Quantifying Foreign Exchange Market Risk at Different Time Horizons

Transcription

1 Universiy of Wollongong Research Online Universiy of Wollongong in Dubai - Papers Universiy of Wollongong in Dubai 004 Quanifying Foreign Exchange Marke Risk a Differen Time Horizons Ramzi Nekhili Universiy of Wollongong, ramzi@uow.edu.au Aslihan Alay-Salih Selcuk Caner Publicaion Deails Nekhili, R., Alay-Salih, A. & Caner, S. 004, 'Quanifying Foreign Exchange Marke Risk a Differen Time Horizons', Proceedings of he Elevenh Annual Mulinaional Finance Sociey (MFS) Conference, pp Research Online is he open access insiuional reposiory for he Universiy of Wollongong. For furher informaion conac he UOW Library: research-pubs@uow.edu.au

2 QUANTIFYING FOREIGN EXCHANGE MARKET RISK AT DIFFERENT TIME HORIZONS Ramzi Nekhili Faculy of Engineering, Deparmen of Indusrial Engineering, Easern Medierranean Universiy, Mersin 10, TURKEY. Tel.: ; fax: address: Aslihan Alay Salih Faculy of Business Admisraion, Bilken Universiy, Ankara, TURKEY. Tel.: ; fax: address: Selcuk Caner Faculy of Business Admisraion, Bilken Universiy, Ankara, TURKEY. Tel.: ; fax: address: Absrac This paper evaluaes he performance of he Value-a-Risk models of he foreign exchange raes a differen ime horizons. I sars wih generaing differen reurns a differen ime horizons from he USD-EUR 5-minue reurns and simulaes six parameric models (Normal-GARCH, Suden-GARCH, Normal-IGARCH, Suden-IGARCH, Ornsein-Uhlenbeck volailiy, and jump) o assess he possible rading losses on 30 minues, 6 hours, 1 hours and daily horizons. Using he variance mehod, Hill esimaor, and he Generalized Pareo Disribuion, VaR forecass are obained. The performance of he seleced VaR models along wih each VaR echnique are evaluaed a 1% and 5% confidence level by calculaing he violaion raio. The resuls show ha, a boh high and low frequencies, he predicive power of he VaR mehods display wide variaion in assessing he foreign exchange risk. Furhermore, if a reurn generaing process considers he ail faness and he sochasic volailiy srucure of he exchange rae reurns, he Exreme Value Theory mehods will no be superior o he sandard variance mehod. Keywords: Value-a-Risk; Variance mehod; Hill esimaor; Exreme Value Theory JEL: C, C41, G10 1

3 1. Inroducion The increasing populariy of Value-a-Risk (VaR) in recen years has played an imporan role in financial risk managemen for banks and fund managers as well as financial marke regulaors. The VaR represens a unique quaniy ha gives he expeced maximum loss over a given planning horizon wih a given confidence level. I serves as a ool o manage and conrol risk for any financial insiuion ha have a rading porfolio. However, he choice of a planning horizon and a confidence level for VaR esimaes is largely arbirary. The ime horizon used o calculae VaR should normally depend on he liquidiy of he securiies in he porfolios and how frequenly hey are raded. 1 The confidence levels or he quaniles ha are useful for capial requiremens range beween 95% and 99%. Inside his range, many VaR values can be produced no only by using a given risk managemen sysem, such as RiskMerics, bu also by relying on differen asse price generaing processes. There is a general consensus in he risk managemen lieraure (e.g. Embrechs e al., 1997) ha any quesion concerning financial marke risk managemen in finance involves quanile (VaR) esimaion. Wihin he ool ki on VaR esimaion, we find a plehora of mehods o assess he VaR of a disribuion of losses and profis. In all hese mehods, he ypical VaR calculaion involves assessing he possible exreme loss resuling from holding a porfolio for a fixed period using he volailiy as a measure of risk over a given period of ime. In doing so, J.P. Morgan has inroduced he RiskMerics analysis in Ocober 1994, and mos companies and banks found i simple o implemen empirically wih minimum compuaional burden. Since hen, a vas body of lieraure developed by comparing some early VaR mehods (e.g. Variance-Covariance mehod and Hisorical Simulaion) wih each oher or wih he benchmark model of RiskMerics (e.g. Allen, 1994). Recenly, he growing need o evaluae exreme risk in financial markes has shifed ineres o EVT (see Embrechs e al. (1997) as example). In fac, EVT provides a useful ool for measuring he marke risk and gives informaion abou he exreme oucomes. Moreover, i provides guidance on he ype of disribuion one should selec. The mos popular EVT models are he ail index esimaor of Hill (1975) and he generalized 1 For an acive rader who is likely o ge a margin call, he curren convenion of 10-day horizon is hardly of any use.

4 Pareo disribuion (GPD). For example, McNeil and Frey (000) found ha GPD is preferable o oher mehods such as sandard GARCH model wih normal and Suden- innovaions, in he sense ha i can incorporae asymmeries in he ails and herefore beer esimae he ail of he disribuion. Their approach consised of fiing a GARCH model o various reurn series and using hisorical simulaion and hreshold mehods from EVT o esimae he disribuion of he residuals. Furhermore, wih differen ime horizons, hey used he square-roo-of-ime scaling of one-day VaR esimaes o obain esimaes for longer ime horizons of 5 or 10 days, and find ha his procedure does no perform well in pracice. As an alernaive, hey proposed using a Mone Carlo mehod based on he fied models o obain beer resuls a differen ime inervals. According o heir resuls, EVT-based mehods along wih wo basic sylized facs, namely sochasic volailiy and fa-ails, play an imporan role in he esimaion of VaR. In he VaR lieraure, here are few sudies ha deal wih he performance of VaR models a he inraday level (e.g. Belrai and Morana, 1999, and Gio, 005) and rarely a differen ime scales. In fac, using inraday reurns for risk managemen purposes can be significan for boh inernal and exernal observers of wha a saisfacory marke risk measure is. Inernally, bank managers need a measure ha allows efficien managemen of he bank's risk posiion. Bank regulaors, on he oher hand, wan o be sure ha a bank's ne worh loss is accuraely measured and ha he bank's capial is sufficien o survive a loss. Therefore, boh bank managers and bank regulaors wan up-o-dae measures of risk. In his conex, his paper ess VaR models of currency reurns a differen ime horizons saring from high frequencies. Our sudy incorporaes no only VaR mehodologies ha are prominen in he lieraure (Exreme Value Theory and he Variance mehod, in conjuncion wih a variey of volailiy models) bu also specific feaures of he currency markes, in paricular he increasing fa-ails wih he ime inerval of exchange rae reurns and he exisence of disconinuiies in he price process. 3 This is also called he exceedances over hreshold model of Pickands (1975). The reader can also refer o a horough analysis of he exremes of daa by Davison and Smih (1990). 3 Alhough we consider risk managemen of currency rade in his sudy, he mehodology applies o he rade of oher financial asses 3

5 Therefore, i is an imporan exercise o look a he effec of various reurn generaing processes and VaR calculaion echniques a various ime horizons for he quanificaion of foreign exchange marke risk, in paricular he USD-EUR marke which is addressed in his sudy. We es he performance of seleced VaR models. We sar wih simulaing a random walk reurn model wih differen volailiy specificaions wih boh normal and Suden- disribuion, and a jump model for he reurns. We proceed by using he VaR compuaion mehods, and esing he performance of hese models a 1% and 5% confidence level. Finally, a realiy check based on minimizing a loss funcion ha compares he acual loss and profi wih he forecased VaR is performed. 3. Marke risk models In his secion we presen four basic sochasic processes ha are used in he lieraure represening he exchange rae reurns process. We consider r, = 1... T as he reurn series where r FX FX ) FX or logged difference of FX, i.e. FX sands for exchange rae. = ( 1 / GARCH(1,1) The GARCH(1,1) specificaion of Bollerslev (1986) for he exchange rae reurns considers he volailiy clusers observed in financial ime series. I is presened as follows, r = σ, 0 σ = α + α + α α 1 1 1, (1) where ε is assumed o follow some probabiliy disribuion wih zero mean and uni variance. We assume wo possible disribuions for he error erms, namely ε is IID N(0,1) and IID Suden-(v), where v represens he degrees of freedom. In mos of he early papers on he inraday exchange rae reurns, he mean has no been aken ino consideraion (see for example Andersen e al., 001). In Nekhili e al. (00), here are no dynamics in he mean of he inraday reurns a 30 minue, 6 hours, 1 hours, and daily horizon. The drif of he coninuous ime process of he reurns is highly insignifican confirming he assumpion ha he expeced reurns are equal o zero for all ime horizons. For his reason, we do no consider a drif in he reurn process and consequenly in he calculaion of he VaR. 3. IGARCH(1,1) 4

6 There is now evidence ha volailiy has a noable degree of persisence in he reurns (see for example Baillie and Bollerslev, 1989). This persisence has been inerpreed wihin he conex of an Inegraed GARCH model as in Baillie e al. (1996) for he exchange rae reurns. The case where α1 in equaion (1) is se equal o IGRACH(1,1) which is presened as follows 4, 1 ζ and α is equal o ζ, consiues he random walk r = σ, σ = a + ζ 1 + ( 1 ζ ) σ, 1 () where ε is IID N(0,1). To es for he possibiliy of fa ails of he reurn disribuion, we also consider ε as IID (0,1,v) wih v degrees of freedom. This model is as imporan for praciioners as mos companies and banks choose o implemen for risk quanificaion. The IGARCH model can be equivalen o he exponenially weighed moving average model (RiskMerics) of J.P. Morgan (1995). This model is in fac a simple form of he IGARCH model where he pre-specified parameer ζ = 0.94 and he inercep a se o zero. 3.3 Ornsein-Uhlenbeck Sochasic volailiy model is anoher imporan reurn generaing process used by Heson (1993). The volailiy of he reurns is assumed o follow a mean-revering Ornsein-Uhlenbeck process defined as follows, r = σ ε, σ = σ e σ 0 0 = σ, 0 a( 0 ) + b(1 e a( 0 ) ) + σ e v 0 a( τ ) dz τ, (3) where σ v is he sandard deviaion of he volailiy, a>0, b>0. The Wiener processes, ε and assumed o be independen. In his empirical work, Heson (1993) allow for a correlaion beween Z, are ε and Z. We may consider his for fuure research. 3.4 Jump process 4 We consider he esimaion of he parameers of he IGARCH(1,1) process for differen ime horizons, and do no consider a fixed parameer of 0.94 as i is usually done by J.P. Morgan. 5

7 To esimae he impac of news on raders, such as he fundamenal macro-economic informaion or he inervenion of domesic and foreign cenral banks, we consider a jump process of he exchange rae reurns as follows, r = ( λθ )( ) + σ ( ε ε ) + Lnκ, (4) 0 0 = 1 n where κ is he jump and assumed o be i.i.d and lognormal wih mean θ and variance δ, and n is he acual number of jumps during he inerval o following a Poisson arrival process wih parameer λ as a mean number of informaion arrivals per uni ime. I is assumed ha upon he arrival of abnormal informaion here is an insananeous jump in he exchange rae of sizeκ, independen ofε. The noise and he Poisson process are infiniely divisible in ime, appropriaely scaled, and have independen incremens. 5. VaR Calculaions In his secion, we show he commonly used VaR esimaion echniques. In he following, we use he variance mehod and he Exreme Value Theory based mehods, namely he Hill esimaor and he Generalized Pareo Disribuion. 5.1 Variance Mehod Variance mehod is he variance-covariance mehod used in porfolio analysis, and in he case of one single asse, as in his case, he USD-EUR exchange rae reurn, i becomes he variance mehod. The variance mehod relaes he VaR wih he variance or he sandard deviaion of he reurns and, by inuiion, he larger he variance of he reurns, he larger he VaR. For insance, in he case of assuming ha he reurns follow a maringale process wihou drif wih r = σ ε where ε follow a disribuion Φ(.) ha is eiher assumed as IID N(0, ) or as IID Suden-(0,,v) wih v degrees of freedom, he VaR for he variance mehod is esimaed by H VaR 1 ( α ) Φ ( α ) σ, σ = (5) σ 6

8 ( ) 1 where Φ α is he ( 1 α ) h H quanile value of he disribuion Φ. VaR denoes he Value-a-Risk a ime wihin a cerain ime horizon H. For insance, if we use a Gaussian disribuion for a confidence ( ) level α of 5% Φ = 1.. For a confidence level of 1%, Φ ( 0.01) = Exreme Value Theory (EVT) Exreme value heory has become an essenial and robus framework o evaluae exreme risks in financial markes. The variance mehod displayed above shows ha he exreme risk is relaed o he variance, bu in he case of fa-ailed disribuions variance is no longer sufficien. Since he ail faness of he exchange rae reurn disribuion is one of is aribues ha characerize he exen of he risk, we use he ail index as a deermining facor in he compuaion of he Value-a-Risk and conras is values wih hose ha are based on differen sochasic processes. Generally speaking, in a model of risk, his approach consiss of selecing a paricular probabiliy disribuion for he daa and esimaing is parameers using empirical daa. This way, he EVT acs in favor of providing he bes ool for esimaing he ail of he disribuion. EVT saes ha he ail disribuion of any ordered daa mus belong o jus hree possible general families, for which he reurn process r is presened as follows: r, Gumbel: Λ() r = exp ( exp ) r R 0, Fréche: Ωω () r = exp r ω ( ), r 0 r > 0, ω > 0 ω exp Weibull: () [ ( r )], ψ ω r = 1, r 0, ω < 0 r > 0 Fréche and Weibull disribuions have only one parameer o esimae, ω, which is called he ail index. The Suden- model and he uncondiional disribuion of ARCH-process boh fall in he domain of his ype of disribuions. As in Gencay e al. (003), if we seω = 1, he densiy of he 5 Suppose we have a hypoheical porfolio consising of $100 million posiion in USD-EUR marke, and consider ha he daily volailiy of USD-EUR reurns is 0.5%. Assuming ha in he VaR calculaion he error erms are normally disribued, he 1-day 99% VaR is $100,000,000 imes imes.36=$1,163,000. 7

9 Weibull disribuion is a hin-ailed disribuion relaive o he normal disribuion. Whereas he densiy of Fréche disribuion also sars from zero bu i has a heavy-ail relaive o he normal disribuion. Finally, he Gumbel disribuion has a ail behavior ha lies beween a hin-ail and a heavy-ail relaive o he normal disribuion. A more general represenaion of hese disribuions is obained by reparameerizing he ail index ω oς = 1/ ω. Therefore, a unified represenaion wih a single parameer is well known as he generalized exreme value disribuion (GEV) 1 ς [ ( 1 + ςr ) ], [- exp () r ], exp if ς 0, H ς () r = (6) exp if ς = 0, where ς is also known as he shape parameer. The case where ς = 0 has o be inerpreed as ς 0 (ς ends o zero), resuling in he Gumbel disribuion. Whenς < 0, we obain he Weibull disribuion, and for ς > 0 he Fréche disribuion. For applicaion in insurance and finance, he Gumbel and he Fréche family urn ou o be he mos imporan models for exremal evens. In fac, he domain of aracion of he Weibull disribuion are he hin-ailed disribuions such as uniform and bea disribuion which do no have much power in explaining financial ime series. For he Gumbel disribuion, he domain of aracion include he normal, exponenial, gamma and lognormal disribuions where only he lognormal disribuion has a moderaely heavy-ail. A modificaion of he GEV disribuion, which considers he behavior of large observaions ha exceed a high hreshold, is now aracing ineres in he finance lieraure, for example McNeil (1999) and Bassi e al. (1998). This new class of disribuions is he generalized Pareo disribuion (GPD). The GPD is a wo parameer disribuion ha relies on he exceedances of observaions over a high hreshold $% u$ wih he following disribuion funcion, -1/ ( 1 + ςr / β ) exp ( r / β ), ς 1, if ς 0, G ς, β () r = (7) 1 if ς = 0, where ς is he shape parameer of he disribuion, β > 0 represens an addiional scaling parameer, and r > 0 when ς 0 and 0 r β / ς whenς < 0. In case where he scale parameer β = 1, he 8

10 disribuion in Equaion 7 is equivalen o Gς, 1( r) = 1+ log H ς ( r), when logh ς ( r) > 1. The GPD ness a number of oher disribuions. Whenς > 0, i becomes he ordinary Pareo disribuion ha is more relevan for financial ime series analysis since i is heavy ailed. Ifς = 0, he GPD corresponds o he exponenial disribuion. Forς < 0, i is well known as Pareo II ype disribuion. A more common case happens whenς < 0. 5, which is valid for high-frequency foreign exchange reurns (Embrechs e al., 1997). In his paper, he quanificaion of he USD-EUR foreign exchange marke risk will be conduced using wo approaches o calculae he VaR. The firs one is he Hill esimaor and he second one is he GPD. Boh of hese approaches esimae firs he shape parameer ς and hen find he VaR Hill Esimaor Le r, = 1... T be he realizaions on exchange rae reurns. By ordering he daa in a descending order, where r > r..., we use he Hill (1975) esimaor for he shape parameer ς and is given by, 1 > r T m 1 1 ˆ = ς ln r ln rm m 1 = 1 (8) where m is he number of order saisics, and ςˆ is he esimaed value ofς. The Hill esimaor is proven o be a consisen esimaor of ς = 1/ ω for fa-ailed disribuions. Differen ail esimaors work well when he sample size is large. However, he esimaion of he ail index is dependen on he choice of he number of order saisics m. In fac, he choice of m represens a problem in ha we do no know how far we can go o selec he order saisic r(m) ha is in he ails. In some ad-hoc mehods, he hreshold level m is obained by arbirarily considering a confidence level and aking he corresponding percenile. Anoher ool in hreshold deerminaion is he Hill-plo. A Hill-plo is consruced such ha esimaed ςˆ is ploed as a funcion of m upper order saisics or he hreshold. A hreshold is seleced from he plo where he parameer ςˆ is fairly sable. However, we find difficuly in searching for he sable porion of he shape parameers for all our simulaed 9

11 reurns a differen ime horizons considered. Therefore, we looked for more efficien echnique o ge opimal esimaes of he order saisics m. Our esimaions are conduced for opimal values of m obained by he boosrap procedure of Danielsson and de Vries (1997). In his echnique, he fac ha he esimaor ςˆ is asympoically normal allows us o minimize is mean squared error o deermine he opimal number of order saisics m. The opimaliy is in he sense ha he bias and variance of he shape parameer esimae vanish a he same rae. The idea is o consruc he boosrap expecaion of ( ˆ ς ς ), and o minimize i wih respec o m. We employ subsample boosrap mehod and ake boosrap resamples of size T1=T 1-s, for some 1>s>0, where T is he sample size. The reason for his sample size reducion is ha i guaranees convergence in probabiliy. Kearns and Pagan (1997), show ha he convergence rae is fixed by m and ha a resample size wih s=1/4 works well. We ake he same resample size o perform our esimaions. The Hill esimaors are calculaed from he opimal number of order saisics obained hrough he subsample boosrap procedure, and using GAUSS programming language. I follows ha he Hill esimaor for Value-a-Risk for a given confidence level α is hen defined as, ςˆ H m Var ( α ) = rm, Tα (9) where T is he sample size, m is he opimal number of order saisics, α is he confidence level, and ςˆ is he esimaed shape parameer Generalized Pareo Disribuion (GPD) The shape parameer ς can also be esimaed from he GPD disribuion by maximum likelihood mehod. The densiy of he GPD disribuion is r ς g () r = 1 + ς, (10) β β 6 For example, suppose ha in daily exchange rae reurns, he sample size T=1000, he opimal number of order saisics m = 900 ha corresponds o a reurn r ( m) = 0. 01, and he esimaed shape parameer ˆ ς = A 5% confidence level, he Value-a-Risk is VaR (0.05) = Tha is he exchange rae reurn will no exceed 1.54 percen in one day 95 percen of he ime. 10

12 and he corresponding log-likelihood funcion is T 1 ς L = T ln( β ) + 1 ln 1 + r, (11) ς = 1 β where T is he sample size. For a given confidence levelα, he VaR is calculaed as follows, ˆ H β ς Var ( ) = + ( / ) ˆ α u αt N u 1), (1) ˆ ς where u is he hreshold posiioned a he ( 1 α ) h sample percenile, T is he number of observaions, N u is he number of exceedances over he hresholdu, and βˆ and ςˆ are he parameer esimaed from he GPD disribuion. McNeil and Frey (000) argue ha he issue of choosing an opimal hreshold does no seem criical for he GPD mehod as i is more imporan for he Hill esimaor mehod. In fac, he GPD quanile esimaor is more sable, in erms of mean squared error, wih respec o he choice of he hreshold. The GPD VaR esimaions are calculaed using EVIS (Exreme Values in S-Plus) sofware of McNeil (1999). 7 The appropriae hreshold value for each reurn process and VaR mehod are chosen according o he confidence levels sudied. For each ime horizon, we ake he upper 1% of he sample for 1% confidence level, and he upper 5% of he sample for 5% confidence level. 8 In our VaR calculaions, we divide he reurns daa, wih T observaions, in wo sub-samples. The firs sub-sample, S E : from 1 T k wih k he lengh of window used for esimaing he VaR from each reurn generaing process, consiues he esimaion sub-sample. The second one, S F : from T k+1 T, represens he forecas sub-sample. Having obained he simulaed disribuions from differen reurn 7 EVIS is downloaded over he inerne a hp:// 8 As an example, suppose ha in daily exchange rae reurns, he hreshold is deermined as 7 percen and esimaed parameers are ˆ β = 0.05 and ˆ ς = Furher suppose ha T=10000 and N u = 500. The VaR a % confidence level isvar (0.01) = [( 0.01) 1] = Tha is he exchange rae reurn will no exceed 19.4 percen in one day 99 percen of he ime 11

13 generaing processes a differen ime horizons, he VaRs are compued using a rolling window procedure. In he case of he variance mehod, VaR are calculaed by finding firs he sandard deviaion of he simulaed reurns Y i wih i=1 T k and hen rolling unil he las T-1 observaion. As T k increases, new simulaed reurns are included bu older ones are removed. Once VaR esimaes are obained, a performance es is run on he forecas sample of he empirical USD-EUR reurns. 6. VaR Performances To assess he VaR model performances, we use he sandard procedure ha is based on he violaion raio or he failure rae (see Jorion, 000). The violaion raio is he number of imes reurns exceed he forecased VaR divided by he number of one-period ahead forecass of VaR. I follows ha, for each H ime horizon, an indicaor variable can be defined as I = 1( r <> VaR ( α )) and = 0 oherwise. The fac ha he acual loss does or does no exceed VaR is a sequence of successes or failures wih H probabiliy p = Pr[ r < VaR ( α )]. Since he reurn observaions are independen, he indicaor T [ I ] = 1, where T is he oal number of forecased VaR, is a Bernoulli process ha follows a I Binomial disribuion. Therefore, we can es he null hypohesis H 0 : p = α agains H 1 : p α, where p is he failure rae, and also consruc a confidence inerval for p a he confidence levelα. For insance, a he 5% level, he confidence inerval is [ p 1.96 p(1 p) / T, p p(1 p) / T ]. A high violaion raio corresponds o an underesimaion of he risk by he VaR model. If he violaion raio is less han α %, he VaR model excessively overpredic he risk. If he violaion raio is high han α %, i implies an excessive underpredicion of risk by he VaR model. To compare he predicive capabiliy of he performing VaR models, we employ a realiy check procedure by minimizing a loss funcion. The loss funcion is calculaed by weighing he difference beween he acual loss or profi and he VaR forecas. I represens he objecive of differen agens in evaluaing risk. 1

14 In fac, some agens rely on he difference beween he risk forecass and heir acual profis and losses. Whereas, some ohers, like Bank of Inernaional Selemens (BIS) regulaors, focus more on he coverage probabiliies. Le's denoe he loss funcion LF wih respec o he confidence levelα, LF (α ) as follows; T 1 H ( ) = l r Var ( α ).( αi + ( 1 α )( 1 I )) LF α, (13) = T l where r is he reurn, l is he lengh of he window, α is he confidence level, and is he indicaor I variable. LF (α ) represens he observed deviaion from he VaR wih he occurrence probabiliyα. Therefore, for an allocaed capial of C dollar, we can evaluae he expeced losses a a cerain rading horizon by muliplying he loss value LF (α ) wih C. Hence, smaller loss values will indicae he bes choice of he reurn process and he VaR mehod. 4. Empirical Applicaion As empirical reurns, we use he Olsen forex daa for he USD-EUR exchange rae. The sample consiss of coninuously recorded 5-minue bid and asks prices from January, 003 hrough November 7, 004 for a oal of 138,816 observaions. Each quoe consiss of a bid and an ask price wih a ime samp o he neares even second. The prices a each 5-minue inerval are obained by linearly inerpolaing from he logarihmic average of he bid and ask for he wo closes icks as in Muller e al. (1990) and Dacorogna e al. (1993). The coninuously compounded prices are he average of he logarihm of he bid and ask prices, [ ln P( bid ) ln P( ask ) ], 1 P = for = 1,..., (14) No o confound he evidence of slow rading paerns over weekends (see Bollerslev and Domowiz, 1993), we removed he weekend quoes from Friday :00 GMT o Sunday :00 GMT. The coninuously compounded 5-minue reurns are calculaed as he log difference of he prices, r = P P 1, for 5 = 1,..., (15) To eliminae he seasonaliy, we filered he raw 5-minue reurns by removing holidays as in Andersen e al. (001). Moreover and o avoid he bias ha can be caused by he buying and selling 13

15 inenions of he quoing insiuions on he price changes observed a high frequencies (see Dacorogna e al., 001), we op o work wih 30-minue aggregaed reurns and aggregae for oher frequencies, namely 6 hour, 1 hour and daily reurns. The number of observaions is respecively, 3135 for 30 minue inerval, 197 for 6 hour inerval, 963 for 1 hour inerval, and 481 for daily inerval. For each daa se, we ake off k=8 monhs from he sample size T, which consiue he forecas sample S F, and we perform maximum likelihood esimaions of he differen reurn models on he remaining daa, which is he esimaion subsample S E. 9 For comparison purposes, we boosrapped he same USD- EUR reurns a differen ime horizons in he same way as i is used in hisorical simulaion. In addiion, we simulaed he uncondiional disribuions of he USD-EUR reurns from he proposed sochasic processes. The simulaion experimens are implemened in GAUSS, where he parameers of each reurn process are fied wih he esimaed ones, o obain uncondiional reurn disribuions. For each simulaion, we generae S=T-k independen samples wih T-k observaions each, and we ake he las observaion from each sample o obain, in fac, T-k number of simulaed reurns. The number of observaions T corresponds o he number of observaions for he 30 minue inerval, and he simulaed 30 minue reurns are aggregaed for oher ime inerval Resuls The resuls of he violaion raios a 1% and 5% confidence levels for he esimaed VaR using he variance mehod (Equaion 5), he Hill esimaor (Equaion 8), and he GPD mehod (Equaion 1) are repored in Tables 1-4. The simulaed reurns obained from he proposed sochasic processes are used a 30 minues, 6 hours, 1 hours, and daily ime horizons. The number of observaions is 3,135 for 30-minue inerval, 1,97 for 6-hour inerval, 963 for 1-hour inerval, and 481 for daily inerval. For each daa se, we ake off more han 8 monhs from he sample size T (for insance k=180 days for daily inervals), which consiue he forecas sample S F. We perform maximum likelihood esimaions 9 The GARCH esimaions show ha aking a forecas sample size of more han 8 monhs from he whole disribuion lead o a decrease in he significance of he ARCH and he GARCH erms. The esimaions are performed using GAUSS programming language along wih TSM and MAXLIK rouines. In some reurn models, we used S-Plus sofware. 10 We esimaed he parameers of he seleced reurn models by rolling over he esimaion sample SE and we found ha here is no big change in he esimaions. The simulaion could be performed each ime we esimae a reurn model bu, since we aggregaed o obain empirical reurns for differen ime horizons, we are doing he same echnique o consruc he simulaed reurns for each ime inerval. 14

16 of he hree VaR models on he remaining daa, which is he esimaion subsample S E. Afer invesigaing he performance of various VaR models, we furher presen he resuls of he realiy check based on he loss funcion LF (α ) (Equaion 13) a 1% and 5% confidence levels. Table 1 displays he violaion raios of he daily reurns for 1% and 5% confidence level. If he reurn generaing process is a GARCH(1,1) wih normal errors, he Hill esimaor performs a 1% confidence level wih a violaion raio of 0.5% which amouns o 0.5% risk underpredicion. Some insiuions may hen no prefer such model because hey would have o allocae more han necessary capial alhough hey can mee heir regulaory requiremens. A 5% confidence level, he variance mehod performs beer wih a violaion raio of 3.8%, which amouns o 1.% risk underpredicion. In he case of assuming a GARCH(1,1) wih Suden- errors as a reurn generaing process, a boh 1% and 5% confidence level, he variance mehod looks beer han he oher VaR mehods. However, i overesimaes he risk a 1% level and underesimaes a 5% level. In using a jump process for he reurns, a boh confidence levels, GPD mehod is considered as he bes alernaive in quanifying he exchange rae reurn's risk. However, using he empirical USD-EUR daily reurns, he GPD mehod performs only a 5% confidence level. In fac, a 1% confidence level and for a relaively small sample size as for he daily reurns, i seems ha here aren enough observaions a he ail of he disribuion for exreme evens. Neverheless, wih boosrapped reurns, he GPD performs well a boh 1% and 5% confidence level. Table displays he VaR resuls using he 1-hour reurns. By assuming a GARCH(1,1) wih normal errors for he reurn process, boh he variance mehod and he Hill esimaor a 1% confidence level have good performances. Alhough hese wo mehods underesimae he risk, he Hill esimaor provides he bes violaion raio wih 1.1%. A 5% confidence level, only he variance mehod performs wih 4.1% violaion raio. In assuming a GARCH(1,1) wih Suden- errors, again as wih he daily reurns he variance mehod performs beer a boh confidence levels. If he reurns are governed by a IGARCH(1,1) wih normally disribued errors, a he 1% level, he GPD performs wih a violaion raio of 0.8%, while a 5% level, he variance mehod has he bes performance wih a violaion raio of 7.%. Assuming a jump reurn model, he GPD performs a 5% level wih a violaion raio of 6.3% which amouns o 1.3% risk underesimaion. For an insiuion, his means ha less 15

17 capial allocaion is needed o mee is capial requiremens. On he oher hand, he GPD mehod does no perform for he empirical USD-EUR 1-hour reurns as i does wih he boosrapped reurns a 1% and 5% confidence level. Therefore, i urns ou ha he variance mehod works beer when we ake ino consideraion he sochasic srucure of he reurn volailiy and he ail faness of he reurn disribuion. Table 3 shows he violaion raios of he 6-hour reurns. We noice ha only he variance mehod performs he bes among he oher VaR mehods. A 1% confidence level and assuming a GARCH(1,1) wih normal errors for he USD-EUR reurns, he variance mehod has a violaion raio of 0.8% and hence overesimaes he risk by only 0.%. Assuming a GARCH(1,1) wih Suden- errors for he USD-EUR reurns, he variance mehod has a violaion raio of 0.8% a 1% level, and 5.5% a 5% level, which amouns of 0.5% of underesimaion of he risk. Using he empirical USD- EUR 6-hour reurns, none of he mehods perform o quanify he USD-EUR marke risk a he considered sample period. A 5% confidence level, he violaion raios of he empirical reurns and he simulaed reurns from a jump process are close o each oher bu boh underesimae he USD-EUR marke risk. In Table 4, we display he violaion raios of he 30-minue reurns. The only bes predicive performance comes from he GPD VaR mehod a 5% level and wih assuming jump reurn model. The corresponding violaion raio is 5.8% which amouns of 0.8% underesimaion of risk. I seems ha a he highes frequency, aking he possible disconinuiy in he reurn process, by modeling jumps, and he fa-ails of he disribuion of he reurns, by using ail esimaion, plays an imporan role in evaluaing he exchange rae reurn risk. However, using oher reurn generaing processes along wih various VaR mehods leads o an overly conservaive way o esimae he risk. This fac comes o suppor he findings of Belrai and Morana (1999) where he high-frequency daa used for quanifying he USD-EUR marke risk has led o an overesimaion of risk a he 5% confidence level. In addiion, he difference in he risk measuremen obained wih daily daa and high-frequency daa is due o he difference in he reurn disribuion a hese frequencies. However, in conras o our findings, hey repor ha for he 30-minue ime horizon, he GARCH(1,1) model performs beer alhough i sill leads o a conservaive risk measuremen. 16

18 Having invesigaed he VaR models, we will nex look a possible exreme losses a he considered ime horizons. This is well-known as a realiy check and where a loss funcion is minimized. This loss funcion is based on weighing he difference beween he acual loss or profi and he VaR forecas. Such funcion represens an objecive for differen agens in evaluaing risk. Tables 5 and 6 give he realiy check respecively a 1% and 5% confidence levels for he daily, 1-hour, and 6-hour simulaed reurns. In hese ables, he reurn generaing processes along wih heir VaR mehods are chosen according o heir previous performances. For example, a daily ime horizon, if he reurn generaing process is a GARCH(1,1) wih normally disribued errors hen he Hill esimaor is he bes VaR model o use as repored in Table 16. Whereas, if he reurn generaing process is GARCH(1,1) wih Suden- disribued errors hen he variance mehod is he bes VaR model o use. These VaR models are hen compared according o heir corresponding loss funcion as in Equaion 13. For an allocaed capial of C dollar, we can evaluae he expeced losses a any rading horizon by muliplying he loss value LF (α ) wih C. Therefore, smaller loss values will indicae he bes alernaive in assuming he reurn process and in using he VaR mehod. For all he horizons considered and a boh he 1% and 5% confidence levels, he boosrapped USD- EUR reurns presen he highes loss funcion among he oher reurns. A daily horizon, a boh 1% and 5% confidence level, a jump reurn model along wih he GPD mehod presens he lowes loss funcion, which is for 1% level and for 5% level. In oher words, for a capial allocaion of $10,000, he esimaed rading loss is of $78.89 a 1% level, and $5.44 a 5% level. A 1-hour horizon, assuming a GARCH(1,1) wih normal errors for he reurns along wih he variance mehod provides he lowes rading loss of $75.58 a 1% level. The same mehod, assuming a IGARCH(1,1) wih normal errors for reurn process, provides he leas rading loss of $35.8 a 5% level. A 6-hour horizon, he lowes rading loss, a 1% level, is amouned o $53.46 by assuming a GARCH(1,1) wih normally disribued errors for he USD-EUR reurns and using he variance mehod. A 30-minue horizon, a 5% level, he leas loss amoun comes wih assuming a jump process for he reurns. These resuls presen cerain facs. The usual VaR resuls on daily daa do no exend o inradaily reurns in he sense ha he predicive performance of some VaR models is unsable a differen rading horizons. A leas in our case, he EVT-based mehods are highly dependen on he rading 17

19 horizon. There is in fac a degradaion of he previous performances of some VaR models a daily and 6 hour rading horizons by going o lower frequencies. This joins he difficuly of relying on a scaling law relaionship beween differen ime horizon VaRs such as he square-roo-of-ime scaling law. Moreover, we would expec ha he ail index echniques ha focus on he predicion of exreme evens o perform beer bu i is clear ha even a lower ail probabiliies he resuls are no so convincing. In addiion, despie he fac ha he EVT-based mehods has proven o be performing wih sock marke daa, i seems ha dealing wih he foreign exchange marke a high frequency represen a difficul empirical ask o confirm some of he empirical findings obained wih sock reurn VaR models. 8. Conclusions Our resuls documen ha he applicaion of he VaR echniques on he foreign exchange marke is highly dependen on he ime horizon and he chosen ail probabiliies. For insance, he variance echnique leads o good predicive power a 6-hour horizon and less so a 1-hour horizon whereas he EVT-based mehods are beer in shorer horizons. A daily horizon, we are confroned wih a variey of choices according o he ail probabiliy and he loss funcion. In fac, wih respec o he loss funcion, he GPD have he bes predicive power a α = 1% and 5%, and hence is he bes mehod o use in he VaR esimaions. A he highes frequency, he 30-minue horizon, mos of he VaR echniques have proven o perform poorly in predicing he risk, wih he noable excepion of he GPD when assuming a jump reurn process. The VaR resuls obained on daily daa do no exend o inradaily reurns in he sense ha he predicive performance of some VaR models is unsable a differen rading horizons. A leas in our case, he EVT-based mehods are highly sensiive o he rading horizon. There is in fac a degradaion of he previous performances of some VaR models a daily and 6-hour rading horizons by going o lower frequencies. This also ells ha compuing daily VaR from high-frequency USD-EUR daa using a cerain scaling law relaionship may be misleading. In fac, here is a debae on using a scaling law beween differen ime horizon VaRs such as he square-roo-of-ime o generae longer horizon VaRs (see McNeil and Frey, 000). Such scaling law could no work because he reurn disribuion behaves differenly a differen ime scales. 18

20 Moreover, we would expec ha he ail index echniques ha focus on he predicion of exreme evens o perform beer bu i is clear ha even a lower ail probabiliies he resuls are no so convincing. In addiion, despie he fac ha he EVT-based mehods has shown o be he bes VaR mehods wih sock marke daa, i seems ha dealing wih he foreign exchange marke a highfrequency represen a difficul empirical ask o confirm some of he empirical findings obained wih sock reurn VaR models. References 1. Andersen, T.G.- Bollerslev, T.- Diebold, F.X.- Labys, P. (001): The disribuion of realized exchange rae volailiy. Journal of he American Saisical Associaion, vol. 96, pp Andersen, T.G.- Bollerslev, T. (1999): Forecasing financial marke volailiy: sample frequency vis-à-vis forecas horizon. Journal of Empirical Finance, vol. 6, pp Allen, M. (1994): Building a role model. Risk, vol. 7, pp Baillie, R.T.- Bollerslev, T. (1989): The message in daily exchange raes: A condiional variance ale. Journal of Business and Economic Saisics, vol. 7, pp Baillie, R.T- Bollerslev, T.- Mikkelsen, H.O. (1996): Fracionally inegraed generalized auoregressive condiional heeroskedasiciy. Journal of Economerics, vol. 74, pp Bassi, F.- Embrechs, P.- Kafezaki, M. (1998): Risk managemen and quanile esimaion. In: R.J. Adler e al. (Eds.), A pracical guide o heavy ails (pp ). Boson, Brickhaeuser. 7. Belrai, A.- & Morana, C. (1999): Compuing value a risk wih high frequency daa. Journal of Empirical Finance, vol. 6, pp Bollerslev, T. (1986): Generalized auoregressive condiional heeroscedasiciy. Journal of Economerics, vol., pp Bollerslev, T.- Domowiz, I. (1993): Trading paerns and prices in he inerbank foreign exchange marke. Journal of Finance, vol. 48, pp Dacorogna, M.M.- Muller, U.A.- Nagler, R.J.- Olsen, R.B.- Pice, O.V. (1993): A geographical model for he daily and weekly seasonal volailiy in he foreign exchange markes. Journal of Inernaional Money and Finance, vol. 1, pp Dacorogna, M.M.- Gencay, R.- Muller, U.A.- Olsen, R.B.- Pice, O.V. (001): An inroducion o high frequency finance, Academic Press, California. 1. Danielsson, J.- de Vries, C. (1997): Tail index and quanile esimaion wih high frequency daa. Journal of Empirical Finance, vol. 4, pp Davison, A.C.- Smih, R.L. (1990): Models of exceedances over high hresholds. Journal of Royal Saisical Sociey, vol. 5, pp Embrechs, P.- Kluppelberg, C.- Mikosh, T. (1997): Modeling exremal evens for insurance and finance. New York, Springer Verlag Book. 15. Gencay, R.- Selcuk, F.- Ulugulyagci, A. (003): High volailiy, hick ails and exreme value heory in Value-a-Risk esimaion. Insurance: Mahemaics & Economics. 16. Gio, P. (005): Marke risk models for inraday daa. European Journal of Finance, forhcoming. 17. Heson, S.L. (1993): A closed-form soluion for opions wih sochasic volailiy wih applicaions o bond and currency opions. Review of Financial Sudies, vol. 6, pp Hill, B.M. (1975): A simple general approach o inference abou he ail of a disribuion. The Annals of Saisics, vol. 3, pp J.P. Morgan. (1995): RiskMerics. Technical Manual. 0. Jorion, P. (000): Value a Risk. New York, McGraw Hill Book. 1. McNeil, A.J. (1999): Exreme value heory for risk managers, Preprin, ETH Zurich.. McNeil, A.J.- Frey, R. (000): Esimaion of ail-relaed risk measures for heeroskedasic financial ime series: an exreme value approach. Journal of Empirical Finance, vol. 7, pp

21 3. Muller, U.A.- Dacorogna, M.M.- Olsen, R.B.- Pice, O.V.- Schwarz, M.S.- Morgenegg, C. (1990): Saisical sudy on foreign exchange raes, empirical evidence of a price change scaling law, an inra-day analysis. Journal of Banking and Finance, vol. 14, pp Nekhili, R.- Salih, A.A- Gencay, R. (00): Exploring exchange rae reurns a differen ime horizons. Physica A, vol. 313, pp Pickands, J. (1975): Saisical inference using exreme order saisics. The Annals of Saisics, vol. 3, pp

22 Table 1: VaR violaion raios for daily reurns 1% confidence level 5% confidence level Reurns Variance Mehod Hill Esimaor GPD Reurns Variance Mehod Hill Esimaor GPD Empirical Empirical * Normal GARCH * 0.0 Normal GARCH 3.8* Suden GARCH 0.5* Suden GARCH 7.7* Normal IGARCH Normal IGARCH Suden IGARCH Suden IGARCH Ornsein-Uhlenbeck Ornsein-Uhlenbeck JUMP * JUMP * * Good performance (he heoreical raio, α, is wihin he confidence inerval). Table : VaR violaion raios for 1 hour reurns 1% confidence level 5% confidence level Reurns Variance Mehod Hill Esimaor GPD Reurns Variance Mehod Hill Esimaor GPD Empirical Empirical Normal GARCH 1.3* 1.1* 0.0 Normal GARCH 4.1* Suden GARCH 0.8* Suden GARCH 6.3* Normal IGARCH * Normal IGARCH 7.* Suden IGARCH Suden IGARCH Ornsein-Uhlenbeck Ornsein-Uhlenbeck JUMP JUMP * * Good performance (he heoreical raio, α, is wihin he confidence inerval). Table 3: VaR violaion raios for 6 hour reurns 1% confidence level 5% confidence level Reurns Variance Mehod Hill Esimaor GPD Reurns Variance Mehod Hill Esimaor GPD Empirical Empirical Normal GARCH 0.8* Normal GARCH Suden GARCH 0.8* Suden GARCH Normal IGARCH Normal IGARCH Suden IGARCH Suden IGARCH Ornsein-Uhlenbeck Ornsein-Uhlenbeck JUMP JUMP * Good performance (he heoreical raio, α, is wihin he confidence inerval). Table 4: VaR violaion raios for 30 minue reurns 1% confidence level 5% confidence level Reurns Variance Mehod Hill Esimaor GPD Reurns Variance Mehod Hill Esimaor GPD Empirical Empirical Normal GARCH Normal GARCH Suden GARCH Suden GARCH Normal IGARCH Normal IGARCH Suden IGARCH Suden IGARCH Ornsein-Uhlenbeck Ornsein-Uhlenbeck JUMP JUMP * * Good performance (he heoreical raio, α, is wihin he confidence inerval). 1

23 Table 5: Realiy Check a 1% confidence level Table 6: Realiy Check a 5% confidence level Reurns VaR Mehod Loss Funcion* Reurns VaR Mehod Loss Funcion* Daily Horizon Daily Horizon Normal GARCH Hill Normal GARCH Variance Suden GARCH Variance Suden GARCH Variance JUMP GPD JUMP GPD Hour Horizon 1 Hour Horizon Normal GARCH Hill Normal GARCH Variance Normal GARCH Variance Suden GARCH Variance Suden GARCH Variance Normal IGARCH Variance 35.8 Normal IGARCH GPD JUMP GPD Hour Horizon 6 Hour Horizon Normal GARCH Variance Suden GARCH Variance 30.4 Suden GARCH Variance Minue Horizon JUMP GPD 7.87 * This value is muliplied by 10 4.