Backtesting the tail risk of VaR in holding US dollar

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1 See discussions, sas, and auhor profiles for his publicaion a: hps:// Backesing he ail risk of VaR in holding US dollar Aricle in Applied Financial Economics February 2009 DOI: / Source: RePEc CITATIONS 3 READS auhor: Woon K. Wong Cardiff Universiy 33 PUBLICATIONS 390 CITATIONS SEE PROFILE All in-ex references underlined in blue are linked o publicaions on ResearchGae, leing you access and read hem immediaely. Available from: Woon K. Wong Rerieved on: 13 May 2016

2 Backesing he Tail Risk of VaR in Holding US Dollar Woon K. Wong Invesmen Managemen Research Uni Paper IMRU

3 Backesing he Tail Risk of VaR in Holding US Dollar Woon K. Wong * This version: 2007/6/18 * Deparmen of Banking and Finance, Tamkang Universiy, 151, Ying-Chuan Road, Tamsui, Taipei Couny, Taiwan, R.O.C. Tel: ; Fax: ; wkw_sgi@mail.ku.edu.w A member of IMRU (for Invesmen Managemen Research Uni ) a Cardiff Business School, Cardiff Universiy, Aberconway Building, Colum Drive, Cardiff, CF10 3EU, Unied Kingdom 0

4 Backesing he Tail Risk of VaR in Holding US Dollar Absrac US dollar is he mos widely held currency in he world. In recen years, however, i suffered huge depreciaion. In his paper, various risk models are used o forecas he Value-a-Risk (VaR) in holding he currency. Being a quanile measure, VaR disregards valuable informaion conveyed by he sizes of ail losses. As a resul, here is ail risk in he use of VaR in pracice. Saddlepoin echnique is used o backes ail risk of VaR by summing all he ail losses. Subsanial downside ail risks are deeced in he US currency, and Asymmeric Power ARCH wih normal inverse Gaussian innovaion is found capable of capuring such risks. Key words: Value-a-Risk; ail risk; foreign exchange; backesing; saddlepoin echnique JEL Classificaion: C53, G32 1

5 1. Inroducion Value-a-Risk (VaR) is now he sandard risk measure sancioned by he Basle Commiee for marke risk capial deerminaion hrough he so called inernal models. 1 One reason for he preference of VaR over oher risk measures is due o he fac ha he backes of VaR can be implemened easily. VaR is defined as he maximum possible loss of a porfolio over a specified period a a given confidence level. Backesing VaR as sipulaed by he Capial Accord basically involves couning he number of excepions or ail losses exceeding VaR. Among academic lieraure, for example, Kupiec (1995) and Chrisoffersen (1998) also use number of excepions as basis for backesing VaR. Unforunaely, frequency-based backess of VaR disregard valuable informaion conveyed by he sizes of ail losses. The oucome is weak es power and inference made can be decepive of he underlying risk. For insance, Berkowiz and O Brien (2002) sudy he rading profis and losses of six large mulinaional banks from January 1998 o March Using backess proposed by Kupiec and Chrisoffersen, hey find he banks inernal VaR models provide sufficien coverage for marke risk. Neverheless, losses were huge when breaches of VaR occurred during he volaile period of fall Tha is, here can be subsanial ail risk in he use of VaR ha is overlooked by risk models. 2 In his paper, I invesigae he ail risk of VaR in holding US dollar and apply Wong s (2007) proposal o backes he risk models by summing he ail losses beyond VaR. A survey conduced by 52 cenral banks in April 2004 showed ha average daily foreign exchange urnover sood a $1.9 billion, a rise of more han 30% compared o 2001 figure; see Galai and Melvin (2004). Being he mos acively raded, US dollar is also he mos widely held currency. In recen years, however, i suffered huge depreciaion agains he world major currencies. From 2002 o 2006, for insance, US dollar has declined by roughly 40%, 30% and 10% agains he Euro, Briish Pound and Japanese Yen respecively. Given he relaively loose moneary policy pursued by 1 Basle Commiee on Banking Supervision (1996a, Secion B.4) sipulaes ha banks may use inernal models o calculae he marke risk capial requiremen, which is defined as he higher of (i) is previous day s 10-day VaR and (ii) an average of he 10-day VaRs over he preceding 60 business days muliplied by a muliplicaion facor of a leas 3. 2 See also Leippold (2004) and Yamai and Yoshiba (2005) regarding furher descripion on he concep of ail risk. 2

6 he US Federal Reserve and he long-susained US curren accoun deficis, i may be an expeced oucome for he US dollar o experience a long erm decline in value; see for example Mann (2002) and Ogawa and Kudo (2004). Wha is of ineres is wheher here are infrequen bu huge losses in holding US dollar, since he Capial Accord emphasizes on having sufficien capial o absorb such huge losses. The aim of his paper is o invesigae if such exreme losses can be capured by risk models widely sudied in he lieraure. In paricular, risk models are rigorously esed by aking ino consideraion he sizes of losses beyond he VaR level. VaR, by is definiion, is concerned wih he value ha is a risk wih only one percen chance of happening. Since he Basle regulaion requires ha backes o be carried ou on pas 250 observaions, here are on average only wo o hree realized losses in a year s sample. Thus backesing based on sizes of losses observed in a small sample is no easy. However, Wong (2007) successfully overcomes he small sample problem by using saddlepoin echnique. Specifically, saddlepoin echnique allows small sample ail probabiliy o be approximaed accuraely for he ail risk (TR) saisic, which is defined as he sum of sizes of all losses in excess of VaR divided by he sample size. Mone Carlo simulaions show ha he echnique yields accurae es size and is very powerful agains excessive ail risk even for small samples. This paper explois he small sample es power of he saddlepoin backes by carrying ou annual as well as full sample backes of ail risk of VaR. Readers are referred o Wong (2007) for furher exposiion of he advanages of he saddlepoin backes. There are wo imporan componens o be modeled in a risk forecas, namely he dynamics of heeroscedasiciy and he disribuion of innovaions. This paper considers he GARCH process proposed by Bollerslev (1986) and he Asymmeric Power ARCH (APARCH) model by Ding e al. (1993) o describe he ime-varying condiional variances of daily reurns of foreign exchange rae reurns. For he innovaions, disribuions fied are sandard normal (n), Suden (), skewed Suden (s) and normal inverse Gaussian (nig). One example of risk model from above combinaions of heeroscedasiciy and innovaion disribuion is GARCH-n used by Berkowiz and O Brien (2002) o forecas US major commercial banks daily VaR. Anoher example is he APARCH-s considered by Gio and Lauren (2003) and Tu and Wong (2006) o forecas VaR for equiy markes. Using frequency-based backess, 3

7 all of hem find he risk models provide saisfacory risks coverage. Neverheless, he saddlepoin backes of ail risk of VaR reveals a differen sory in he case of foreign exchange rae reurns. For annual backess, GARCH-n fails o provide sufficien risk coverage for he hree mos acively raded currencies, i.e. Euro, Japanese Yen and Briish Pound. Also, significan downside ail risks are observed; US dollar reurns versus Euro and Briish Pound are highly skewed o he lef, and APARCH-s fails o explain all of he skewness presen in he ails of reurns disribuion. Recenly, normal inverse Gaussian disribuions were found o provide very accurae fiing for he ails of financial reurns; see Eberlein e al. (1998) and Jensen and Lunde (2001). Forsberg and Bollerslev (2002) in paricular find he disribuion fis very well for foreign exchange rae reurns. Consisen wih hese recen findings, APARCH-nig is found o be he bes risk model in providing accurae risks coverage. This paper is organized as follows. Secion 2 inroduces he saddlepoin echnique proposed by Wong (2007) for backesing VaR based on ail losses. Secion 3 considers risk models for forecasing VaR of foreign exchange rae reurns whereas he associaed backes resuls are repored in Secion 4. Finally, Secion 5 concludes. 2. Backesing Tail Risk of VaR This secion inroduces he small sample asympoic echnique proposed by Wong (2007) o backes he ail risk of VaR. The ail risk (TR) es saisic is firs presened. The required p-value of backes is obained under normal null hypohesis. For non-normal null hypohesis, one can firs ransform he empirical disribuion ino a normal one as suggesed by Berkowiz (2001) and Kerkhof and Melenberg (2004), and hen apply he saddlepoin resuls described in his secion. The Mone Carlo simulaion sudies carried ou by Wong shows ha he saddlepoin mehod is very accurae and powerful even for small sample size. This secion ends wih a discussion on he relaionship beween ail risk and ES in risk managemen backesing. 2.1 The ail risk saisic Le r be he random profi or loss of a porfolio over a holding period and q be 4

8 he α quanile of r, i.e. VaR a 1 α confidence level. Through ou his paper, a loss is represened by a negaive value, so ha q = VaR is less han zero. Now define a random variable X wih realizaion x such ha r q if r < q, x = (1) 0 if r q. Given a sample of T reurns r, K, r }, he ail risk of VaR can be simply calculaed as he arihmeic mean of x: { 1 T T 1 TR = x = x. (2) T Noe ha x (,0] and he expecaion of X is he heoreical value of TR. Tail risk as defined above is relaed o VaR and expeced shorfall 3 following manner (see Inui and Kijima (2005) and Wong (2007)): = 1 (ES) in he 1 α TR = ES VaR, (3) where for marke risk, Basle II ses α = Thus TR refers o he risk a he par of ail ha is disregarded by VaR. Though TR is closely relaed o ES, here is a suble difference beween hem in backesing. TR ells a risk manager he oal sum of ail losses a rading porfolio may incur over a period whereas ES is he expeced loss when VaR is breached. In his regard, he saddlepoin backes of ail risk of VaR is similar o a frequency-based backes of VaR in ha while he laer couns he number VaR breaches, he former sums all he ail losses. Therefore, backesing he ail risk of VaR can be regarded as a direc and complemenary counerpar o ha of he Basle s backesing procedure for VaR. 2.2 Normaliy assumpion The required p-value of he saddlepoin backes is obained under he null assumpion ha r is normally disribued for he following wo reasons. Firs, as poined ou by Loz and Sahl (2005), he daily reurns of a well diversified porfolio are ofen fied wih a normal disribuion hanks o he workings of he cenral limi heorem. So he saddlepoin backes can be applied direcly o see wheher he normaliy assumpion holds. Second, a non-normal null hypohesis can always be urned ino a normal one by applying an appropriae ransformaion o he empirical disribuion. The idea is ha if he observed reurns are faer-ailed han he model 3 Expeced shorfall is defined as he average of losses exceeding VaR; see Arzner (1997, 1999). 5

9 presumes, he ransformed daa will be also faer-ailed han he normal hypohesis. From now onwards, unless saed oherwise, le r be he porfolio reurn wih a sandard normal disribuion and denoe is CDF and PDF by Φ and φ respecively. Since q refers o he α-quanile of a sandard normal disribuion, q = Φ 1 ( α). I is no difficul o show ha he mean of X, which is expecaion of TR, is given by and is variance is µ X 2 /2 q e = E( X ) = qα, (4) 2π σ 2 X = var( qe q /2 2 2 X ) = α + q α + µ X 2 2π. (5) Given a sample { x } of size T, he well known cenraliy heorem saes ha he sample mean, x, ends o normaliy as T ends o infiniy: x µ X T σ X d N(0,1). (6) Since he Basle Commiee se α = 0.01 for marke risk measuremen, he sample { x } are mosly zeroes. This renders above limiing resul wih large approximaion errors for sample sizes we would encouner in pracice. 2.3 The saddlepoin echnique In order o provide an accurae mehod o calculae he p-value for saisical backes of ail risk, Wong (2007) resors o he saddlepoin echnique as follows. Firs, consider an IID sandard normal random sample { r } of size T, consruc { x } according o (1). The PDF of X can be defined in erms of φ as φ ( x + q) f ( x) = 1 α if x < 0, if x = 0, and he corresponding cumulan generaing funcion is given by The saddlepoin ϖ saisfying 2 ( exp( q + /2) Φ( q ) + α ) K ( ) = ln 1. (8) (7) K (ϖ ) = x (9) plays a crucial role in he expansion of he rue densiy of TR (= x ) by means of 6

10 seepes decen; see Daniels (1954, 1987). Now define η = ϖ T K (ϖ ) and ς = sgn( ϖ ) 2T ( ϖ x K( ϖ )), where sgn(s) equals o zero when s = 0, or akes he same sign of s when s 0. According o Lugannani and Rice (1980), he ail probabiliy of exceeding he sample mean x can be obained by saddlepoin echnique as 1 1 P ( X x) = 1 Φ( ς ) + φ( ς ). (10) η ς To obain he ail probabiliy in pracice, he saddlepoin ϖ is obained by solving for in (9) so ha he required cumulan and is derivaives can be evaluaed a = ϖ. Then cumulan derivaives are used o calculae η and ς as defined above, which in urn are used o calculae he required ail probabiliy given by (10). Finally, i is noed ha when x = 0, P ( X 0) = Hypohesis esing For hypohesis esing, i is more inuiive o use he sandardized es saisic ~ 1 = T σ X (( x) ( )) z µ X, (11) in which he ail risk is represened by a posiive value. The advanage of considering he sandardized es saisic is ha for large sample, is criical values are approximaely hose of a sandard normal disribuion. While he likelihood raio backess of Kupiec (1995) and Berkowiz (2001) are wo-ailed ess, he saddlepoin backes based on (11) can be eiher one- or wo-ailed. For a wo-ailed backes, he null and alernaive hypoheses are H : ~ z 0 versus H : ~ z 0. 0 = 1 To carry ou a one-ailed regulaory backes o check wheher a risk model provides sufficien ail risk coverage, he null and alernaive hypoheses become H : ~ z 0 versus H : ~ z > Since for hypohesis esing, porfolio losses are represened by posiive numbers, he risk model is said o capure he ail risk of VaR if ~ z is zero of negaive. On he oher hand, if he alernaive hypohesis is acceped, hen he risk model is described as failing o capure he ail risk of VaR. 2.5 Comparing o oher size-based backess 7

11 Though various backess of ES have been proposed, noiceably he censored Gaussian approach of Berkowiz (2001) and he funcional dela mehod of Kerkohf and Melenberg (2004), i is useful o consider he backes of ail risk of VaR for he following reasons. Firs, VaR is now ubiquious in risk managemen. I is unlikely ha VaR will be replaced by expeced shorfall. 4 Since he Basle Commiee sipulaes ha VaR be backesed by couning he number of excepions, i is meaningful o carry ou he proposed backes based on he sum of sizes of all excepions. Second, he Basle Commiee requires VaR o be complemened by oher risk analyses such as sress esing; see Basle Commiee on Banking Supervision (1996b). While sress esing offers rigorous and comprehensive scenarios sudy, by is very naure, i lacks uniformiy and someimes can even be subjecive. 5 In his conex, he saddlepoin backes of ail risk of VaR plays an excellen complimenary role in risk managemen since i saisically ess he sizes of ail losses for possible inadequacy of risk models. Third, boh he censored Gaussian and funcional dela approaches require large sample for heir limiing resuls o approximae well. Since regulaory backess are carried ou on samples as small as 250 observaions, hese wo approaches are no accurae enough. Finally, Berkowiz s censored Gaussian approach is a es of join hypohesis of zero mean and uni variance. I is a wo-sided es and hus is no suiable for regulaory backesing. This is because financial flucuaions by heir naure are very complicaed and i is likely ha a perfec model can never be fied. While a risk manager may no know he perfec model, i is feasible ha he can always allow for possible inadequacy in her risk model by marking up he risk forecas. Such pracice is eviden from Berkowiz and O Brien (2002) who observe ha banks VaR forecass are conservaive. This is also in line wih he one-ailed definiion and he corresponding backesing procedure of VaR by he Capial Accord; see Basle Commiee on Banking Supervision (1996a). 4 Expeced shorfall is a coheren risk measure ha saisfies subaddiiviy, homogeneiy, monooniciy and ranslaion invariance properies. VaR is no subaddiive and is hus no a coheren risk measure. For his reason, some academics sugges ha VaR could be replaced by ES; see Arzner (1997, 1999) for furher deails. 5 For example, in addiion o cerain specified sress scenarios he bank porfolios are required o subjec o, banks are also expeced o develop heir own sress ess which hey idenify as mos adverse for he porfolios concerned. 8

12 3 Risk models Here I consider wo ypes of heeroscedasiciy and four differen disribuions of innovaion processes, yielding a oal of eigh combinaions of risk models for forecasing currency risk. 6 The wo heeroscedasiciy models are GARCH by Bollerslev (1986) and Asymmery Power ARCH (APARCH) by Ding e al. (1993). The innovaion processes are IID and have zero mean and uni variance. They are normal (n), Suden (), Skewed Suden (s) and normal inverse Gaussian (nig). Some examples in applying hese risk models include he use of GARCH-n by Berkowiz and O Brien (2002) in forecasing VaR of rading accouns a six US major commercial banks. Gio and Lauren (2003) and Tu and Wong (2005) use APARCH-s o forecas VaR for long and shor rading posiions in various equiy markes. For normal inverse Gaussian disribuion, Forsberg and Bollerslev (2002) find he disribuion fis very well foreign exchange rae reurns daa. Also, Jensen and Lunde (2001) find APARCH-nig as a sochasic volailiy model fis ails of equiy reurns well, whereas Eberlein e al. (1998) use nig o forecas equiy VaR. Now le r be he foreign exchange rae reurn on day. The reurn is assumed o follow an AR(1) process in he mean level, so i can be wrien as r = µ + ε = a + a r + ε, (12) ε = σ z. (13) The specificaions of he heeroscedasic dynamics and innovaions processes are now described below. 3.1 Model specificaion The GARCH dynamic proposed by Bollerslev (1986) is saed as follows: = ω αε 1 βσ 1 σ. (14) The second heeroscedasic process, APARCH, incorporaes asymmery effec and flexible power law in σ. I can be specified as follows: 6 The risk models considered in his paper are for daily VaR forecass. For longer horizon of monhly forecass, readers are referred o Sarore e al. (2002). 9

13 δ δ δ σ = ω + α( ε 1 αnε 1) + βσ 1, (15) where α n allows for 1 ε o have an asymmeric effec on σ, and δ specifies he ype of power law in σ. All innovaions are sandardized IID processes wih zero mean and uni variance. The sandard normal IID process is denoed as z ~ N(0,1). Le υ be he degree of freedom parameer and he Suden process is wrien as ( 0,1, υ). For he Skewed Suden innovaions, z ~ s (0,1, υ, ξ ) and he PDF funcion is given by f s 2s m g[ ξ ( sz + m) υ] if z <, 1 ξ + ξ s ( z ξ, υ) = (16) 2s 1 m g[ ξ ( sz + m) υ] if z, 1 ξ + ξ s where g ( υ) is he sandardized Suden densiy. While υ is he degree of freedom parameer ha describes faness a he ails, ξ is he skewed parameer which is less (greaer) han 1 if he disribuion is skewed o he lef (righ). The oher wo parameers, m and s 2 non-sandardized skewed Suden: are respecively he mean and variance of he oherwise Γ(( υ 1) / 2) υ 2 1 m = ( ξ ξ ), (17) π Γ( υ / 2) s = ( ξ + ξ 1) m. (18) Finally, here are four parameers for he normal inverse Gaussian innovaions. The parameers are µ, ν, γ and κ which specify respecively he locaion, scale, skewness and resriced by seing flaness of he disribuion. The locaion and scale parameers are v = κ ( 1 ρ 2 ) 3/2 and = ρv ( 1 ρ 2 ) 1/2 where ρ = γ κ µ ha he process has zero mean and uni variance. In his seing, he sandardized normal inverse Gaussian process have only wo free parameers and is wrien as 2 2 1/2 z ~ nig(0,1, γ, κ). Le h ( z) = κ[ v + ( z µ ) ], hen he densiy of nig( 0,1, γ, κ) is, so given by f NIG /2 ( π h( z) ) κ v exp{ v ( κ γ ) + γ ( z µ )} K ( κ h( z) ) ( z) = 1, (19) where K ( ) is he modified Bessel funcion of he hird kind. 1 x 10

14 3.2 VaR calculaions and backess If a European invesor holds US dollar, he faces downside risk of US currency (or a fall in EUR/USD foreign exchange rae). Specifically, he downside risk of holding a long posiion in US dollar can be represened by VaR as where VaR α = µ + Zασ, (20) Z α is he lef α-quanile of he corresponding innovaion disribuion. For he upside risk of a shor posiion in US currency, he corresponding VaR is given by = + 1 VaR 1 α µ Z ασ. (21) Under he Basle regulaion, α is se a 1% for marke risk measuremen. For sandard normal disribuion, for insance, Z Z = The 1% quanile of he α = 1 α sandardized Suden disribuion depends on degree of freedom parameer, and readers are referred o Gio and Lauren (2003) for he quanile formula of he sandardized skewed Suden disribuion. Numerical mehods are required o obain he quanile values of he sandardized normal inverse Gaussian disribuion by using he fac ha he long- and shor-posiion VaR can be obained respecively from P < VaR α Ω ) = α and P VaR α Ω ) = α, ( r 1 where Ω 1 is he informaion se a ime -1. ( r < To carry ou he saddlepoin backes, i is required o ransform he empirical disribuion ino a sandard normal one if we have a non-normal null hypohesis. As an example, if VaR is forecased using APARCH-, hen under he null hypohesis, sandard normal random deviaes can be obained using Φ 1 ( ( z υ)) where G ( υ) is he sandardized Suden CDF. Readers are referred o Lamber and Lauren (2002) for he corresponding ransformaion of Skewed Suden disribuion. Finally, for he normal inverse Gaussian innovaion process, one has o resor o numerical mehod o obain he associaed probabiliy value p of an observed innovaion z. Normal deviae is hen simply given by Φ 1 ( p). G 4. Backess of foreign exchange rae risk models 11

15 4.1 Daa Four currencies versus US dollar are invesigaed in his paper. They are Euro, Japanese Yen, Briish Pound and Canadian dollar. The daa are obained daily from Daasream and range from 3 January 1994 o 12 January 2007, a oal of 3,365 observaions. 7 Basic saisics of daily reurns of he foreign exchange raes are provided in Table 1. < Inser Table 1 > All exchange raes are measured in unis of foreign currency per US dollar and reurns are calculaed as differences of log of foreign exchange raes in percenage poins. Thus we can see from Table 1 ha he nominal reurns of holding US dollar in he sample period are negaive excep for Japanese Yen, i.e. US dollar appreciaed in his period only agains Japanese Yen. More imporanly, all invesors ouside US holding he currency face risk of large losses. This is eviden from he absolue minima being larger han maxima as well as negaive skewness saisics for all four currencies. 4.2 Parameer esimaion All four foreign exchange daily log reurns are fied wih he eigh risk models described in Secion 3. Due o consrain of space, Table 2 only abulaes he esimaed parameers of APARCH-s and APARCH-nig based on full sample. < Inser Table 2 > Consisen wih lieraure ha foreign exchange markes are highly efficien, coefficiens a he mean level ( a 0, a 1 ) are no significan. Also, shocks do no have asymmeric effecs on fuure volailiy, excep for he case of APARCH-nig fied on Canadian dollar. Though he skewness parameers for boh risk models ( ln ξ, κ ) are significan only for Euro and Yen, he esimaed coefficiens are all negaive which indicaes he risk of exreme losses in holding US currency. In all cases, he disribuions of innovaion processes are faer-ailed han a normal one. 4.3 Backes on full forecas sample Kupiec backes For purpose of backesing, ou-of-sample daily VaR forecass are generaed from 7 Prior o he formaion of Euro, he currency is calculaed as weighed average of consiuen European counry currencies. 12

16 1 January 1999 o 17 January Specifically, 1,000 observaions prior o 1 January 1999 are firs used o esimae he risk models and hen produce en one-day-ahead VaR forecass. The esimaion window is rolled forward en rading days and a new se of model parameers are obained. These newly esimaed parameers are in urn used o forecas nex en one-day-ahead VaRs. In his manner, he process is coninued unil he las day of he sample is reached. Frequency-based Kupiec backess are applied o he full forecas sample and he resuls are shown in Table 3. 8 < Inser Table 3 > Figures abulaed in Table 3 are excepion or failure raes and * (**) indicaes he null hypohesis of risk coverage is rejeced a 5% (1%) significance level. From Panel A in which failure raes for downside VaR are shown, GARCH-n and APARCH-n are found o fail he backess for all four currencies, a resul ha is consisen wih mos lieraure ha normal ail is oo hin o capure he exreme risk. Panel B provides he backes resuls for upside VaR which reveal ha GARCH- and APARCH- are rejeced for Euro and Yen currencies. Under he null hypohesis of correc coverage, empirical excepion rae equals one. Since he empirical failure raes are significanly less han one, i can be said ha boh risk models provide oo much coverage. In conras, regardless of ype of heeroscedasiciy, risk models wih Skewed Suden or normal inverse Gaussian innovaions forecas saisfacory VaRs for holding boh long and shor posiions in US dollar. Saddlepoin backes Table 4 presens he sandardized es saisic ~ z for backesing he ail risk of VaR forecased by he risk models described in Secion 3. As sample size T ends o infiniy, ~ z ends o a sandard normal deviae under he null hypohesis. A oo large z~ value means ha he risk model fails o provide sufficien coverage for ail risk of VaR. On he oher hand, a significanly negaive ~ z saisic means ha he empirical ail is oo hin and hus providing excessive coverage for he ail risk of VaR. < Inser Table 4 > Panel A presens he saddlepoin backess of ail risk of VaR for holding a long 8 Chrisoffersen s (1998) independence and condiional coverage backess have also been applied o he risk forecass. Empirical resuls indicae much of dependence srucures have been removed by he risk models and he condiional coverage backess produce resuls qualiaively similar o Kupiec backess. 13

17 he risk model a 5% (1%) level. 9 < Inser Table 5 > posiion in US dollar. I can be seen ha all risk models wih normal innovaions are rejeced for no providing sufficien coverage for he ail risk of VaR. A firs glance, his oucome is qualiaively similar o he Kupiec backess shown in Table 3. Closer observaion reveals ha seven of he rejecions made by saddlepoin backess are a 1% significance level whereas only 3 rejecions are made a 1% significance level a Kupiec case. This is consisen wih he fac ha Kupiec backes can be poor in es power even for large sample since i uses a frequency-based approach. Evidence of higher es power by saddlepoin backess are found in Panel B, where VaR is forecased for holding shor posiions in US dollar. For Euro, Japanese Yen and Briish pounds, mos risk models are found o provide excessive ail risks coverage. This illusraes he difficuly in modeling and forecasing he ails of skewed reurns in pracice. For Canadian dollar, GARCH and APARCH wih normal innovaions do no even provide sufficien ail risk coverage hough Kupiec backess accep hese wo risk models. Ou of all he backess, APARCH wih nig innovaions provide saisfacory risk forecass for all currencies. 4.4 Annual backess I now apply annual VaR backesing o he esimaed risk models along he line of Basle rules. The es is one-sided and is carried ou on a year-o-year basis. I is remarked here ha a year s daily reurns is oo small a sample size for a wo-sided es. Specifically, a 99%-coverage for marke risk implies a probabiliy of for having zero excepion in 250 rading days, higer han he significance level of 0.05 in rejecing a null hypohesis. For his reason as well as consrain of space, I repor here only he resuls for one-ailed Bernoulli and saddlepoin backess of GARCH-n and APARCH-nig risk models. Given T reurns wih b losses exceeding VaR, he p-value of a one-ailed Bernoulli es is given by P( B b) where B is disribued as Binomial ( T, α). Therefore given 250 reurns, 6 (8) excepions warran a rejecion of 9 Some aricles define he 5% significance level b as he smalles ineger such ha P ( B b) > For B ~ Binomial (250, 0.01), b = 5. However, his definiion is no adoped by his paper because P ( B = 5) =

18 Table 5 repors he backes resuls of VaR forecased by GARCH-n from 1999 o Consisen wih he resuls shown in Table 3 and 4, Panel A reveals ha significan downside ail risks are deeced if GARH-n is used as a risk model for Euro, Japanese Yen and Briish Pound. In paricular, 1999 and 2000 are wo years in which he GARCH-n is rejeced for no providing sufficien coverage for all hree currencies. While he downside ail risk for Euro-denominaed invesors subsides from 2001 onwards, Yen-denominaed invesors have o face subsanial ail risks in 2002 and Overall, here are 5 Bernoulli rejecions, all a 5% significance level whereas saddlepoin backess yield 9 rejecions a 1% significance levels. On he backess of VaR for holding shor posiions in US dollar, Panel B reveals an ineresing oucome ha differs from full sample backess. Insead of excessive coverage, one finds hree independen cases of insufficien coverage: wo by saddlepoin echnique and one by Bernoulli mehod. < Inser Table 6 > Finally, Table 6 presens he backes resuls when APARCH-nig risk model is used. Consisen wih he full sample case, risk forecass by APARCH-nig perform well when backesing is carried ou annually. Only one Bernoulli rejecion and one saddlepoin rejecion in each side of he reurns disribuion is observed. However, a rejecion a 1% significance level ou of 32 backess is saisically no significan he Binomial probabiliy of having one success wih 1% rae ou of 32 rials is as high as Therefore, ogeher wih he empirical backes resuls of APARCH-nig in he full sample case, I conclude ha he risk model capures well he ail risk of VaR in foreign exchange raes. 5. Conclusion I is now well known ha VaR has cerain undesirable heoreical properies as a risk measure. In paricular, VaR is a quanile measure ha disregards informaion conveyed by he sizes of ail losses, giving rise o ail risk in he use of VaR in pracice. However, VaR is now he risk measure sancioned by he Basle Commiee and is wide usage is difficul o be changed. Therefore, i is useful o backes he ail risk of VaR in a manner ha is complimenary o he backesing procedure sipulaed by 15

19 Basle II. Specifically, he Basle rules coun he number of excepions whereas in his paper VaR is backesed by summing he sizes of ail losses. The aim of his paper is o invesigae and backes he ail risk of VaR in holding US dollar. Risk models ha accoun for heeroscedasiciy, skewed and fa-ailed reurns are used o fi four major currencies, namely Euro, Japanese Yen, Briish Pound and Canadian dollar versus US dollar. The VaRs for holding long and shor rading posiions in US dollar are forecased and backesed using he frequency-based Bernoulli and Kupiec ess as well as he saddlepoin es based on sizes of ail losses. Possibly due o relaively loose moneary policy pursued by Federal Reserve and huge US curren accoun deficis in recen years, he empirical findings of his paper reveal subsanial downside risks in holding he US dollar. Moreover, hese downside ail risks canno be capured by GARCH and APARCH models wih normal innovaions. Though Suden disribuion provides sufficien coverage for downside risk, is ail is oo hick for invesors who hold shor rading posiions in US dollar his suggess ha he ail of US dollar reurns is hicker on he lef (or upside) since Suden disribuion is symmeric. The skewness in he reurns of US currency is also apparen from he basic saisics and risk models (in paricular, he skewed parameers) of he foreign exchange raes reurns. Wheher he recen skewed US dollar reurns are caused by he loose moneary policy and large curren accoun deficis cerainly meris furher research. One conclusion of his paper is ha he skewed Suden innovaions are able o fi mos of he skewed reurns bu APARCH wih normal inverse Gaussian innovaions provide he bes VaR forecass ha pass all saddlepoin backess. Anoher observaion worh menioning is ha he backes resuls are less affeced by he ype of heeroscedasiciy han he ail specificaion of innovaion process. For example, boh GARCH- and APARCH- provide oo much cover for invesors holding shor posiions in US dollar whereas GARCH-nig and APARCH-nig provide comparable risk forecass (GARCH-nig fails wo backess only marginally a 5% significance level). This finding can be beer undersood if one considers he fac ha VaR is se a he far end of disribuion ail by he Capial Accord, so ha he oucome of a backes hinges more on ail specificaion raher han ype of heeroscedasiciy. 16

20 Since he saddlepoin backes is a small sample asympoic echnique ha uilizes informaion conveyed by sizes of excepions, i is accurae and powerful even for small samples. This renders annual backess along he line of Basle rules feasible, and annual backess find subsanial ail risks in he hree mos acively raded currencies (Euro, Yen and Pound) during 1999 and There is also evidence ha Yen-denominaed US dollar holder coninues o face subsanial downside risk. These risks are neverheless well capured by he APARCH-nig risk model. Reference Arzner, Philippe, Freddy Delbaen, Jean M. Eber, and David Heah, 1997, Thinking coherenly, Risk 10(11), Arzner, Philippe, Freddy Delbaen, Jean M. Eber, and David Heah, 1999, Coheren measures of risk, Mahemaical Finance 9(3), Basle Commiee on Banking Supervision, 1996a, Amendmen o he Capial Accord o Incorporae Marke Risk. Basle Commiee on Banking Supervision, 1996b, Supervisory Framework for he use of Backesing in conjuncion wih he Inernal Models Approach o Marke Risk Capial Requiremens. Bollerslev, T., 1986, Generalized auoregressive heeroscedasiciy, Journal of Economerics 31, Berkowiz, Jeremy, 2001, Tesing densiy forecass, wih applicaions o risk managemen, Journal of Business and Economic Saisics 19, Berkowiz, Jeremy, and James O Brien, 2002, How accurae are Value-a-Risk models a commercial banks? Journal of Finance 57, Chrisoffersen, Peer F., 1998, Evaluaing inerval forecass, Inernaional Economic Review 39, Daniels, H. E., 1954, Saddlepoin Approximaions in Saisics, Annals of Mahemaical Saisics 25, Daniels, H. E., 1987, Tail Probabiliy Approximaions, Inernaional Saisical Review 55, Eberlein, Erns, Ulrich Keller, and Karsen Prause, 1998, New insighs ino smile, mispricing and value a risk: he hyperbolic model, Journal of Business 71,

21 406. Forsberg, L. and T, Bollerslev, 2002, Bridging he Gap beween he Disribuion of Realized (ECU) Volailiy and ARCH modeling: The GARCH-NIG model, Journal of Applied Economerics 17, Galai, G. and M. Melvin, 2004, Why has FX rading surged? Explaining he 2004 riennial survey, BIS Quarerly Review, December, Gio, Pierre, and Sebasien Lauren, 2003, Value-a-Risk for long and shor rading posiions, Journal of Applied Economerics 18, Inui, K. and M. Kijima, 2005, On he significance of expeced shorfall as a coheren risk measure, Journal of Banking and Finance 29, Jensen, M. B. and A. Lunde, 2001, The NIG-S&ARCH model: a fa-ailed, sochasic, and ARCH volailiy model, The Economerics Journal 4(2), Lamber, P. and S. Lauren, 2002, Modeling skewness dynamics in series of financial daa using skewed locaion-scale disribuions, unpublished working paper, available from hp:// Kerkhof, Jeroen and Berrand Melenberg, 2004, Backesing for risk-based regulaory capial, Journal of Banking and Finance 28, Kupiec, Paul H., 1995, Techniques for verifying he accuracy of risk measuremen models, Journal of Derivaives 3, Leippold, Markus, 2004, Don rely on VaR, Euromoney, November, London. Loz, Chrisopher, and Gerhard Sahl, 2005, Why value a risk holds is own? In: Euromoney, February 2005, London. Lugannani, R., and S. O. Rice, 1980, Saddlepoin approximaion for he disribuion of he sum of independen random variables, Advanced Applied Probabiliy 12, Mann, C. L., 2002, Perspecives on he U.S. curren accoun defici and susainabiliy, Journal of Economic Perspecives 16, Ogawa, E. and T. Kudo, 2004, How much depreciaion of he US dollar for susainabiliy of he curren accouns? RIETI Policy Symposium on June, Available from hp:// Sarore, D., Trevisan, L. Trova, M. and F. Volo, 2002, US dollar/euro exchange rae: a monhly economeric model for forecasing, European Journal of Finance 8, Tu, Anhony H., and Woon K. Wong, 2005, Value-a-Risk for Long and Shor 18

22 Posiions of Asian Sock Markes, 2005 Inernaional Conference on Business and Finance Proceedings, Tamkang Universiy, Taiwan, R.O.C. Wong, W. K., 2007, Backesing Value-a-Risk based on ail losses, 2007 Far Easern Meeing of Economeric Sociey, Taipei, available from hp:// Yamai, Yasuhiro, and Toshinao Yoshiba, 2005, Value-a-risk versus expeced shorfall: A pracical perspecive, Journal of Banking and Finance 29,

23 Table 1: Basic saisics The daily currency daa are obained from Daasream and range from 3 January 1994 o 12 January 2007, a oal of 3,365 observaions. All exchange raes are measured in unis of foreign currency per US dollar and reurns are calculaed as differences of log of foreign exchange raes in percenage poins. EUR YEN BPD CAD Mean Median Minimum Maximum Sandard deviaion Skewness Excess kurosis Jarque-Bera es Probabiliy

24 Table 2: Risk model parameers Risk models are esimaed based on full sample of 3,365 observaions using maximum likelihood mehod. APARCH-s is saed below as r µ + ε = a + a r + ε, ε = σ z, = δ δ δ σ = ω + α( ε 1 α nε 1 ) + βσ 1 z s (0,1, z ~ nig(0,1, γ, κ where and ~ υ, ξ ). The innovaion process in APARCH-nig is given by ). Negaive ξ and γ skewed parameers indicae he reurns are skewed o he lef. Boh esimaed υ and κ reveal faness in he ails of disribuions ha are non-normal. EUR YEN BPD CAD Parameers Panel A: APARCH-s a a ω * α 0.035* 0.039* 0.029* 0.046* β 0.965* 0.950* 0.964* 0.949* α n δ 1.535* 1.847* 2.112* 1.943* υ 7.800* 5.568* 7.505* * ln ξ * * Likelihood Ljung-Box (20) Parameers Panel B: APARCH-nig a a ω 0.015* 0.008* α 0.051* 0.036* 0.043* 0.046* β 0.920* 0.936* 0.944* 0.939* α n * δ 1.523* 2.261* 1.534* 2.197* κ 1.605* 1.274* 1.515* 2.108* γ * * Likelihood Ljung-Box (20)

25 Table 3: Kupiec backes of VaR Frequency-based Kupiec backess are carried ou on full sample of 2,358 VaR forecass. Under he null hypohesis of correc coverage, excepion raes equal o one. * and ** indicae rejecions of he null hypohesis a 5% and 1% significance levels respecively. An empirical excepion rae ha is significanly larger (smaller) han one implies ha an insufficien (excessive) coverage by he risk model. EUR YEN BPD CAD Panel A: Kupiec backes VaR for holding long posiions in US dollar Model Excepion raes GARCH-n 1.612** 1.654** 1.527* 1.442* GARCH GARCH-s GARCH-nig APARCH-n 1.569* 1.527* 1.866** 1.484* APARCH APARCH-s APARCH-nig Panel A: Kupiec backes VaR for holding shor posiions in US dollar Model Excepion raes GARCH-n GARCH ** 0.509** GARCH-s GARCH-nig APARCH-n APARCH ** 0.551* APARCH-s APARCH-nig

26 Table 4: Saddlepoin backes of ail risk of VaR Saddlepoin backes of ail risk of VaR are carried ou on full sample of 2,358 VaR forecass. For large sample, es saisic z ~ given by (11) is approximaely sandard normal under he null hypohesis. The required p-value of backes is calculaed using saddlepoin echnique given by (10). A z ~ value ha is significanly posiive (negaive) implies ha an insufficien (excessive) coverage of ail risk of VaR by he risk model. * (**) indicaes rejecion of he null hypohesis a 5% (1%) significance level. EUR YEN BPD CAD Panel A: Backes of ail risk of VaR for holding long posiions in US dollar Model ~ z saisics GARCH-n 7.172** ** 5.060** 2.549* GARCH GARCH-s GARCH-nig APARCH-n 7.643** 9.889** 6.714** 3.068** APARCH APARCH-s APARCH-nig Panel B: Backes of ail risk of VaR for holding shor posiions in US dollar Model ~ z saisics GARCH-n ** GARCH ** ** ** GARCH-s * ** ** GARCH-nig * * APARCH-n ** APARCH ** ** * APARCH-s * * APARCH-nig

27 Table 5: Annual backess of GARCH-n VaR based on ail losses VaRs forecased by GARCH-n are backesed annually by using Bernoulli mehod and saddlepoin echnique which sums all ail losses exceeding VaR. Boh ess are one-sided wih he null hypohesis of correc risk coverage. B refers o he number of excepions whose p-value is given by he Binomial probabiliy of P( B b) wih 250 rials and 0.01 rae of success. Saddlepoin es saisic ~ z is defined by (11) whose p-value is calculaed using (10). * (**) indicaes rejecion of he null hypohesis a 5% (1%) significance level. EUR YEN BPD CAD Panel A: Saddlepoin backes of VaR for holding long posiion in US dollar YEAR B z ~ B z ~ B z ~ B z ~ * 7.710** 6* 3.345** ** * 6.819** ** ** * 7.951** ** * ** Panel B: Saddlepoin backes of VaR for holding shor posiion in US dollar YEAR B z ~ B z ~ B z ~ B z ~ ** * **

28 Table 6: Annual backess of APARCH-nig VaR based on ail losses VaRs forecased by APARCH-nig are backesed annually by using Bernoulli mehod and saddlepoin echnique which sums all ail losses exceeding VaR. Boh ess are one-sided wih he null hypohesis of correc risk coverage. B refers o he number of excepions whose p-value is given by he Binomial probabiliy of P( B b) wih 250 rials and 0.01 rae of success. Saddlepoin es saisic ~ z is defined by (11) whose p-value is calculaed using (10). * (**) indicaes rejecion of he null hypohesis a 5% (1%) significance level. EUR YEN BPD CAD Panel A: Saddlepoin backes of VaR for holding long posiion in US dollar YEAR B TR B TR B TR B TR * 2.209* Panel B: Saddlepoin backes of VaR for holding shor posiion in US dollar YEAR B TR B TR B TR B TR ** *