARCH (1,1) model o forecas he volailiy. Table repors he esimaion resuls of he TGARCH(1,1) models for observaions 1 o 891. I indicaes ha, in he TGARCH (1,1) model, he persisence ha can be 1 measured by he sum of ARCH and GARCH parameers, i.e., ( α1 + β1 + δ ), is close o uniy for each series. This may indicae spurious high volailiy persisence. Furhermore, we use he saisic of CUSUM of squares o roughly examine if here migh be srucural changes in he variance process. If he hypohesis of no srucural change fails o accep, i implies ha srucural changes may exis and ha regime-swiching models are appropriae for esimaing he volailiy process. In his paper, we use he CUSUM of squares es as a simple diagnosic for he sabiliy of he variance process. As wih he CUSUM of squares es, movemen ouside he criical lines is suggesive of parameer or variance insabiliy. According o Figure 1, he resuls of CUSUM-squares saisic sugges ha, for each index reurn, here exiss variance insabiliy. 9 I implies ha he spuriously high persisence migh be relaed o srucural changes. Therefore, we furher se a wo-regime SWARCH model o esimae and forecas he volailiy. 10 Table 3 repors he esimaion resuls of he SWARCH-L model using 9 A formal saisical es for he null hypohesis of no-regime swiching has been proposed in Hansen (199). In his paper, we are focused on he problem of he excessive GARCH forecass in volaile periods and hus consider he SWARCH model o allow for regimes wih differen volailiy levels. We use he CUSUM-squares saisic as a simple diagnosis for he possibiliy of differen volailiy regimes. 10 We use wo regimes and do no consider models wih more regimes, because we wan o explore wheher he inroducion of regimes help solve he spurious high persisence problem wih he GARCH forecass and i urns ou ha wo regimes are sufficien for ha. 17

19 observaions 1 o 891. I shows ha he saying probabiliy in each SWARCH-L model is no high (0.4~0.6), especially for he high-volailiy regimes. This implies ha he duraion of high-volailiy is no long. Tha is, he effecs of shocks don always las persisenly. To compare wih VaR calculaions calculaed from parameric volailiy models, we implemen wo alernaive nonparameric approaches, including he hisorical simulaion (hereafer, HS) and TGARCH-adjused hisorical simulaion (adjused HS). 11 The criical parameer in he HS models is he window widh ha is used in esimaion. We repor resuls for wo cases: 500 and 1000 days. The resuls show ha VaR forecass from he radiional HS approach are fixed for a long period. These forecass would no change unil a grea loss occurs or he losses deviae away from he mean of he moving windows. For he case of 500-day window, we use he prior 500 observaions o esimae he TGARCH model, and hen use he variance forecas for day 89, he firs ou-of-forecas observaion, o adjus he variance for implemening he adjused HS. Similarly, we move forward hrough ime, generaing ou-of-sample adjused-hs. Table 4 repors he saisical summary of VaR forecass from compeing VaR models In average, excep for he FTSE 100, VaR forecass from he SWARCH-L model are higher han any oher measures. Relaive performances of models compared are given in Table 5. For he 95% VaR, SWARCH-L model produces he highes MRB, meaning is VaR forecass are much higher han he average over all models compared. MRB of he oher models are beween and 0.16, showing ha he differences across hese models are relaively small. For he 99% VaR, MRB are higher in each model, beween -0.0 and 0.9, which shows ha he forecass from hese VaR models are no similar. 11 Following Hull and Whie s (1998) procedure we implemen he HS adjused by he TGARCH volailiy. 18

20 While MRB measure he average deviaion of a model from he average VaR across models, RMSRB measures he dispersion of a specific VaR model deviaing from he average VaR across models. According o Table 4, he resuls of RMSRB indicae ha he SWARCH model produces higher VaR esimaions a a given dae for boh he 95% and 99% VaRs. The resuls of correlaion beween he VaR forecas and he realized reurn sugges he superior performance of he SWARCH-L model. The SWARCH-L model exhibis he highes correlaion among all models for he four index reurns examined. Besides, VaR calculaed from he TGARCH and TGARCH-adjused HS models performs relaively beer han ha from he EWMA and hisorical simulaion models. Unsurprisingly, he HS models have he lowes correlaion. This shows ha he hisorical-simulaed VaR is unable o well rack changes in risk over ime. The benchmark of score based on he binary loss funcion is 5 for he 95% VaR and 5 for he 99% VaR. For he 95% VaR, We found he score of he SWARCH-L model is lower han 5, and he score of he adjused HS model is mos close o 5. The scores of he EWMA, TGARCH, and HS models are much higher, indicaing here are much more excepions exceeding VaR. For he 99% VaR, he resuls are similar, bu he SWARCH-L and HS (1000 days) models performs he bes for he Nikkei 5 reurns. The score based on he quadraic loss funcion measures he magniude of excepions. For he 95% VaR, he score of he SWARCH-L model is he lowes. We believe ha i is resuled by is fewer excepions. We also found ha he HS achieves he highes score alhough i exhibis he same number of excepions wih he EWMA model (for NIKKEI 5) or even fewer excepions han he TGARCH model (for FTSE 100). As for he 99% VaR, he score of adjused HS model is relaively lower han he oher models. 19

21 For he HS models, we use observaion periods of 500 and 1,000 days moving window. The performance under differen observaion periods is no grealy dissimilar. Adjused-HS model consisenly performs beer han he hisorical model across each crierion. The HS model ends o produce higher scores of loss funcion, implying i underesimaes risk. Besides, is esimaions have he lowes correlaion beween acual oucomes among all models. Tha is, i has poor abiliy o adjus risk measures over ime. According o he number of acual loss exceeding VaR, we calculae each model s LR saisics for PF es. For he 95% VaR, only hisorical simulaion models Pr r, VaR, = a he 99% for CAC 40 series rejec he hypohesis of ( ) 5% + T < + T confidence level. However, for he 99% VaR, he LR saisics of boh he EWMA and TGARCH mehods exremely exceed he 1% criical value of χ (1) = Excepionally, he SWARCH-L and HS models do no perform well for he FTSE 100 series. For he S&P 500, NIKKEI 5, and CAC 40, he PF es canno rejec he Pr r, VaR, = under he SWARCH-L, HS, and adjused hypohesis of ( ) 1% + T < + T HS models. This is probably because he PF ess generally indicae he coverage probabiliy is correc for mos models, especially for he 95% VaR. In summary, he SWRCH-L model ends o produce oo few excepions, alhough he PF es does no rejec is accuracy. The srengh of he SWARCH-L model is is efficiency o rack he evoluion of risk in erms of is highes correlaion. 5. Conclusion This paper evaluaes he forecasing performance of he SWARCH model based on a sysemaic evaluaion for he corresponding VaR forecass. VaR has been widely used o quanify and conrol he marke risk, and he beer forecas of volailiy help 0

22 improving he VaR forecass. The esimaion resuls show ha he high degree of persisence esimaed from he widely used GARCH models can be adjused by allowing regime swiches in he ime-varying volailiy. By evaluaing ou-of-sample VaR forecass via relaive performances based on cerain loss funcions and he hypohesis esing based on he LR saisic, we conclude ha he SWARCH-L model ouperforms alernaive compeing models, including he RiskMeric or EWMA model, TGARCH model, HS model, and adjused HS model. In his paper we are focused on he problem of he excessive GARCH forecass in volaile periods and hus consider he SWARCH model o allow for regimes wih differen volailiy levels. I is lef o he fuure research o examine if a SWARCH model wih more regimes, or a Markov swiching GARCH model can explain he dynamics of ime-varying volailiy beer. Besides, we only examine he performance of daily VaR forecass. Furhermore, Cerain insiuions, however, care heir rading risk under longer holding periods. I is also commendable o evaluae VaR forecass of each model under differen horizons. 1

23 References Alexander, C. O. and C. T. Leigh (1997), On The Covariance Marices Used in Value a Risk Models, Journal of Derivaives, 4, Ang, A. and G. Bekaer (00), Regime Swiches in Ineres Raes, Journal of Business and Economic Saisics, 0, Cai, J. (1994), A Markov Model of Uncondiional Variance in ARCH, Journal of Business and Economic Saisics, 1, Duffie, D. and J. Pan (1997), An Overview of Value a Risk, Journal of Derivaives, 4, Efron, B. (1979), Boosrapping Mehods: Anoher Look a he Jackknife, Annals of Saisics, 7, 1-6. Engle, R. F. and S. Manganelli (1999), CAViaR: Condiional Auoregressive Value a Risk by Regression Quaniles, UCSD Discussion Paper. Glosen, L. R., R. Jagannahan, and D. Runkle (1993), On he Relaion beween he Expeced Value and he Volailiy of he Nominal Excess Reurn on Socks, Journal of Finance, 48, Gray, S. F. (1996), Modeling he Condiional Disribuion of Ineres Raes as a Regime-swiching Process, Journal of Financial Economics, 4, 7-6. Hamilon, J. D. and R. Susmel (1994), Auoregressive Condiional Heeroskedasiciy and Changes in Regime, Journal of Economerics, 64, Hansen, B. E. (199), The Likelihood Raio Tes under Nonsandard Condiions: Tesing he Markov Swiching Model of GNP, Journal of Applied Economerics, 7, S61-S8. Hendricks, D. (1996), Evaluaion of Value-a-risk Models Using Hisorical Daa,

24 Federal Reserve Bank of New York Economic Policy Review,, Ho, L. C., P. Burridge, J. Cadle, and M. Theobald (000), Value-a-Risk: Applying he Exreme Value Approach o Asian Markes in he Recen Financial Turmoil, Pacific-Basin Finance Journal, 8, Hull, J. and A. Whie (1998), Incorporaing Volailiy Updaing ino The Hisorical Simulaion Mehod for Value-a-Risk, Journal of Risk, 1, J.P. Morgan/Reuers (1996), RiskMerics Technical Documen (4h ediion), New York. Jorion, P. (000), Value a Risk, nd ediion, McGraw-Hill. Kupiec, P. H. (1995), Techniques for Verifying The Accuracy of Risk Measuremen Models, Journal of Derivaives, 3, Longin, F. M. (000), From Value a Risk o Sress Tesing: The Exreme Value Approach, Journal of Banking and Finance, 4, Lopez, J. A. (1998), Mehods for Evaluaing Value-a-Risk, Federal Reserve Bank of New York Economic Policy Review, Penza, P. and Bansal, V. K. (001), Measuring Marke Risk wih Value a Risk, John Wiley & Sons, Inc. 3

25 Table 1 Descripive Saisics of Index Reurns (%) Saisics S&P 500 NIKKEI 5 FTSE 100 CAC 40 Mean Median Maximum Minimum Sd. Dev Skewness Kurosis Jarque-Bera * * * * ρ (1) * * Q(15) * 31.69* * 31.97* Q (15) * 361.7* * * Jarque-Bera is he Jarque-Bera saisic for normaliy. ρ(1) indicaes he firs order auocorrelaion in reurns. Q(15) and Q (15) repor values of he Ljung-Box Q saisic for up o 15h-order auocorrelaion in reurn and squared reurns, respecively. *: Significan a he 1% level of significance. 4

26 Table Esimaion Resuls of he TGARCH(1,1) Model (Observaions 1 o 891) r c0 + φ + ε where ε = r φr 1 ε Ω 1 ~ N (0, σ ) = r 1 σ w d, d 1 if ε 0, and d 0 oherwise. = + αε 1 + βσ 1 + δ 1 ε 1 1 = 1 < Parameer S&P 500 NIKKEI 5 FTSE 100 CAC * * 0.08* c0 (0.014) (0.0) (0.016) (0.0) φ 0.054** ** (0.019) (0.00) (0.019) (0.0) w 0.011** 0.053** 0.006** 0.064** (0.003) (0.016) (0.00) (0.018) α ** 0.013* 0.07** (0.010) (0.00) (0.006) (0.00) β 0.96** 0.898** 0.955** 0.890** (0.01) (0.015) (0.009) (0.0) δ 0.106** 0.15** 0.050** 0.09** (0.01) (0.03) (0.014) (0.08) = Q(15) and Q (15) indicae he Ljung-Box saisics for upo 15-h order auocorrelaion in sandardized residuals and squared sandardized residuals, respecively. Bollerslev-Wooldridge robus sandard errors are repored in parenheses. **: Significan a he 1% level of significance. *: Significan a he 5% level of significance. 5

27 Table 3 Esimaion Resuls of he SWARCH-L (,) Model (Observaions 1 o 891) r c0 + φ + ε where ε = r E( r Ω 1) ε Ω 1 ~ N (0, σ ) = r 1 σ = g σ s = 1 for sae 1, s = for sae ; g 1 is se o equal o 1, and he ~ s ransiional probabiliy is Pr( s j 1 s j s 1 i = p i, j = 1, = Ω ) = Pr( = = ) ij σ w d d 1 if ε 0, and d 0 oherwise. ~ = + α1ε 1 + αε + δ 1 ε 1 1 = 1 < Parameer S&P 500 NIKKEI 5 FTSE 100 CAC ** * 0.059** c0 (0.015) (0.034) (0.015) (0.03) φ ** 0.054** * (0.00) (0.019) (0.00) (0.01) w 0.18** 0.333** 0.39** 0.413** (0.06) (0.088) (0.034) (0.077) 0.135** 0.068* 0.149** 0.160** α1 (0.043) (0.031) (0.039) (0.044) 0.39** 0.75** 0.171** 0.133** α (0.04) (0.04) (0.033) (0.031) δ 0.36** 0.57** 0.63** 0.148** (0.07) (0.064) (0.05) (0.059) 5.96** 5.606** 3.176** 4.83** g (0.711) (0.913) (0.54) (0.466) ˆp 11 ˆp 0.406** (0.081) 0.40** (0.088) 0.414** (0.108) 0.434** (0.111) Sandard errors are repored in parenheses. **: Significan a he 1% level of significance. *: Significan a he 5% level of significance ** (0.104) 0.35** (0.103) 0.414** (0.113) 0.443** (0.089) 1 = 6

28 Table 4 VaR Calculaions IIndex Reurn S&P 500 NIKKEI 5 FTSE 100 CAC 40 95% VaR Mean Sd. Dev. Mean Sd. Dev. Mean Sd. Dev. Mean Sd. Dev. EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days) % VaR Mean Sd. Dev. Mean Sd. Dev. Mean Sd. Dev. Mean Sd. Dev. EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days)

29 Table 5 Performances of VaR Forecass Panel A. S&P 500 MRB RMSRB Correlaion Binary Loss Funcion Quadraic Loss LR(PF Tes) Funcion 95%VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 dsays) %VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days) Panel B. FTSE 100 MRB RMSRB Correlaion Binary Loss Funcion Quadraic Loss Funcion LR(PF Tes) 95%VaR EWMA TGARCH SWARCH-L Hisorical (500 days) Hisorical (1000 days) Adjused HS (500 days) Adjused HS (1000 days) %VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days)

30 Table 5 (Coninued) Performances of VaR Forecass Panel C. NIKKEI 5 MRB RMSRB Correlaion Binary Loss Quadraic Loss LR(PF Tes) 95%VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 dsays) Adjused HS (1000 days) %VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days) Panel D. CAC MRB RMSRB Correlaion Binary Loss Quadraic LR(PF Tes) Funcion Loss Funcion 95%VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days) %VaR EWMA TGARCH SWARCH-L HS (500 days) HS (1000 days) Adjused HS (500 days) Adjused HS (1000 days)

31 Figure 1 CUSUM of Squares Tes S&P 500 NIKKEI CUSUM of Squares 5% Significance CUSUM of Squares 5% Significance FTSE 100 CAC CUSUM of Squares 5% Significance CUSUM of Squares 5% Significance Noe: The saisic of CUSUM of squares is used o es for he sabiliy of he variance process. As wih he CUSUM of squares es, movemen ouside he criical lines is suggesive of parameer or variance insabiliy. 30