GARCH Processes and Value at Risk: An Empirical Analysis for Mexican Interest Rate Futures 1

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1 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) INVESTIGACIÓN / RESEARCH GARCH Processes and Value a Risk: An Empirical Analysis for Mexican Ineres Rae Fuures 1 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. 2 1 The views expressed in his paper are hose of he auhor only and do no necessarily reflec hose of Banco de México or is saff. I am hankful o an anonymous referee and Carlos Muñoz Hink for useful commens. 2 Ph.D. Banco de México, Dirección General de Invesigación Económica. Tecnológico de Monerrey, Campus Ciudad de México. gbenavid@banxico.org.mx ABSTRACT. In his research paper GARCH processes are applied in order o esimae Value a Risk (VaR) for an ineres rae fuures porfolio. According o several documens in he lieraure, GARCH models end o overesimae VaR because of volailiy persisence. The main objecive here is o pu o es if GARCH models acually overesimae VaR. The analysis is carried ou for several ime-horizons for he above menioned asse, which has rading a he Mexican Derivaives Exchange. To analyze he VaR wih ime horizons of more han one rading day Real-World Densiies (RWD) are esimaed applying GARCH processes. The resuls show ha GARCH models are relaively accurae for ime horizons of one rading day. However, he volailiy persisence capured by hese models is refleced wih relaively high VaR esimaes for longer ime horizons. In erms of Risk Managemen his is considered undesirable given ha no-opimal amouns of capial mus be se aside in order o mee Minimum Capial Risk Requiremens for fuures porfolios. These resuls have also implicaions for shor-erm ineres rae forecass given ha RWD are esimaed. Keywords: Boosrapping, GARCH, ineres raes, Mexico, Value a Risk, volailiy persisence. RESUMEN. En el presene rabajo de invesigación se uilizan procesos GARCH para esimar el valor-enriesgo (VaR) de un porafolio hipoéico de fuuros de asas de inerés. De acuerdo con algunos documenos en la lieraura (Brooks, e. al. 2000), los modelos GARCH ienden a sobreesimar el VaR debido a que capuran la persisencia en la volailidad. El principal objeivo del presene rabajo es poner a prueba si los modelos GARCH en realidad sobreesiman el VaR. El análisis se lleva a cabo para diferenes horizones en el iempo para el acivo previamene mencionado, el cual iene negociación en el Mercado Mexicano de Derivados (Mexder). Para analizar el VaR con horizones de iempo de más de un día de negociación (rading day) densidades del mundo-real son esimadas con procesos GARCH y simulaciones Boosrapping. Los resulados muesran que los modelos GARCH son relaivamene cereros para horizones de un día de negociación. Sin embargo, la volailidad persisene capurada por ese ipo de modelos se refleja con esimados de VaR relaivamene alos para horizones de iempo mayores (Ej. de diez días de negociación ó más). En érminos de análisis de riesgos lo anerior es considerado subópimo ya que canidades innecesarias de capial endrían que desinarse para cubrir los requerimienos mínimos de capial en riesgo (Minimum Capial Risk Requiremens). Lo anerior para una posición (cora ó larga) en un porafolio de fuuros de asas de inerés. Los méodos aquí explicados pueden servir para pronosicar asas de inerés, ya que esas se esiman a ravés de esimaciones de densidades del mundo-real. La invesigación aquí realizada iene implicaciones para bancos cenrales, ya que se obienen predicciones de disribuciones de asas de inerés y se analizan esimaciones de VaR. Eso úlimo es relevane para un Banco Cenral considerando su función de supervisor financiero. Palabras clave: GARCH, Mexico, persisencia en la volailidad, remuesreo, asas de inerés, Valor en Riesgo. (Recibido: 11 de ocubre de Acepado: 14 de diciembre de 2007) 92

2 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. 1. INTRODUCTION Nowadays i is imporan o measure financial risks in order o make beer-informed decisions relevan o risk managemen. I is well documened ha volailiy is a measure of financial risk. Measuring financial volailiy of asse prices is a way of quanifying poenial losses due o financial risks. An imporan ool for his measure is o esimae volailiy-based Value a Risk (VaR). Nowadays, here are several mehods applied in order o obain a volailiy-based Value a Risk. Among he mos popular ones are he use of ARCH models. The main objecive of his paper is o analyze if Auoregressive Condiional Heeroscedasiciy (ARCH-ype) models are accurae o predic risks caused by ineres rae volailiy from a Value a Risk perspecive. The idea is o analyze if volailiy persisence inheren in his ype of models affecs VaR. Volailiy persisence in his projec refers o he financial volailiy ha akes a long ime o die away. This is done by considering a heoreical porfolio of ineres rae (Cees 91-day and TIIE 28-day) fuures. VaR is esimaed using ARCH-ype models and hen heir accuracy is formally esed wih back-esing (Kupiec: 1995, Jorion: 2000, 2001). The procedure is o find ou how accurae is he VaR wih daily ineres rae fuures observaions. The ime horizons considered are from on rading day up o six monhs or equivalen in rading days. For one rading day a parameric approach is applied. For en rading days and more Boosrapping simulaions (Enfron: 1982) are carried ou (non-parameric approach). If he number of daily violaions or excepions is reasonable according o VaR models performance crieria (Jorion: 1998), hen he models are considered accurae. Oherwise, he ARCH-ype models are rejeced. According o Pérignon, Deng and Wang (2006) banks normally over esimae VaR. The n-day forecas horizon is also inerpreed as he probabiliy ha fuure ineres rae will be wihin cerain saisical confidence inerval i.e. he 95% confidence inerval VaR. I is expeced ha hese resuls could have implicaions for forecass abou he fuure range of Mexican ineres raes. The layou of his paper is as follows. The lieraure review is presened in Secion 2. The moivaion and conribuion of his work are presened in Secion 3. The models are explained in Secion 4. Daa is deailed in Secion 5. Secion 6 presens he descripive saisics. The resuls are analysed in Secion 7. Finally, Secion 8 concludes. 2. LITERATURE REVIEW Hisorical volailiy is described by Brooks (2002) as simply involving calculaion of he variance or sandard deviaion of reurns in he usual saisical way over some long period (ime frame). This variance or sandard deviaion may become a volailiy forecas for all fuure periods (Markowiz: 1952). However, in his ype of calculaion here is a drawback. This is because volailiy is assumed consan for a specified period of ime. Nowadays, i is well known ha financial prices have ime-varying volailiy i.e. volailiy changes hrough ime (he volailiy ha i is considered here is he condiional volailiy of a financial asse and no necessarily he uncondiional one. I am hankful o Vicor Guerrero for asking me o clarify his poin). I is well documened ha non-linear ARCH models can provide accurae esimaes of ime-varying price volailiy. Jus o menion a few papers see for example, Engle (1982), Taylor (1985), Bollerslev, Chou and Kroner (1992), Ng and Pirrong (1994), Susmel and Thompson (1997), Wei and Leuhold (1998), Engle (2000), Manfredo e al. (2001), ec (For an excellen survey abou applicaions of ARCH models in Finance he reader can refer o Bollerslev, Chou and Kroner (1992)). Noneheless, here is a growing lieraure of he implicaions of non-linear dynamics for financial risk managemen (Brock e al.:1992; Hsieh: 1993). In he ligh of his opic some researchers have exended he work for he applicaion of ime-varying volailiy models, specifically ARCH-ype models, in VaR esimaions (Brooks, Clare and Persand: 2000; Manfredo: 2001; Engle: 2003; Gio: 2005; Mohamed: 2005; among ohers). Mos of hese findings enhance he use of ime-varying models in risk managemen applicaions using VaR. Even hough, here are several research papers, which used hese ypes of models for financial ime series here is, however, no works ha have analysed VaR for ineres rae fuures in an emerging economy. This is considered a gap in he lieraure. 3. MOTIVATION AND CONTRIBUTION Previous works have applied non-linear models wihin a VaR framework in order o esimae Minimum Capial Risk Requiremens (MCRRs) (Hsieh: 1991; Brooks, Clare and Persand: 2000). MCRR is defined as he minimum amoun of capial needed o successfully handle all bu a pre-specified percenage of possible losses (Brooks, Clare and Persand: 2000). This concep is relevan o banks and bank regulaors. For 93

3 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) he laer i is imporan o require banks o mainain enough capial so banks could absorb unforeseen losses. These regulaory pracices go back o he original Basle Accord of Even ough here is a broad agreemen abou he need of MCRRs here is, however, significanly less agreemen abou he mehod o calculae hem. According o Brooks, Clare and Persand (2000) he mos well known mehods are he Sandard/Inernaional Model Approach of he Basle Accord (1988), he Building-Block Approach of he EC Capial Adequacy Direcive (CAD), he Comprehensive Approach of he Securiies Exchange Commission (SEC) of he US, he Pre-commimen Approach of he Federal Reserve Board (FED) and he Porfolio Approach of he Securiies and Fuures Auhoriy of he UK. By esimaing he VaR of heir financial porfolios banks are able o calculae he amoun of MCRRs needed o mee bank supervision requiremens. According o Basel Bank Supervision Requiremens, banks have o hold capial (as a precauionary acion) a leas hree imes he equivalen o he VaR for a ime horizon of 10 rading days a he 99% confidence level. In his projec he works of Hsieh (1991) and Brooks, Clare and Persand (2000) are exended. The exension here is ha MCRRs are esimaed for fuures conracs ha have no been applied for his ype of analysis and ha he null hypohesis ha ARCH-ype models overesimae VaR is esed. This also has implicaions for ineres rae forecass. By esimaing Real-World densiies i is possible o have an idea of fuure ineres rae range-levels wih cerain saisical confidence. For example, if a 95% confidence level VaR wih a ime horizon of one monh is applied, i is possible o quanify he range of possible ineres raes in one monh wih 95% saisical cerainy. Also, i is possible o quanify wha are he chances of observing hose exreme values i.e. one in weny (hose ouside he 95% inerval in a parameric and non-parameric disribuion). These findings conribue wih new knowledge o he exising academic lieraure on he use of ime-varying volailiy models in VaR esimaes. The resuls could be for he ineres of agens involved in making risk managemen decisions relaed o ineres rae forecass. These groups of persons could be privae bankers, policy makers, invesors, fuures raders, cenral bankers, academic researchers, among ohers. 4. THE MODELS 4.1 GARCH Specificaion The volailiy of he ime series under analysis is esimaed wih hisorical daa. I is known ha ARCH models (Engle: 1982) are accurae esimaors of imevarying volailiy. A well known model wihin he family of ARCH models is he univariae Generalized Auoregressive Condiional heeroscedasiciy, GARCH(p, q) model. This model is esimaed applying he sandard procedure as explained in Bollerslev (1986) and Taylor (1986) (he ARCH-ype models presened in his paper were esimaed using Eviews compuer language). The formulae for he GARCH(p, q) are presened below. For he model here are wo main equaions. These are he mean equaion and he variance equaion: mean equaion, and he variance equaion, (1) (2) Where: Δy = firs differences of he naural log (logs) of he series under analysis a ime (he ineres rae spo or fuures-index), e is he error erm a ime, I -1 is he informaion se a ime -1, s 2 = variance a ime and -j for s 2 m, w, are parameers and N(0, s -j. 2 ) is i, i for he assumpion ha he log reurns are normally disribued. In oher words, assuming a consan mean m (he mean of he series y ) he disribuion of e is assumed o be Gaussian wih zero mean and variance s 2 The parameers are esimaed using maximum. likelihood mehodology applying he Marquard algorihm This algorihm modifies he Gauss-Newon algorihm by adding a correcion marix o he Hessian approximaion. This allows o handle numerical problems when he ouer producs are near singular hus, increases he chance of improving convergence of he parameers. The objecive log-likelihood funcion o be maximized is he following:, (3) where θ is he se of parameers (μ, ω, α, β) esimaed i i 94

4 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. ha maximize he objecive funcion ln L(θ). z represens he sandardized residual calculaed as. Δy - μ σ 2 The res of he noaion is he same as expressed previously. Considering ha he assumpion of normaliy in he residuals saed above does no hold (as i is common wih financial high frequency price daa), he Bollerslev and Wooldridge (1992) mehodology is used in order o esimae consisen sandard errors. Wih his mehod he resuls have robus sandard errors and covariance. The esimaed coefficiens are reliable once hey are posiive, saisically significan and he sum of he α + β < 1 (oherwise he series are considered explosive or equivalenly non-mean revering, Taylor: 1986). 4.2 The VaR model The VaR is a useful measure of risk (Value a Risk is normally abbreviaed as VaR. The small leer a differeniaes his abbreviaion o ha of Vecor Auoregressive Models, which are usually abbreviaed as VAR (wih a capiol A)). I was developed in he early 1990s by he corporaion JPMorgan. According o Jorion (2001) VaR summarizes he expeced maximum loss over a arge horizon wih a given level of confidence inerval. Even hough i is a saisical figure, mos of he imes is presened in moneary erms. The inuiion is o have an esimae of he poenial change in he value of a financial asse resuling from sysemic marke changes over a specified ime horizon (Mohamed: 2005). I is also normally used o obain he probabiliy of losses for a financial porfolio of fuures conracs. Assuming normaliy, he VaR esimae is relaively easy o obain from GARCH models. For example, for a one rading day 95% confidence inerval VaR he esimaed GARCH sandard deviaion (for he nex day) is muliplied by If he sandard deviaion forecas is, les say, , he VaR is approximaely 1.07%. To inerpre his resul i could be said ha an invesor can be 95% sure ha he or she will no lose more han 1.07% of asse or porfolio value in ha specific day. However, a problem wih a parameric approach is ha if he observed asse reurns depar significanly from a normal disribuion he applied saisical model may be incorrec o use (Dowd: 1998). So, as i was said, when using VaR models i is necessary o make an assumpion abou he disribuion of he reurns. Alhough normaliy is ofen assumed, i is known in pracice ha for price reurns series normaliy is highly quesionable (Mandelbro: 1963, Fama: 1965, Engle: 1982, 2003). For ime horizons of more han one rading day (en, hiry, niney and one hundred and eighy rading days), he boosrapping mehodology of Enfron (1982) will be applied The boosrap is a resampling mehod for inferring he disribuion of a saisic, which is derived by he daa in he populaion sample. This is normally esimaed by simulaions. I is said o be a nonparameric mehod given ha i does no draw repeaed samples from well-known saisical disribuions. On he oher hand, Mone Carlo simulaions draw repeaed samples from assumed disribuions. In his research projec he boosrap mehodology was implemened using Eviews compuer language. The fac ha he reurns of he series are non-normally disribued moivaes he use of a non-parameric procedure as he boosrapping. The procedure used in Hsieh (1993) and Brooks, Clare and Persand (2000) is considered here. In he laer hey empirically esed he performance of ha VaR model for fuures conracs raded in he London Inernaional Financial Fuures Exchange (LIFFE) (hese fuures conracs were he FTSE-100 sock index fuures conrac, he Shor Serling conrac and he Gil conrac). A similar paradigm is applied here for ineres rae-indexed (ineres rae) fuures conracs. Thus, a hypoheical porfolio of ineres rae fuures is considered and MCRRs will be esimaed. These esimaed MCRRs values for he ineres rae porfolio are compared o he observed (hisorical) ineres raes. This analysis allows o evaluae how accurae are he ARCH-ype models in erms of esimaing MCRRs for ineres rae-indexed fuures. Ye, anoher objecive is o analyze he performance of hese in erms of how accurae are hey for providing an upper hreshold for ineres raes i.e. wha are he saisical chances ha ineres rae will be high enough o be ouside he upper (posiive) confidence inerval. In order o calculae an appropriae VaR esimae i is necessary o find ou he maximum loss ha a posiion migh have during he life of he fuures conrac. In oher words, by replicaing wih he boosrap he daily values of a long fuures posiion i is possible o obain he possible loss during he sample period. This will be obained wih he lowes replicaed value. The same reasoning applies for a shor posiion. Bu in ha case he highes possible loss will be obained wih he highes replicaed value As i is well known in fuures marke mechanics decreases in fuures prices mean 95

5 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) losses for long posiions and increases in fuures prices mean losses for shor posiions. Following Brooks, Clare and Persand (2000) and Brooks (2002) he formulae is as follows. The maximum loss (L) is given by L = (P P ) x Number of conracs (4) 0 1 where P represens he price a which he conrac is 0 iniially bough or sold; and P is he lowes (highes) 1 simulaed price for a long (shor) posiion, respecively, over he holding period. Wihou loss of generaliy i is possible o assume ha he number of conracs held is one. Algebraically, he following can be wrien, From Equaion 5 he following can be expressed, (8) (9) when he maximum loss for he long posiion is obained. For he case of finding he maximum possible loss for he shor posiion he following formula applies, (5) Given ha P is a consan, he disribuion of L will 0 depend on he disribuion of P. I is reasonable o 1 assume ha prices are lognormally disribued (Hsieh: 1993) i.e. he log of he raios of he prices are normally disribued The log of he raios of he prices can be represened as ln(p /P ). However, his assumpion 1 0 is no considered here. Insead he log of he raios of he prices is ransformed ino a sandard normal disribuion following JPMorgan Risk-Merics (1996) mehodology. This is done by maching he momens of he log of he raios of he prices disribuion o a disribuion from a se of possible ones known (Johnson: 1949). Following Johnson (1949) a sandard normal variable can be consruced by subracing he mean from he log reurns and hen dividing i by he sandard deviaion of he series, (6) The expression above is approximaely normally disribued. I is known ha he 5% lower (upper) ail criical value is (1.645). To find he fifh percenile hen he following applies, (7) Cross-muliplying and aking he exponenial he case for he long posiion is, (10) The MCRRs of he shor posiion can be inerpreed as an upper hreshold for ineres rae. This will be he hreshold of more ineres given ha in he Mexican economy i was common o observe increases in ineres raes. MCRRs for boh posiions are repored in his paper. The simulaions were performed in he following way. The GARCH model was esimaed wih he boosrap using he sandardized residuals from he whole sample (insead of residuals aken from a normal disribuion as i was wrien in Equaion 1). The ineres rae variable was simulaed, wih he boosrap as well, for he relevan ime horizon (10, 30, 90 and 180 rading days) wih 10,000 replicaions. The formula used was r 1 = r -1 e reurnt (where ineres rae is defined as r and could be he fuures or spo price. The res of he noaion is he same as specified above). From he ineres rae simulaions he maximum and minimum values were aken in order o have he MCRRs for he shor and long posiions respecively. 5. DATA SOURCES The daa consiss of daily spo and fuures closing prices of he ineres rae obained from Banco de México and MEXDER respecively. The daa was downloaded from boh insiuions Web Pages (Banco de México s Web page is hp:// (he Web page is also available in English). The MEXDER web page is hp:// ). Two ypes of ineres raes are considered: Cees 91-days and TIIE 28-days. The firs one is calculaed from Mexican Governmen Bonds and he second one is an equilibrium rae 96

6 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. calculaed according o Mexican commercial banks borrowing and lending ransacions. The sample size is 951 daily observaions from 1 s January 2003 o 29 h Sepember The sample period was chosen according o availabiliy of ineres rae fuures daa and high volume of rading was a year of relaively high rading for Mexican ineres rae fuures. According o he Fuures Indusry Associaion hese ypes of conracs were rank fifh in he world in erms of volume of rading. In oher words, hese are highly liquid fuures conracs. The ineres rae conracs used here are he closes ones o mauriy. They have delivery daes for up o en years. The MEXDER is relaively new compared o oher derivaives exchanges around he world. I began operaions in DESCRIPTIVE STATISTICS This secion presens he descripive saisics for he realized (observed) volailiies of he ineres rae Cees and TIIE and he forecas volailiy from he models. Prior o fiing he GARCH model an ARCH-effecs es was conduced for he series under analysis. This was done in order o see if hese ypes of models are appropriae for he daa (Brooks: 2002). The es conduced was he ARCH-LM following he procedure of Engle (1982). These ess were conduced by using ordinary leas squares regressing he logarihmic reurns of he series under analysis agains a consan. The ARCH-LM es is performed on he residuals of ha regression. The es consiss on regressing, in a second regression, he square residuals agains a consan and lagged values of he same square residuals. The null hypohesis is ha he errors are homoscedasic. An F-saisic was used in order o es he null. The es was carried ou wih differen lags 2 o 10. All have he same qualiaive resuls. Only he cases for 2 lags are repored. According o he resuls boh series under sudy have ARCH effecs. Under he null of homoscedasiciy in he errors he F-saisics were for he Cees and for he TIIE (he criical value a 95% confidence level is 3.84 for 948 degrees of freedom). Boh saisics clearly rejec he null in favour of heeroscedasiciy on hose errors. The parsimonious specificaion GARCH(1,1) was chosen according o resuls obained from informaion crieria (Akaike Informaion Crierion and Schwarz Crierion ess). The model parameers were posiive and saisically significan a he 1% level. The sum of α 1 + b 1 was less han one. Diagnosic ess on he models were applied o ensure ha here were no serious misspecificaion problems. The Auocorrelaion Funcion as well as he BDS es were applied on he sandardized residuals obained from he forecas models. Boh show ha hese residuals were i.i.d. (hese resuls are available upon reques). Table 1 shows he descripive saisics for he realized volailiy and he volailiy from he forecasing models. Figures 1 and 2 presens he logs of boh ineres rae series and heir respecive realized volailiies for 97

7 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) he ime frame under analysis. The daily realized volailiy is defined as he log-reurn squared. As i can be observed in Table 1 he four momens of he disribuion of he Cees series are he ones wih higher values (he realized volailiies and he volailiy forecass). The disribuions from boh series are highly skewed and lepokuric indicaing non-normaliy of he reurns and he forecas esimaes. 98

8 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. 7.1 Parameric mehod 7. RESULTS Once he nex-day volailiy esimae is obained he 95% confidence inervals are creaed by muliplying by he forecased condiional sandard deviaion (from he GARCH model). An analysis is made abou he number of imes he observed ineres rae spo reurn was above ha 95% hreshold. This is formally known as a violaion or an excepion. Again, he posiive par (righ ail of he disribuion) is he one of mos ineres given ha i is posiive ineres rae wha i causes more concern o relaively high ineres rae economies hus, he ineres on predicing i. Alhough for some economies i may be of ineres he significan decreases in ineres raes. For ha case i is imporan o see he negaive side of he disribuion (lef ail of he disribuion). This is equivalen o aking a long posiion on he porfolio. Figure 3 and 4 shows he spo ineres rae reurns and he fuures confidence inervals. I can be observed ha he Cees ineres rae spo reurns were mosly wihin he 95% confidence level for he daily forecass. However, here were violaions in 25 days, which represen 2.62% of he oal number of observaions. Considering ha a 95% confidence level is applied he model i should no exceed he VaR more han 5% (Jorion: 2001). The null hypohesis in his case is no o rejec he model because i has fewer han 5% violaions. The siuaion is qualiaively he same when TIIE series are used o calculae he 95% confidence inervals. Figure 4 shows he same ineres rae spo reurns bu wih confidence inervals consruced wih he TIIE ineres rae. For his case he number of violaions is 30, which represens 3.15% of he oal number of observaions. Again, he model is no rejeced. Applying he Kupiec es as explained by Jorion (2000), he non-rejecion region (inerpolaing) is 11 < x < 47. So, he model is no rejeced for boh series under sudy. 7.2 Boosrapping simulaions The mehodology o carry ou he simulaions was explained in Secion V.2 above. Wih he simulaions i is possible o esimae Real-World Densiies simulaed wih an ARCH model (for more informaion abou Real-World Densiies esimaed wih ARCH model simulaions he reader can refer o Taylor (2005)). These are basically predicive densiies 99

9 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) esimaed in a given day for a specific dae in he fuure. Tables 3 and 4 show he hisograms and Figures 5 and 6 presen he Real-World densiies for he Cees and TIIE series. Simulaions were carried ou for 29/09/06 and he Jump-off period was 18/09/ 06. I can be observed in he figures ha he Cees curve shows he higher maximum value and higher skewness and kurosis. As shown wih he real daa he Cees series is considerably more volaile han he TIIE series (see Table 1). This is also consisen wih he informaion given in he above menion hisograms (Tables 3 and 4). The high volailiy observed in he Cees fuures is also refleced wih high volailiy persisence in ARCH simulaions. As he ime horizon increases so he confidence inervals calculaed wih he simulaions. This can be observed in Figure 7. In ha graph here is no even of violaion or excepion. This is synonymous of overesimaed VaRs. The upper and lower bounds are higher compared wih hose for one rading day. Having a model ha shows no excepions could be cosly for some porfolio invesors, especially for banks. This is because unnecessary amouns of capial mus be se aside in order o mee MCRRs. This is an opporuniy cos of capial. Table 4 presens he VaR for he boosrap simulaions performed for he Cees and TIIE series respecively. The numbers of n-days ahead considered in he simulaions were 10, 30, 90 and 180 rading days. The simulaions were carried ou applying he GARCH(1,1) model. Considering he fac ha he ineres rae reurns show auocorrelaion i is necessary o do he boosrap adjusing for an auocorrelaed process (I am hankful o Alejandro Díaz de León and Daniel Chiquiar for poining his ou. I also wan o hank Arnulfo Rodríguez for his assisance in helping me o incorporae he Poliis and Romano (1994) mehodology in he Eviews compuer code). The procedure posulaed by Poliis and Romano (1994) is applied here. This is basically a mehod in which he auocorrelaed reurns are grouped in o non-overlapping blocks. For his case he size of hese blocks is fixed during he esimaion (i is also possible o have differen size blocks, which vary randomly. For a more deailed explanaion please refer o Poliis and Romano (1994)). Wih he boosrap he blocks are resample. During he simulaion of he ineres rae he GARCH simulaed residuals (plus he original esimaed parameers) are aken from he 100

10 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. 101

11 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) resample blocks. The inuiion is ha if he auocorrelaions are negligible for a lengh greaer han he fixed size of he block, hen his moving block boosrap will esimae samples wih approximaely he same auocorrelaion srucure as he original series (Brownsone and Kazimi: 1998). Thus, wih his procedure he auocorrelaed process of he residuals is almos replicaed and i is possible o obain a more accurae simulaed ineres rae series. From Table 4 i can be observed ha for one rading day shor posiions (shor posiions given in fourh column) he MCRRs are abou 6.11% and 2.32% for he Cees and TIIE series respecively. The inerpreaion of hese figures is ha we can be 95% cerain ha we will lose no more han 6.11% for Cees or 2.32% for TIIE of porfolio value for he nex rading day. As he number of he rading days increases so he VaR ime horizon. In oher words, for en rading days we will be 95% cerain ha we will lose no more han 21.41% for Cees or 5.20% for TIIE of porfolio value for he nex en rading days. I is imporan o poin ou ha he fac ha Cees show higher variance, skewness and kurosis (see descripive saisics in Tables 1 and 2) is refleced wih higher VaR esimaed and consequenly wih higher MCRRs. As he ime horizon is increased he VaR esimaes increase considerably. For he case of he Cees series he MCRRs for 180 rading days goes as far as %. This does conras wih he MCRRs for he TIIE series ha for he same ime horizon he MCRRs is only abou 24.08%. The fac ha ARCH-ype models end o overesimae he VaR because of volailiy persisence is eviden. As i can be observed for boh series in Table 4 he MCRRs quickly increase o high levels as he ime horizon increases for some relaively few days. For he case of he Cees series he MCRRs increase is even higher. As explained before he explanaion o his phenomenon is relaed wih Cees having significanly higher values for he higher momens han hose for he TIIE series. The evidence here suggess rejecion of he null hypohesis ha ARCH-ype models do no overesimae VaR. In his sense, hese resuls are consisen wih Brooks, Clare and Persand (2000). In porfolio analysis he overesimaion is considered cosly. This is because unnecessary quaniies of capial are se aside o mee MCRRs, which in his case are unnecessary high. 102

12 Procesos GARCH y Valor en Riesgo: Un Análisis Empírico de Fuuros de Tasas de Inerés Mexicanas Guillermo Benavides P. 103

13 PANORAMA SOCIOECONÓMICO AÑO 25, Nº 35, p (Julio-Diciembre 2007) 8. CONCLUSIONS In his research projec an analysis of Mexican shorerm ineres rae volailiy was presened. The research on his projec differs from ha found in he lieraure in ha ineres rae fuures are examined in order o draw conclusions abou ARCH-ype models overesimaing Value a Risk (VaR). High VaR will give no opimal Minimum Capial Risk Requiremens (MCRRs). This is considered cosly given ha invesors need o se aside more capial o mee MCRRs. The resuls show ha GARCH processes can be accurae o esimae MCRRs for one-rading day ahead ime horizons. However, for ime horizons of more han en rading days he MCRRs were relaively high because of he volailiy persisence capured by ARCH-ype models. In erms of forecasing shor-erm ineres raes he esimaion of he predicive Real-World densiies provided confidence inervals, which can give insighs abou he expeced range for fuure ineres raes. In oher words, i is possible o be 95% sure ha he ineres rae will fall wihin a specific confidence inerval. I is recommended o exend he presen research work conrolling for over nigh volailiy in he GARCH model during he esimaion procedure. 104

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