A PARAMETRIC MODEL TO ESTIMATE RISK IN A FIXED INCOME PORTFOLIO. Pilar Abad Sonia Benito

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1 A PARAMETRIC MODEL TO ESTIMATE RISK IN A FIXED INCOME PORTFOLIO Pilar Abad Sonia Benio FUNDACIÓN DE LAS CAJAS DE AHORROS DOCUMENTO DE TRABAJO Nº 3/6

2 De conformidad con la base quina de la convocaoria del Programa de Esímulo a la Invesigación, ese rabajo ha sido someido a evaluación exerna anónima de especialisas cualificados a fin de conrasar su nivel écnico. ISBN: La serie DOCUMENTOS DE TRABAJO incluye avances y resulados de invesigaciones denro de los programas de la Fundación de las Cajas de Ahorros. Las opiniones son responsabilidad de los auores.

3 A Parameric Model o Esimae Risk in a Fixed Income Porfolio * Pilar Abad D. Economería y Esadísica Universia de Barcelona Diagonal, , Barcelona, Spain (and Universidad de Vigo) Sonia Benio D. de Análisis Económico II Universidad Nacional de Educación a Disancia Senda del rey nº 11 84, Madrid, Spain Absrac: In his paper we propose a mehodology ha le us o calculae he variance and covariance marix of a very large se of ineres rae changes a a very low compuaional cos. The proposal uses he parameizaion of ineres raes ha underlies he model of Nelson and Siegel (1987) o esimae he yield curve. Saring wih ha model, we are able o obain he variance-covariance marix of a vecor of k ineres raes by esimaing he variance of he principal componens of he four parameers of he model. We used he mehodology we propose o calculae risk in a fixed income porfolio, in paricular o calculae Value a Risk (VaR). The resuls of he paper indicae ha he applicaion of our mehod o calculae VaR provides a precise measure of risk when compared o oher parameric mehods. JEL: E43, G11. Keywords: Value a Risk, Marke risk, Nelson and Siegel mehod. * We are graeful o Alfonso Novales for helpful commens and suggesions. Financial suppor from he Spanish Minisry of Educaion and Science and FEDER (SEJ5-3753/ECO) and Naional Plan of Scienific Research, Developmen and Technological Innovaion (BEC3-3965) is acknowledged. 1

4 1. Inroducion One of he mos imporan asks facing financial insiuions is he evaluaion of he degree o which hey are exposed o marke risk. This risk appears as a consequence of he changes in he marke prices of he asses ha compose heir porfolios. One way o measure his risk is o evaluae he possible losses ha can occur from changes in marke prices. This is precisely wha he VaR (value a risk) mehodology does. This mehodology has been very widely used recenly, and i has become a basic ool for marke risk managemen of many invesmen banks, rading banks, financial insiuions and some non-financial corporaions. Also, he Basel Commiee on Banking Supervision (1996) a he Bank for Inernaional Selemens uses VaR o require financial insiuions such as banks and invesmen firms o mee capial requiremens o cover he marke risk ha hey incur as a resul of heir normal operaions. The VaR of a porfolio is a saisical measure ha ells us wha is he maximum amoun ha an invesor may lose over a given ime horizon and wih a given probabiliy. Alernaively, he VaR of a porfolio can be defined as he amoun of funds ha a financial insiuion should have in order o cover he porfolio losses in almos all circumsances, excep for hose ha occur wih a very low probabiliy. Alhough VaR is a simple concep, is calculaion is no rivial. Formally, VaR ( α %) is he percenil α of he probabiliy disribuion of he changes in value of a porfolio, ha is, i is he value for which α % of he values lie o he lef on he disribuion. Consequenially, in order o calculae VaR we mus firsly esimae he probabiliy disribuion of he changes in value of he porfolio. Several mehods have been developed o do his: Mone Carlo Simulaion, Hisorical Simulaion, Parameric Models, and Sress Tesing. See Jorion() o ge a general vision of his mehodologies. Among all of hese, he mos widely used mehods are hose based on he parameric approach, or on variance and co-variance. We can see some applicaions of his mehod in Morgan(1995), García-Donao a all(1) Geno(1), Geno(), Benio and Novales(5), Alex NcMain(1). The parameric approach is based on he assumpion ha he changes in value of a porfolio will follow a known disribuion, which is generally assumed o be Normal. Under such an assumpion, he only relevan parameer for he calculaion of VaR is he variance condiional on he changes in value of he porfolio, assuming ha on average hese are zero.

5 The esimaion of his variance is no rivial, since i requires esimaing he variance covariance marix of he asses ha make up he porfolio. The esimaion of his marix poses wo ypes of problems: (1) a dimensionaliy problem and () a viabiliy problem. The firs appears due o he large dimension of he marix, which makes i difficul o esimae. For example, in order o esimae he variance of he reurn of a porfolio ha is made up of five asses, i is necessary o esimae five variances and fifeen covariances, ha is a oal of weny variables. This problem becomes especially imporan in fixed income porfolios in which he value depends on a large number of differen ineres raes, for differen ime horizons. The second problem has o do wih he difficuly of esimaing he condiional covariances if one uses sophisicaed models, such as mulivariae GARCH models. The esimaion of such models is boh very cosly in erms of compuaion, and is also generally no even possible when he dimension of he marix is greaer han hree. I is for his reason ha hese models have no been a all popular for financial managemen. In he recen lieraure, hese problems are ackled using he assumpion ha here exis common facors in he volailiy of he ineres raes, and ha hese same facors explain he changes in he emporal srucure of he ineres raes (TSIR). Under hese wo assumpions, i becomes heoreically possible o obain he variance-covariance marix of a wide range of ineres raes using a facor model of TSIR. For example, Alexander (1) and Geno () show ha if we begin wih a principal componens model (Alexander 1) or a regression model (Geno, ), hen we can ge he variance-covariance marix from a vecor of ineres raes a a low calculaion cos. The presen paper proposes an alernaive mehod of esimaing he variancecovariance marix of ineres raes a a low compuaional cos. We ake as our saring poin he model of Nelson and Siegel (1987), which was developed iniially o esimae he TSIR. This model provides an expression of he ineres raes as a funcion of four parameers. Saring wih his model, we can obain he variance-covariance marix of he ineres raes by calculaing he variances of only four variables he principal componens of he changes in he four parameers. This paper coninues as follows. In secion we presen he mehod proposed o esimae he variance-covariance marix for a large vecor of ineres raes a a low compuaional cos. The nex secions evaluae he proposed mehod for a sample of daa from he Spanish marke. In secion 3 we briefly describe he daa ha we use, and we apply 3

6 he proposed mehod o obain he variance-covariance marix of a vecor of ineres raes. In secion 4 we evaluae he proposed mehodology o calculae he VaR in fixed income porfolios, and we compare he resuls wih hose ha are obained from sandard mehods of calculaion. Finally, secion 5 presens he main conclusions of he paper.. A parameric model for esimaing risk. In his secion we presen a mehodology o calculae he variance-covariance marix for a large vecor of ineres raes a a low compuaional cos. To do his we ake as our saring poin he model proposed by Nelson and Siegel (1987), designed o esimae he yield curve (TSIR). The Nelson and Siegel formulaion specifies a parsimonious represenaion of he forward rae funcion given by: m m m m 1e τ e τ ϕ = β + β + β (1) τ This expression allows one o accommodae he differen forms ha may characerise (level, posiive or negaive slope, and greaer or lower curvaure) as a funcion of four parameers ( β, β1, β and τ ). Bearing in mind he fac ha he spo ineres rae for a erm of m can be expressed as he sum of he insananeous forward ineres raes from up o m, ha is, by inegraing he expression ha defines he insananeous forward rae: m r ( ) m = ud ϕ u () we obain he following expression for he spo ineres rae for a erm of m: m m m τ τ τ τ τ τ τ r ( m) = β β1 e + β1 + β βe β e (3) m m m m Equaion (3) shows ha spo ineres raes are a funcion of only four parameers. Consequenially he changes in hese parameers are he variables ha deermine he changes in he ineres raes. Using a linear approximaion we can esimae he change in he zero coupon rae of erm m from he following expression: 4

7 ( ) dr m d β, ( ) ( ) ( ) ( ) β r 1, m r m r m r m d β β β τ d β 1, dτ In a mulivariae conex, he changes in he vecor of ineres raes ha make up he TSIR can be expressed by generalizing equaion (4) in he following way: (4) dr = G d β (5) dr dr dr dr k, dβ = dβ, dβ1, dβ, d τ and where = [ (1) (). ( )] ' G r(1) r(1) r(1) r(1) β β1 β τ r() r() r() r() = β β1 β τ..... r( ) ( ) ( ) ( ) k r k r k r k β β1 β τ This approximaion (equaion (5)) has been used wih some success in ineres rae risk managemen for fixed income asses (Gómez, 1999) and for porfolio immunizaion (Gómez, 1998). In he conex of his model, and using expression (5), we can calculae he variancecovariance marix of a vecor of changes in he k ineres raes using he following expression: ' var( dr ) = G Ψ G (6) where: var( β, ) cov( β, β1, ) cov( β, β, ) cov( β, τ) var( β1, ) cov( β1, β, ) cov( β1, τ ) Ψ = var( β, ) cov( β, τ) var( τ ) A his poin we noe ha we have arrived a an imporan simplificaion in he dimension of he variance-covariance marix ha we need o esimae. Noe ha for a vecor of k ineres raes, insead of having o esimae k(k+1)/ variances and covariances, we only 5

8 need o esimae 1 second order momens. However, he problem associaed wih he difficuly of he esimaion of he covariances sill remains. Bu we can sill simplify he calculaion of he variance-covariance marix even furher, by applying principal componens o he vecor of he changes in he parameers ( d β ). In his way, he vecor of changes in he parameers of he model of Nelson and Siegel (1987) can be expressed as: dβ = AF (7) F f f f f and = 1,, 3, 4, β β β β β1 β1 β1 β β β β β τ τ τ τ a a a a a a a a A = a a a a a a a a where F is he vecor of principal componens associaed wih he vecor d β and A is he marix of consans ha form he eigenvecors associaed wih each one of he four eigenvalues of he variance-covariance marix of he changes in he parameers of he Nelson and Siegel model ( d ). β Subsiuing equaion (7) ino equaion (5) and given ha each principal componen is orhogonal o he res, we can express he variance-covariance marix of he ineres raes as follows: * *' var( dr) = G ΩG (8) var( f1, ) var( f, ) where: Ω = and G var( 3, ) * G A. f var( f4, ) Therefore, equaion (8) gives us an alernaive mehod o esimae he variancecovariance marix of he changes in a vecor of k ineres raes using he esimaion of he four principal componens of he changes in he parameers of he Nelson and Siegel (1987) model. In his way, he dimensionaliy problem associaed wih he calculaion of he covariances has been solved. In he following secions we evaluae his mehod, boh o calculae he variance marix of a vecor of ineres raes, and o calculae he VaR of fixed income porfolios. 6

9 3. Esimaing he variance-covariance marix 3.1. The daa To examine he mehod proposed in his paper, we esimae a daily erm srucure of ineres raes using acual mean daily Treasury ransacions prices. The original daa se consiss of daily observaions derived from acual ransacions in all bonds raded on he Spanish governmen deb marke. The daabase of bonds raded on he secondary marke of Treasury deb covers he period from Sepember, o Ocober, We use his daily daabase o esimae he daily erm srucure of ineres raes. We fi Nelson and Siegel s (1987) exponenial model for he esimaion of he yield curve and minimise price errors weighed by duraion. We work wih daily daa for ineres raes a 1,,, 15 year mauriies. 3.. The resuls In his secion we examine his new approach o variance and covariance marix esimaion. The firs secion begins by comparing he changes in he esimaed and observed ineres raes. The changes in ineres raes are modelled by equaion (5), and hen we compare hese changes wih he observed ones 1. Then we esimae he variance-covariance marix of a vecor of 1 ypes of ineres rae, using he mehodology proposed in he previous secion, and we compare hese esimaions (Indirec Esimaion) wih hose obained using some habiual univariae procedures (Direc Esimaion). Boh in direc and indirec esimaion we need a mehod for esimaing variances and covariances. For he case of indirec esimaion he esimaion mehod gives us he variances of he four principal componens of he changes in he parameers of he Nelson and Siegel model, which allow us o obain, from equaion (8), he variance-covariance marix of he ineres raes. In order o esimae he variance-covariance marix of he ineres raes changes and he variance of he principal componens, we use wo alernaive measures of volailiy: exponenially weighed moving average (EWMA) and Generalized Auoregressive Condiional Heeroskedasiciy models (GARCH). (1) Under he firs alernaive, he variance-covariance marix is esimaed using he RiskMerics mehodology, developed by J.P. Morgan. RiskMerics uses he so called 1 The sofware we used in his applicaion is MATLAB. 7

10 exponenially weighed moving average (EWMA) mehod. Accordingly, he esimaor for he variance is: N 1 j var( dx) = (1 λ) λ ( dx j dx ) (9) j= he esimaor he covariance is: cov( dx dy ) N 1 j (1 λ ) λ ( dx j dx)( dy j dy) (1) = j= J.P. Morgan uses he exponenially weighed moving average mehod o esimae he VaR of is porfolios. On a widely diversified inernaional porfolio, RiskMerics found ha he value λ =.94 wih N = produces he bes backesing resuls. In his paper, we use boh of hese values. Therefore, we obain he direc esimaions of he variance-covariance marix (D_EWMA) of he ineres raes from equaions (9) and (1) where x and y are he ineres raes a differen mauriies. For he case of indirec esimaion of he variancecovariance marix (I_EWMA), we use equaion (9) o obain he variances of he principal componens (where x are now hese principal componens) and, from here, equaion (8) gives us he relevan marix. () The EWMA mehodology, which is currenly used for he Riskmerics TM daa, is quie accepable for calculaing VaR measures, bu some auhors sugges ha one alernaive is o use variance-covariance marices obained using Mulivariae Generalized Auoregressive Condiional Meeroskedasiciy Models (GARCH). Neverheless, he large variance-covariance marices used in VaR calculaions could never be esimaed direcly using a full mulivariae GARCH model, because he compuaional complexiy would be insurmounable. For his reason we only compue he variances of ineres raes changes using univariae GARCH models and do no compue he covariance. Given ha indirec esimaion (I_GARCH) does no require he esimaion of covariances, we esimae he condicional variance of he principal componens of he changes in he parameer of he Nelson and Siegel model using univariae GARCH models. In he sub-secion wo of his secion, we compare he alernaive esimaions of he variance-covariance marix described above. This comparison is summarised in Table 1. 8

11 Table 1. Type of variance-covariance marix esimaion Type of variance models EWMA GARCH Type of Direc Esimaion D-EWMA D-GARCH * esimaion Indirec Esimaion I-EWMA I-GARCH * We have no esimaed mulivariae GARCH model because of he compuaional complexiy are insurmounable, so ha only presen he resul of he variances which have been esimaed unsing univariae GARCH models. Wha is relevan is ha he esimaion of he variance-covariance marix using he mehodology proposed here (indirec esimaion) involves a minimum calculaion cos, since i is only necessary o esimae he variance of four variables (he principal componens of he daily changes in he parameers of he Nelson and Siegel model) Comparing he changes of ineres raes Firsly, we have evaluaed he abiliy of he model ha we propose here o esimae he daily changes in a vecor of ineres raes. To do his, we compare he observed ineres raes wih heir esimaions from equaion (5). In Illusraion 1 in he Appendix, we show he scaer diagrams ha relae he observed changes wih he esimaed changes in ineres raes a 1, 3, 5 and 1 years. As can be seen, independenly of he period considered, he relaionship is very close. In Table we repor some descripive saisics of he errors of esimaion of he ineres rae. The average error is very small, abou five basic poins for all mauriies. This error represens, in relaive erms,.5% of he ineres raes. Furhermore, we observe oo ha boh he average error and he sandard deviaion are very similar in all period lenghs so ha he accurae of he model seems good for all mauriies. 9

12 Table. Esimaion errors in ineres raes. Descripive saisics. 1 year years 3 years 4 years 5 years 6 years 7 years 8 years 9 years 1 years Mean (a) Sandard deviaion Maximum error Minimum error Noe: The sample period is from 1/9/1995 o 9/1/1997. The errors (and all saisics) are expressed in basic poins. (a) The average error is calculaed in absolue value. Therefore, hese resuls imply ha he degree of error commied when esimaing he changes in zero coupon raes using equaion (5) are pracically non-exisen. In wha follows, we evaluae he differences in he esimaion of he variance-covariance marix using he differen alernaives Comparing he esimaions of variance-covariance marix In Illusraion we show he condiional variances of he ineres raes a 1, 3, 5 and 1 years, as esimaed using he exponenially weighed moving average mehod, boh direcly and indirecly: D_EWMA versus I_EWMA. In Illusraion 3 we show he direc esimaion of he condiional variances of hese same ineres raes using he GARCH (D_GARCH) models, and he indirec esimaion of he same daa (I_GARCH). As can be seen in boh illusraions, in mos of he ime horizons considered, he variances esimaed using he mehod proposed in his paper are very similar o he direc esimaes. The descripive saisics of he differences beween he sandard deviaions ha are esimaed using boh procedures are repored in Table 3. We compare he direc and indirec esimaion mehods using an EWMA model in panel (a), and using a GARCH model in panel (b). Panel (a) shows ha he differences in absolue value for EWMA specificaion oscillae beween.6 and 1 base poin. This average difference represens beween 1% and % of he size of he esimaed series. Panel (b) of Table 3 also shows ha he average difference in absolue value for GARCH specificaion is quie small, even hough greaer han hose of panel (a). However, as a percenage of he esimaed condiional variance series, hese differences are smaller han hose of panel (a). In boh comparisons, we can noe ha he range of differences beween each pair of esimaions is far greaer for a one year rae han for he oher horizons. 1

13 We also noe ha he range of he esimaion error is also greaer for he case of one year han for he oher ineres raes (Table ). Table 3. Differences in he esimaion of he sandard deviaion of ineres raes. Descripive saisics. 1 year years 3 years 4 years 5 years 6 years 7 years 8 years 9 years 1 years Panel (a): Comparing D_EWMA vs. I_EWMA. Mean (a) Sandard deviaion Maximum error Minimum error Panel (b): Comparing D_EGARCH vs. I_EGARCH. Mean (a) Sandard deviaion Maximum error Minimum error Noe: Sample period from 9/9/1995 o 9/1/1997 (515 observaions). I_EWMA indirec esimaion (equaion (8)) and D_EWMA direc esimaion. Riskmerics mehodology (EWMA). I_GARCH: indirec esimaion (equaion (8)) and D_GARCH direc esimaion. Condiional auoregressive volailiy models (GARCH). (a) The average of he differences has been calculaed in absolue value. Differences measured in base poins. We now compare he covariances esimaed direcly wih hose obained from he procedure suggesed in his paper. As we have menioned above, given he exreme complexiy of he GARCH mulivariae model esimaions, he direc esimaion of he covariances was only done using EWMA models. Illusraion 4 shows he esimaed covariances beween he differen pairs of ineres raes, using boh procedures: D_EWMA versus I_EWMA. As can be seen in he graphs, he esimaed covariances have very similar behaviour, alhough we can noe ha here are greaer differences han for he variances. In Table 4 we repor some of he descripive saisics of he esimaed covariances. The average absolue value difference is very small, beween.7 and.19. However, his does represen abou 4% of he average esimaed covariance. 11

14 Table 4. Differences in he esimaion of covariances beween ineres raes. Descripive saisics. Comparing D_EWMA vs. I_EWMA 1 year 3 years 5 years 3 years 5 years 1 years 5 years 1 years 1 years Mean (a) Sandard deviaion Maximum error Minimum error Noe: Sample period from 9/9/1995 o 9/1/1997 (515 observaions). I_EWMA indirec esimaion (equaion (8)) and D_EWMA direc esimaion. Riskmerics mehodology (EWMA). (a) The average difference is calculaed in absolue value. To sum up his secion, we have shown ha he procedure proposed in his paper o esimae he variance-covariance marix of a large vecor of ineres raes generaes resuls ha are quie saisfacory, above all as far as variances are concerned. For he case of covariances, we have deeced some differences ha could be imporan. In he following secion we evaluae wheher hese differences are imporan for risk managemen. To do his, we apply he mehodology o he calculaion of Value a Risk (VaR) in several fixed income porfolios. 4. Esimaing he Value a Risk In his secion we evaluae he uiliy of he proposed mehod for risk managemen of fixed income porfolios, by consrucing a parameric measure of VaR as an indicaor of he risk of a given porfolio Value a Risk The VaR of a porfolio is a measure of he maximum loss ha he porfolio may suffer over a given ime horizon and wih a given probabiliy. Formally, he VaR measure is defined as he lower limi of he confidence inerval of one ail: ( ) where α is he level of confidence and ( τ ) over he ime horizon τ. Pr Δ V τ < VaR = α (1) Δ V is he change in he value of he porfolio 1

15 The mehods ha are based on he parameric, or variance-covariance, approach sar wih he assumpion ha he changes in he value of a porfolio follow a Normal disribuion. Assuming ha he average change is zero, he VaR for one day of porfolio j is obained as: VaR j, ( α%) = σ dv, j k α % (11) where k α % is he α percenile of he Sandard Normal disribuion, and he parameer o esimae is he sandard deviaion condiional upon he value of porfolio j ( σ dvj, ). In a porfolio ha is made up of fixed income asses, he duraion can be used o obain he variance of he value of porfolio j from he variance of he ineres raes in he following way (Jorion, ): ' σ dv, = Dj, ΣD j j, (1) where Σ is he variance-covariance marix of he ineres raes and D j, is he vecor of he duraion of porfolio j. This vecor represens he sensiiviy of he value of he porfolio o changes in he ineres raes ha deermine is value. In his secion, value a risk measures are calculaed and compared. In he parameric approach, we use he esimaions of he variance-covariance marix as obained in he previous secion (see Table 1). Table 5 illusraes he four measures of VaR ha we obain from he four variance-covariance models: Table 5. Type of VaR measures Direc Esimaion Indirec Esimaion Type of variance- covariance marix esimaion D_EWMA Type of VaR measure VaR_D_EWMA D_GARCH VaR_D_GARCH * I_EWMA I_GARCH VaR_I_EWMA VaR_I_GARCH * We did no compue VaR_D_GARCH because of he impossibiliy o esimae a mulivariae GARCH model wih 1 variables. In he case of he firs VaR measure, VaR_D_EWMA, he VaR is obained by direcly esimaing Σ wih an EWMA model. This is a popular approach o measuring marke risk, and i is used by JP Morgan (RiskMeric TM ). The second VaR measure, 13

16 VaR_D_GARCH, is also obained by direcly esimaing he variance-covariance marix, bu in his case he second order momens are esimaed using GARCH models. This VaR measure has no been calculaed, given ha he large variance-covariance marices used in VaR calculaions could never be esimaed direcly using a full mulivariae GARCH model, because he compuaional complexiy would be insurmounable. The final wo VaR measures are calculaed by esimaing he variance-covariance marix of he ineres raes using he procedure described in Secion. We can esimae he variance-covariance marix of ineres raes indirecly, by subsiuing equaion (8) ino equaion (1) o obain a new expression for he variance of he changes in he value of he porfolio: * *' ' m m' σ dv, = Dj, G Ω G Dj, = Dj, Ω j D (13) In indirec esimaion, Ω is a diagonal marix ha conains on is principal diagonal he condiional variance of he principal componens of he changes in he four m parameers of he Nelson and Siegel model, and D j, is he modified vecor of duraions of porfolio j (of dimension 1x4) which represens he sensiiviy of he value of he porfolio o changes in he principal componens of he four parameers of he Nelson and Siegel model. In he VaR_I_EWMA, we use an EWMA model o esimae he variance of he principal componens; and we use a GARCH model o esimae hese variances in he case of he calculaion of he VaR_I_GARCH measure. 4.. The porfolios In order o evaluae he procedure proposed in his paper for calculaing VaR we have considered 4 differen porfolios made up of heoreical bonds wih mauriies a 3, 5, 1 and 15 years, consruced from real daa from he Spanish deb marke. In each porfolio, he bond coupon is 3.%. The period of analysis is from 9/9/1995 o 9/1/1997, which allows us o perform 516 esimaions of daily VaR for each porfolio. In order o esimae he daily VaR we have assumed ha he characerisics of each porfolio do no change over he daes of he period of analysis: he iniial value of he porfolio, he mauriy dae and he coupon rae. In his way, he resuls are comparable over he enire period of analysis since we avoid boh he pull o par effec (he value of he bonds ends o par as he mauriy dae of he bond approaches) and he roll down effec (he volailiy of he bond decreases over ime). 14

17 4.3. Comparing VaR measures In his secion value a risk measures are compared. For all porfolio considered we calculae daily VaR a a 5%, 4%, 3%, % and 1% confidence level. Firsly, before formally evaluaing he precision of he VaR measures under comparison, we examine acual daily porfolio value changes as implied by daily flucuaions in he zero cupon ineres rae and compare hem wih he 5% VaR. In Illusraion 5 we show he acual change in a 1 year porfolio ogeher wih he VaR a 5% for he hree measures of VaR ha we consider: VaR_D_EWMA (Figure 1), VaR_I_EWMA (Figure ) and VaR_I_GARCH (Figure 3). In Figures 1 and we observe ha he value of he porfolio falls below he VaR on more occasions han in Figure 3. In all case, he number of imes ha he value of he porfolio falls below he VaR is closer o is heoreical level. This resul is also eviden in he oher porfolios ha we consider, bu ha we have no repored due o space consideraions. This preliminary analysis suggess ha he esimaions of VaR ha are obained from boh models, boh direcly and indirecly are very precise. However, a more rigorous evaluaion of he precision of he esimaions is required. We hen compare VaR measures he acual change in porfolio value on day +1, denoed as Δ V + 1. If Δ V + 1 < VaR, hen we have an excepion. For esing purposes, we define he excepion indicaor variable as I = if Δ V < VaR + 1 if ΔV VaR a) Tesing he Level The mos basic es of a value a risk procedure is o see if he saed probabiliy level is acually achieved. The mean of he excepion indicaor series is he level of he procedure ha is achieved. If we assume he probabiliy of an excepion is consan, hen he number of excepions follows he binomial disribuion. Thus i is possible o form confidence inervals for he level of each VaR measure (see Kupiec (1995)). + 1 (14) 15

18 Table 6. Tesing he Level Number of excepions Confidence VaR measures 3- years 5-years 1-years 15-years inervals a he 95% level VaR_D_EWMA (1%) (1-1) VaR_D_EWMA (%) (5-17) VaR_D_EWMA (3%) (8-3) VaR_D_EWMA (4%) (1-3) VaR_D_EWMA (5%) (17-36) VaR_I_EWMA (1%) (1-1) VaR_I_EWMA (%) (5-17) VaR_I_EWMA (3%) (8-3) VaR_I_EWMA (4%) (1-3) VaR_I_EWMA (5%) (17-36) VaR_I_GARCH (1%) (1-1) VaR_I_GARCH (%) (5-17) VaR_I_GARCH (3%) 7 * (8-3) VaR_I_GARCH (4%) 11 * (1-3) VaR_I_GARCH (5%) 13 * (17-36) Noe: Sample period 9/9/1995 o 9/1/1997. Confidence inervals derived from he number of excepions follows he binomial disribuion (516, x%) for x=1,, 3, 4 and 5. An * indicaes he cases in which he number of excepions is ou of he confidence inerval, so ha, we obain evidence o rejec he null hypohesis a he 5% level ype I error rae. Table 6 shows he level ha is achieved and a 95% confidence inerval for each of he 1-day VaR esimaes. An * indicaes he cases in which he number of excepions is ou of he confidence inerval, so ha, we obain evidence o rejec he null hypohesis a he 5% confidence level. For he hree measures and almos all porfolios considered, he number of excepions is inside he inerval confidence, so ha he VaR esimaion (direc and indirec) seems o be good. We find jus only hree cases in which he number of excepions is ou of he confidence inerval. This happen for VaR_I_GARCH measure for 3%, 4% and 5% confidence level of he porfolio a 3 years. In hose cases he number of excepions are much lower han he heoreical level, so ha i seems ha his measure is overesimaing he risk of shor-erm porfolio. 16

19 b) Tesing Consisency of Level We wan he level of he VaR ha is found o be he saed level on average, bu we also wan o find he saed level a all poins in ime. One approach o esing he consisency of he level is o use he Ljung-Box pormaneau es (Ljung and Box, 1978) on he excepion indicaor variable of zeros and ones. When using Ljung-Box ess, here is a choice of he number of lags in which o look for auocorrelaion. If he es uses only a few lags bu auocorrelaion occurs over a long ime frame, he es will miss some of he auocorrelaion. Conversely should a large number of lags be used in he es when he auocorrelaion is only in a few lags, hen he es won be as sensiive as if he number of lags in he es mached he auocorrelaion. Differen lags have been used for each esimae in order o ry o ge a good idea of he auocorrelaion. Table 7 shows he Ljung-Box saisics a lags of 4 and 8. 17

20 Table 7. Tesing Consisency of Level Lags 3- years 5-years 1-years 15-years VaR_D_EWMA (1%) 4, (,995), (,995),5 (,971),8 (,936) 8,41 (1,),41 (1,) 1,6 (,998) 5,7 (,75) VaR_D_EWMA (%) 4,9 (,99),8 (,936) 1,19 (,879) 1,19 (,879) 8,59 (1,) 4,98 (,76) 13,6 * (,93) 4,1 (,848) VaR_D_EWMA (3%) 4 4,13 (,389) 1, (,91) 1,64 (,8),3 (,681) 8 4,85 (,773) 4,46 (,814) 11, (,189) 5,41 (,713) VaR_D_EWMA (4%) 4,96 (,565),16 (,77),3 (,681) 5,9 (,58) 8 5,6 (,79) 4,41 (,819) 7,3 (,54) 7,17 (,518) VaR_D_EWMA (5%) 4,66 (,617),16 (,77) 4,44 (,349) 6,4 (,196) 8 1,66 (,) 6,59 (,581) 7,9 (,444) 7,33 (,5) VaR_I_EWMA (1%) 4,9 (,99),66 (,956),66 (,956) 3,37 (,498) 8,59 (1,) 5,91 (,657) 5,91 (,657) 6,77 (,56) VaR_I_EWMA (%) 4,4 (,983),3 (,681),3 (,681) 8,57 * (,73) 8,8 (,999) 5,7 (,78) 4,63 (,796) 1,9 (,7) VaR_I_EWMA (3%) 4, (,7),16 (,77) 4,84 (,35) 4,8 (,369) 8 7,79 (,454) 4,34 (,86) 6,71 (,568) 5,47 (,76) VaR_I_EWMA (4%) 4,44 (,656) 1,8 (,864) 3,3 (,59) 4,3 (,43) 8 6,75 (,564),6 (,956) 4,48 (,81) 6,8 (,557) VaR_I_EWMA (5%) 4 1,93 (,748),75 (,945) 3,53 (,474) 3,78 (,436) 8 5,55 (,698),49 (,96) 7,87 (,447) 14,77 * (,64) VaR_I_GARCH (1%) 4,7 (,999), (,995), (,995),13 (,998) 8,15 (1,),41 (1,),41 (1,),6 (1,) VaR_I_GARCH (%) 4, (,995),66 (,956),4 (,983), (,995) 8,41 (1,) 5,91 (,657) 9,68 (,88) 19,69 * (,1) VaR_I_GARCH (3%) 4,4 (,983),66 (,956),51 (,643) 3,37 (,497) 8,8 (,999) 5,91 (,657) 6,15 (,63) 9,15 (,33) VaR_I_GARCH (4%) 4 1, (,91) 1,4 (,843),9 (,68),9 (,68) 8 4,4 (,8) 5,4 (,753) 5,8 (,77) 5,93 (,655) VaR_I_GARCH (5%) 4,51 (,643) 1,95 (,746),19 (,71) 1,95 (,745) 8 5,5 (,75) 3,66 (,886) 3,91 (,865) 3,67 (,886) Noe: Sample period 9/9/1995 o 9/1/1997. The Ljung-Box Q-saisics on he excepion indicaor variable and heir p-values. The Q-saisic a lag 4 (8) for he null hypohesis ha here is no auocorrelaion up o order 5 (1). An * indicaes ha here is evidence o rejec he null hypohesis a he 5% level ype I error rae. We only deec he exisence of auocorrelaion in he porfolios a 1 years wih he VaR_D_EWMA (%) esimae, in he porfolio a 15 years wih he measures VaR_I_EWMA (%) and (5%) and he porfolio a 15 years wih he measures VaR_I_GARCH (%). In general, he resuls of he Ljung-Box comparison indicae ha auocorrelaion is no presen. When we consider oher lags, ha are no repored here in he 18

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