Academy of Economic Studies Bucharest Doctoral School of Finance and Banking Dissertation Paper

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1 Academy of Ecoomic Studies Bucharest Doctoral School of Fiace ad Bakig Dissertatio Paper Measurig market risk: a copula ad extreme value approach Supervisor PhD. Professor Moisă Altăr M. Sc. Studet Stâgă Alexadru Leoard July 2007

2 Abstract This paper presets a methodology for measurig the risk of a portfolio composed of assets with heteroscedastic retur series. I order to obtai good estimates for Value-at-Risk ad Expected Shortfall, the model tries to capture as realistically as possible the data geeratig process for each retur series ad also the depedece structure that exists at the portfolio level. For this purpose, the idividual retur series are modelled usig GARCH methods with semi-parametric iovatios ad the depedece structure is defied with the help of a Studet t copula. The model built with these techiques is the used for the simulatio of a portfolio retur distributio that allows the estimatio of the risk measures. This methodology is applied to a portfolio of five Romaia stocks ad the accuracy of the risk measures is the tested usig a backtestig procedure. 2

3 Cotets Cotets... 3 A. Itroductio... 4 B. Literature Review... 7 C. Methodology... 9 I. GARCH models... 9 II. Extreme Value Theory (EVT) models.... Limitig distributios of the maxima Limitig distributio of exceedaces over a threshold... 3 III. Copula models... 5 IV. Measures of risk... 9 D. Applicatio... 2 I. Data... 2 II. Estimatio ad results GARCH models Extreme Value Theory (EVT) models Copula models Portfolio simulatio Measures of risk Backtestig E. Coclusios F. Refereces Appedix I GARCH Appedix II EVT Appedix III Copula Appedix IV Simulatio ad measures of risk

4 A. Itroductio The fiacial istitutios with sigificat amouts of tradig activity proved to be very vulerable to extreme market movemets ad, i time, the measuremet of market risk became a primary cocer for regulators ad also for iteral risk cotrol. For example, U.S. baks ad bak holdig compaies with a importat tradig portfolio are subject to market risk requiremets. They have bee required to hold capital agaist their defied market risk exposures, ad, the ecessary capital is a fuctio of baks' ow risk estimates. I this cotext, Value-at-risk (VaR) has emerged as oe of the most used risk measure i the fiacial idustry, mostly because of its simplicity ad ituitive iterpretatio. Value at Risk measures the worst loss to be expected of a portfolio over a give time horizo at a give cofidece level. Although a clear defiitio of VaR may be give, this measure of risk does t have a uique method of estimatio because its accuracy highly depeds o the ability to idetify the true portfolio loss distributio. Simple models of estimatio, like Historical Simulatio or Variace-Covariace failed to give accurate high cofidece level estimates but are used ofte because of their low computig power demads. More complex models based o Mote Carlo simulatio have the advatage of flexibility i modellig the loss distributio ad the potetial of beig more accurate but they are difficult to compute for very complex portfolios with a high umber of risk factors. Although VaR offers a simple ad ituitive way of evaluatig market risk, Artzer et al. (997, 998) have criticized it as a measure of risk for two mai reasos. First they proved that VaR is ot ecessarily subadditive ad secodly, this measure gives oly a upper limit o the losses give a cofidece level, but it tells othig about the potetial size of the loss if this upper limit is exceeded. I order to solve these two issues, they propose the use of the socalled expected shortfall or tail coditioal expectatio istead of VaR. Expected shortfall measures the expected loss give that the loss exceeds VaR; i mathematical terms it ca be writte as E [L L > VaR]. This property represets the beefits of diversificatio for a portfolio, the risk derived from the portfolio (x + y) is lower tha (or equal to) the risk derived from the sum of the risk of the idividual securities x, y 4

5 This paper aims at computig accurate estimates for both measures of risk by usig a flexible modellig techique i order to build the loss distributio of the portfolio. The first stage of this process represets the modellig of the retur series for each idividual stock. Oe possible solutio for this issue is based o o-parametric methods (empirical methods), that make o assumptios cocerig the ature of the empirical distributio fuctio. However, these techiques have several drawbacks, for example they caot be used to estimate out of sample quatiles ad the kerel based estimators usually perform poorly i the smoothig of tails (Silverma, 986). Aother possible optio would be the use of parametric methods for describig the etire distributio of the series. Empirical evideces have show that the distributios of fiacial returs series exhibit fat tails ad sometimes egative skews (Zagari 996). For this reaso, the ormal distributio, i spite of its popularity, it is ot cosidered a good choice as its symmetry ad expoetially decayig tail does t seem to be supported by data. A alterative to the ormal distributio may be cosidered the Studet-t distributio as it displays polyomial decay i the tails ad thus havig heavier tails tha the ormal oe. Hece, it may be able to capture the observed excess kurtosis although it maitais the hypothesis of symmetry. A third possibility would be the use of extreme value parametric methods for describig the tails of the distributio ad parametric (ex. gaussia, studet-t) or oparametric methods (ex. kerel smoothig) for the iterior of the distributio. These types of tools permit a high flexibility because the parameters for each tail ca be estimated separately ad thus, both the excess kurtosis ad the skewess of the fiacial series ca be icorporated ito the model. The methodology used i this paper takes advatage of the flexibility provided by the third method i order to capture the data geeratig process for each fiacial series. More specifically it uses extreme value theory for the estimatio of the tail parameters ad a kerel smoothig techique for buildig the iterior of the distributio. Oce the tools used for the costructio of the returs series distributio are defied, the ext step cosists of choosig the distributio that should be aalyzed. I the cotext of market risk maagemet, the aalysis of both the coditioal ad ucoditioal returs distributio provides useful iformatio. However, the coditioal returs distributio takes ito accout the curret volatility backgroud ad forms the basis for short term risk evaluatio, thus beig the mai iterest of market risk maagemet. 5

6 The aalysis of ucoditioal tails provides additioal iformatio about risk ad ca be used for the estimatio of the magitude of a rare adverse evet that ca lead to a importat loss. This kid of iformatio may be used for stress testig scearios ad log term risk estimatio. Because short-term risk evaluatio is a primary cocer of market risk maagemet, the aalysis of the coditioal retur distributio is the mai focus of this paper ad it is based o the assumptio that returs follow a statioary time series process with stochastic volatility structure. This premise is supported by empirical evidece regardig the presece of stochastic volatility i the fiacial time series (Frey, 997) ad implies that returs are ot ecessarily idepedet over time. The specific methodology used for describig volatility dyamics is based o the Geeralized Autoregressive Coditioal Heteroskedasticity (GARCH) methods ad has a additioal advatage of providig a iid iovatios series that ca be directly modelled by usig a semi-parametric distributio with tails described by extreme value theory. The estimatio of the GARCH models for each returs series ad the costructio of the semi-parametric distributios based o the iovatios represets the first stage of the portfolio risk evaluatio. At the secod stage of the process the depedece structure betwee assets is defied with the help of copula methods. A joit distributio fuctio for risk factors cotais a descriptio of the margial distributio for each idividual factor ad also a descriptio of their depedece structure. The copula methods provide a mechaism for isolatig the depedece structure of the portfolio from the idividual margis of the assets ad as a cosequece it provides flexibility i modellig the portfolio as a whole. The compositio of the portfolio has to be take ito cosideratio i order to choose a specific copula for modellig. There is empirical evidece that equity markets ted to be more correlated i volatile times (Logi, 2000) ad this implies that the depedece structure should allow for high tail depedece amog assets. Ufortuately the Gaussia depedece structure, relyig oly o the otio of correlatio, does t allow for extreme co-movemets regardless of potetially large magitudes i correlatio betwee the uderlyig idividual assets. Because of these drawbacks, the depedece structure used i this paper is based o the Studet-t copula that relies o the otio of correlatio but, i additio it is also characterized by a parameter (Degrees of Freedom DoF) that cotrols the tail depedece (extreme co-movemets) of margial distributios. 6

7 The fial stage of the risk evaluatio process is based o the parameters estimated i the previous stages ad cosists of the simulatio of a portfolio coditioal returs distributio that ca be used for Value-at-Risk ad Expected Shortfall estimatio. I the followig sectios of this paper, the methodology of the risk evaluatio process is preseted i detail ad the results of each itermediary stage are displayed. The fial part of this aalysis presets a evaluatio of the methodology by usig a backtestig procedure i order to test the accuracy of the risk measures. B. Literature Review The modellig of fiacial retur distributios usig extreme value theory is applied ad tested i several research papers, both from a geeral market risk perspective ad from a more specific fiacial sector perspective. Daielsso ad De Vries (997b) propose a semi-parametric method for VaR estimatio, where the ucoditioal retur distributio is described by a combiatio of oparametric historical simulatio ad extreme value theory. The authors build their model based o the assumptio that extreme returs occur ifrequetly, ad do ot appear to be related to a particular level of volatility or exhibit time depedece. As a cosequece they propose the ucoditioal loss distributio as a base for Value-at-Risk estimatio. However, the research coducted by McNeil ad Frey (2000) i their 2000 article Estimatio of Tail- Related Risk Measures for Heteroskedastic Fiacial Time Series: a Extreme Value Approach cotradicts the assumptio of Daielsso ad De Vries regardig the superiority of the VaR estimates obtaied from the ucoditioal distributio ad prove that a coditioal approach agaist the curret volatility backgroud is better suited for VaR estimatio. Kaj Nystrom ad Jimmy Skoglud, agreed o this matter i their 200 article o uivariate Extreme Value Theory, GARCH ad Measures of risk ad believe that i order to measure portfolio risks it is importat to correctly idetify a model for the risk factors. Although they both combie GARCH models to estimate the curret volatility ad the Extreme Value Theory for estimatig the tail of the iovatio distributio of the GARCH model, Nystrom ad Skoglud itroduce the assumptio of asymmetry. This cocept is importat i the cotext of the GARCH model as it o loger assumes that egative ad positive shocks have the same impact o volatility. 7

8 Oe importat similarity betwee the two articles is give by the assumptio of a Studet s t distributio for the iovatio series. Although the Studet-t distributio cotiues to assume the symmetry hypothesis, it also makes it possible to capture the observed kurtosis. This is a importat part of the model, as Nystrom ag Skoglud emphasize, because the distributios of the fiacial series are ofte characterized by excess kurtosis ad egative skewess. So by usig the ormal distributio approximatio, the risk of high quatiles is severely uderestimated ad it is for this reaso that the authors chose a alterative to this distributio i the form of a Studet-t distributio. After havig looked at empirical evidece, Nystrom ad Skoglud came to the coclusio that, i the case of daily risk measuremet, while the ormal model ideed teds to uderestimate the lower tail ad overestimate the upper oe, the t distributio also has its flaws, i the sese that it actually teds to overestimate both tails. I relatio to the distributio of residuals, McNeil ad Frey, while agreeig i favour of a t distributio istead of a ormal oe, they believe GPD-approximatio to be a much better model for whe the tails are asymmetric. If the tails of the distributio of residuals were symmetric the the t distributio, they argue, is a good alterative. Embrechts et al (999) i their article Extreme Value Theory as a Risk Measuremet Tool proouced themselves i favour of usig a parametric estimatio techique which is based o a limit result for the excess distributio over high threshold, a techique also preferred by McNeil ad Frey. This is a techique which will be explaied i greater detail later i the methodology of this paper. 8

9 C. Methodology I. GARCH models I a geeralized autoregressive coditioal heteroscedasticity (GARCH) model, returs are assumed to be geerated by a stochastic process with time-varyig volatility. This implies that the coditioal distributios chage over time i a autocorrelated way ad the coditioal variace is a autoregressive process The GARCH model was itroduced by Bollerslev (986) ad it cosists of two equatios, the coditioal mea equatio that explais how the expected value of the retur chages over time ad the coditioal variace equatio that describes the evolutio of the coditioal variace of the uexpected retur process. A ARMA(m,) model describes how the retur chages over time (the coditioal mea equatio): y t = c + m φi yt i + θ jε t j + i= j= ε t I this model it is assumed that ε t is idepedet ad idetically distributed with mea zero ad variace σ 2. GARCH exteds the ARMA model by assumig that ε t = z t σ t, where z t is idepedet ad idetically distributed with mea zero ad uit variace ad z t σ t are stochastically idepedet. The dyamics of σ 2 t, the coditioal variace at time t, is described by the secod equatio of GARCH model, ad the represetatio of this equatio for GARCH(p,q) is: σ 2 t = k + p i= G σ i 2 t i + q j= A ε j 2 t j The coefficiets of the GARCH model must respect some costraits i order to avoid the possibility that the volatility becomes egative or the process o-statioary. q j = p A j + G i= i < ; for a statioary volatility process 9

10 0 0 0 > k G A j j ; for positive volatility I some equity markets it ca be observed that volatility is higher if the market is fallig tha if the market is risig. The volatility respose to a large egative retur is ofte greater tha it is to a large positive retur of the same magitude. Oe possible reaso for this effect may be explaied by the debt/equity ratio. Whe the equity price falls the debt remais costat i the short term, so the debt/equity ratio icreases, the compay becomes more highly leveraged ad so the future of the firm becomes more ucertai. The asymmetry i volatility clusterig caused by the leverage effect ca be captured with asymmetric GARCH models like GJR-GARCH, itroduced by Gloste, Jagaatha ad Rukle (993). = = = = q j j t j t j q j j t j p i i t i t Sg L A G k ε ε σ σ < = j t j t j Sg t ε ε = = = < + + p i q j j i q j j L G A 2 ; for a statioary volatility process > j j j j L A k G A ; for positive volatility I estimatig the retur series with GARCH models it is commoly assumed that the iovatio series (z t ) has a stadard ormal distributio. This premise relies o the fact that the excess kurtosis of the retur distributio ca be partially captured by the GARCH model. However, it is possible that some of the excess kurtosis to remai uexplaied ad as a 0

11 cosequece the assumptio of ormality for the iovatios might ot be valid (McNeil, 2000). Bollerslev (986) proposed the use of the t-distributio for the iovatio series i order to better explai the excess kurtosis of the fiacial series. II. Extreme Value Theory (EVT) models Extreme Value Theory (EVT) was coceived as the probabilistic theory for studyig rare evets (i.e. realizatios from the tails of a distributio) ad it is maily used for the parametric modellig of the tails of a distributio. Because EVT eeds iformatio oly about extreme evets i order to model the tails, it is ot ecessary to make a particular assumptio about the shape of the etire distributio i order to use the theory. Furthermore, because EVT is a parametric techique it ca be used to estimate out of sample quatiles by extrapolatio. The EVT uses two approaches i order to study the extreme evets. The first method (block maxima) is used to describe the distributio of miimum or maximum realizatios of a process. I order to apply this method the data sample is divided ito blocks ad the maximum value from each block is cosidered a extreme evet. These values are the extracted from the sample data ad modelled separately by fittig them to a Geeralized Extreme Value (GEV) distributio. The secod method (peak-over-threshold) is used for modellig the distributio of exceedaces over a particular threshold. I order to idetify the extreme values, this method sets a threshold over which all realizatios of the process are cosidered extreme. After settig this critical value, the observatios of the sample data that are larger tha the threshold are extracted ad the exceedaces are computed (exceedace = extreme value threshold). Fially, i order to describe the extreme evets, the exceedaces are fitted to a Geeralized Pareto Distributio. Both EVT methods have a parameter that is used for idetifyig the extreme values of the process that is aalyzed. This parameter has to be fixed before the extreme data ca be fitted to a certai distributio. I the case of block maxima methods this parameter is the size of the block ad i the case of peak-over-threshold methods the value of the threshold has this role.

12 . Limitig distributios of the maxima. The extreme value theory is applied i order to describe the limitig distributios of the sample maxima. This cocept is similar to the cetral limit theorem that sets the ormal distributio as the limitig distributio of sample averages. The EVT describes this family of limitig distributios uder a sigle parameterizatio kow as the geeralized extreme value (GEV) distributio. If r t, t =, 2,...,, is a ucorrelated sample of returs with the distributio fuctio F(x) = P{r t x}, which has variace σ 2 ad mea μ 2, we deote the sample maxima 3 of r t by M = r, M 2 = max(r, r 2 ),, M = max(r,..., r ), where 2. Let R deote the real lie, if there exists a sequece c > 0, d R ad some o-degeerate distributio fuctio H such that M d c d H the H belogs to oe of the followig three families of distributios: Gumbel: Λ ( x) = e e x, x R Fréchet: Φ 0, x 0 ( x ) = { α e x α, x > 0, α > 0 Weibull: Ψ α ( x ) = x α e, x 0, α < 0, x > 0 2 We assume for coveiece that μ = 0 ad σ 2 =. 3 The sample maxima is mi(r,..., r ) = max( r,..., r ). 2

13 The Fisher ad Tippett (928) theorem suggests that the limitig distributio of the maxima belogs to oe of the three distributios above, regardless of the origial distributio of the observed data. If we cosider ξ= /α (vo Mises, 936) ad Jekiso, 955), Fréchet, Weibull ad Gumbel distributios ca be expressed as a uified model with a sigle parameter. This represetatio is kow as the geeralized extreme value distributio (GEV): H ξ ( x) = e e ( + ξx ) e x ξ if ξ 0, +ξx > 0 if ξ = 0 where ξ= /α is a shape parameter ad α is the tail idex. The class of distributios of F(x) where the Fisher-Tippett theorem holds is quite large. Oe of the coditios (Falk, Hüssler, ad Reiss 994) is that F(x) has to be i the domai of attractio of the Frechet distributio (ξ > 0), which i geeral is true for the fiacial time series. Gedeko (943) shows that if the tail of F(x) decays like a power fuctio (heavy-tailed distributios like Pareto, Cauchy, Studet-t), the it is i the domai of attractio of the Fréchet distributio. 2. Limitig distributio of exceedaces over a threshold The limitig distributio of exceedaces over a threshold is a member of the family of extreme value distributios. I order to estimate the parameters of this limitig distributio, first we have to idetify the extreme values of the sample data. If we take a sample of observatios, r t, t =, 2,..., with a distributio fuctio F(x) = Pr{r t x} ad we set a high-threshold u, the the exceedaces over this threshold occur whe r t > u for ay t i t =, 2,...,. A excess over u is defied by y = r i u (peak-over-threshold method). For a high threshold u, the probability distributio of excess values of r over threshold u is defied as: 3

14 } Pr{ ) ( u r y u r y F u > = Give that r exceeds the threshold u, this represets the probability that the value of r exceeds the threshold u by at most a amout y. This coditioal probability may be writte as: ) ( ) ( ) ( ) Pr( }, Pr{ ) ( u F u F u y F u r u r y u r y F u + = > > = Because r > u ad x = y + u, we ca also write the followig expressio: ) ( ) ( )] ( [ ) ( u F y F u F x F u + = The theorem of Balkema ad de Haa (974) ad Pickads (975) shows that for sufficietly high threshold u, the distributio fuctio of the excess may be approximated by the Geeralized Pareto Distributio (GPD) The GPD ca be defied as: ( ) = + = 0, 0,,, ξ σ υ ξ ξ υ σ ξ ξ σ υ if x if e x G [ ] < 0,, 0,, ξ υ ξ ξ σ υ υ if if x where ξ= /α is a shape parameter ad α is the tail idex, σ is the scale parameter, ad υ is the locatio parameter. Whe υ = 0 ad σ =, the represetatio is kow as the stadard GPD. The relatioship betwee the limitig distributio of exceedaces (stadard GPD) ad the limitig distributios of the sample maxima (GEV) is: 4

15 G ( x) = + log H ( x) if log H ( x) > ξ ξ ξ Whe ξ > 0, GDP takes the form of the ordiary Pareto distributio which is a heavytail distributio, ad as a cosequece it is very useful for the aalysis of fiacial series. If ξ >0, E[X k ] is ifiite for k /ξ ad i order to have a fiacial series with fiite variace, ξ must be less tha 0.5. Whe ξ = 0, the GPD correspods to the thi-tailed distributios ad for ξ < 0 it correspods to fiite-tailed distributios. For ξ > 0.5 the GPD model ca be estimated with the maximum-likelihood method because i this case maximum-likelihood regularity coditios are fulfilled ad the maximum-likelihood estimates are asymptotically ormally distributed (Hoskig ad Wallis 987). Oe importat aspect whe applyig EVT is the choice of the threshold value. This value must be set low eough i order to have a sufficiet umber of exceedaces for computig accurate estimates of the tail parameters with the ML method. At the same time the threshold must be set high eough i order to have the GPD as the limitig distributio of the exceedaces. Ufortuately there is o atural estimator of the threshold ad thus, its value must be set more or less arbitrarily. I practice, istead of assumig that the tail of the uderlyig distributio begis at the threshold u, we ca choose a fractio k/ of the sample data which is cosidered to be the tail of the distributio, hece implicitly choosig also the threshold value. McNeil ad Frey (2000) ad Nyström ad Skoglud (200) coducted Mote-Carlo experimets i order to evaluate the properties of the ML estimator for various distributios ad sample sizes. The results show that the ML estimator is almost ivariat to the threshold value if k is set betwee 5-3% of the sample data. III. Copula models The essetial idea behid the copula approach is that a joit distributio ca be decomposed ito margial distributios ad a depedece structure represeted by a fuctio called copula. Usig a copula, margial distributios that are estimated separately ca be combied i a joit risk distributio that preserves the origial characteristics of the margials. 5

16 A real advatage of usig copula fuctios for the descriptio of depedece structures cosists i the ability to combie differet types of margial distributios (parametric or o-parametric) ito a joit risk distributio. At the same time, the joit distributios created usig copulas ca have a depedece structure described by more tha a simple correlatio matrix (e.g. the t-copula has a additioal tail depedece parameter - degrees of freedom). Defiitio (Copula) A fuctio followig properties: C :[0,] [0,] is a -dimesioal copula if it satisfies the a) For all u i [ 0,], C(,...,, ui,...,) = ui b) For all u [ 0,], C( u..., u ) = 0 if at least oe ui = 0 i c) C is grouded ad -icreasig Sklar s theorem: Give a d-dimesioal distributio fuctio G with cotiuous margial cumulative distributios F,..., F d, the there exists a uique -dimesioal copula :[0,] d C [0,] such that for x R : G x,..., x ) = C( F ( x ),..., F ( x )) (c) ( Moreover, if F, F 2,,F are cotiuous, the C is uique. Sklar s Theorem is a fudametal result cocerig copula fuctios ad basically it states that ay joit distributio ca be writte i terms of a copula ad margial distributio fuctios. If F is a uivariate distributio fuctio the the geeralized iverse of F is defied as F ( t) = if{ x R : F( x) t} for all t [0,] ad usig the covetio if{φ } = 6

17 Corollary Let G be a -dimesioal distributio fuctio with cotiuous margials F,..., F d ad a -dimesioal copula C. The for ay u [0, ], C( u,..., u ) = G( F ( u ),..., F ( u )) (c2) Note: without the cotiuity assumptio, this relatio may ot hold (Nelse 999). The copula liks the quatiles of the two distributios rather tha the origial variables, so oe of the key properties of a copula is that the depedece structure is uaffected by a mootoically icreasig trasformatio of the variables. Theorem (copula ivariace) Cosider cotiuous radom variables (X,..., X ) with copula C. If g,..., g : R R are strictly icreasig o the rage of X,..., X, the (g (X ),..., g (X )) also have C as their copula. Remark By applyig Sklar s theorem ad by exploitig the relatio betwee the distributio ad the desity fuctio, we ca easily derive the multivariate copula desity c(f (x ),.,F (x )) associated with a copula fuctio C(F (x ),.,F (x )) f ( x,..., x ) where we defie: [ C( F ( x ),..., F ( x ))] F ( x )... F ( x ) = * f i ( xi ) = c( F ( x ),..., F ( x ))* i= i= c ( F ( x ),..., F ( x )) f ( x,..., x ) f ( x ) = i = i i f ( x ) (c4) Defiitio (Normal-copula) Let R be a symmetric, positive defiite matrix with diag(r) = ad let Φ R deote the stadard multivariate ormal distributio with correlatio matrix R 4. The the Multivariate Gaussia Copula is defied as: i i (c3) ( u,..., u ; R) = Φ ( Φ ( u ), Φ ( u ) Φ ( u )) C u, 2 R 2,..., (c5) 4 Give a radom vector X = (X,...,X ) we defie the stadardized ormal joit desity fuctio f(x) with corrwlatio matrix R, as follows: f ( x)= R exp x x R 2 2 2π 2 ( ) 7

18 where Φ - (u) deotes the iverse of the ormal cumulative distributio fuctio. The associated multiormal copula desity is obtaied by applyig equatio (c4): c ( Φ( x ),..., Φ( x )) = f exp x R x ( ) ( ) 2 x x 2 R 2,..., 2π = gaussia ( x ) = 2 exp x i i 2π 2 gaussia f i i i = ad hece, fixig u i = Φ(x i ), ad deotig with ζ = (Φ - (u ),., Φ - (u )) the vector of the gaussia uivariate iverse distributio fuctios, we have c ( u, u,..., u ; R) = exp ς ( R I ) ς 2 R 2 2 (c6) Defiitio (Studet t-copula) Let R be a symmetric, positive defiite matrix with diag(r) = ad let T R,v deote the stadard multivariate Studet s t distributio with correlatio matrix R ad v degrees of freedom 5. The the multivariate Studet s t copula is defied as follows: ( u,..., u ; R, ν ) T ( t ( u ), t ( u ) t ( u )) C u, 2 = R. ν ν ν 2,..., ν (c7) where t v - (u) deotes the iverse of the Studet s t cumulative distributio fuctio. The associated Studet s t copula desity is obtaied by applyig equatio (4) 5 Give a radom vector X=(X,...,X) with a joit stadardized multiormal distributio with correlatio matrix R ad a χ v 2 distributed radom variable S, idepedet from X, we defie the stadardized multivariate Studet s t joit desity fuctio with correlatio matrix R ad v degrees of freedom, as the joit distributio fuctio of the radom vector EMBED Equatio.3 Y = (,..., ) : f ( y) Γ 2 = Γ ν 2 ( ν + ) * ( πν ) 2 R 2 y R * + ν y 2 X S / ( ν + ) v X S / v 8

19 ( u u,..., u, R, ν ) c f ( x,..., x ) Studet ( x ) ν + Γ ν Γ ς R ς + ν Studet 2 2 = = R 2, 2 * * ν + f ν ν + i= i i 2 Γ Γ ς 2 i= + i v where ζ = (t v (u),., t (u )). ν ν + 2 (c8) IV. Measures of risk The market risk represets the ucertaity of observig a evet i the future that could lead to a importat portfolio loss. I this cotext, a measure of risk is a fuctio that takes as a argumet the distributio that characterizes the risk factor ad returs a scalar value that describes the potetial risk implied. The key aspect of measurig risk resides i the ability to correctly idetify the distributios of the risk factors. Eve thought the correspodece betwee the distributio of the risk factor ad a scalar could be expressed i differet ways, oly a part of all these potetial fuctios are appropriate idicators of risk. Artzer et al. (997, 998) proposed the theory of coheret risk that captures the desired properties of a risk measure. If x is a set of real-valued radom variables (e.g. the loss distributio of a equity) ad the fuctio ω is a real-valued risk measure, the ω should respect the followig properties i order to be cosidered coheret: Positive homogeeity. This property basically states that if we icrease the quatity of a certai equity i our portfolio we should also have a liear icrease i the risk ivolved ad ot a diversificatio effect. ω(λx) = λω(x). 9

20 Subadditivity. This property represets the advatages of portfolio diversificatio. The risk of a portfolio (x + y) should be lower tha (or equal to) the sum of the risk of the idividual securities (x,y) ω(x + y) ω(x) + ω(y). Mootoicity. This property implies that a higher risk should be cosidered for a higher retur. x y ω(x) ω(y). Traslatioal ivariace. This property states that the iclusio of uits of a risk-free asset with returs r i the portfolio should lower the risk of the portfolio. ω(x + r) = ω(x). Value-at-Risk Value at Risk measures the worst loss to be expected of a portfolio over a give time horizo at a give cofidece level. If we mark losses with a positive sig ad gais with a egative sig we ca estimate Value at Risk by takig the relevat quatile q α of the coditioal distributio. VaR α = qα Although VaR offers a simple ad ituitive way of evaluatig risk Artzer et al. (997, 998) have criticized it as a measure of risk for two mai reasos. Firstly they showed that VaR is ot ecessarily subadditive ad as a cosequece it is ot a coheret measure of risk ad secodly, this measure gives oly a upper limit o the losses give a cofidece level, but it tells othig about the potetial size of the loss if this upper limit is exceeded. 20

21 Expected Shortfall The Expected Shortfall (ES) of a asset or a portfolio is the average loss give that VaR has bee exceeded. where r t is the retur at time t ES ( α) = E[ r r > VaR ( α)] t t t t Although ES is a coheret measure of risk, its accuracy also depeds o the ability to idetify the true loss distributio of the portfolio. D. Applicatio I. Data The risk evaluatio techiques described earlier i the paper are applied to a portfolio of five Romaia equities traded o the Bucharest Stock Exchage (symbols: SIF, SIF2, SIF3, SIF4, SIF5). These particular stocks were selected due to their high market liquidity, a log time series with very few missig values ad high volatility periods that ca help evaluate the accuracy of the model with backtestig procedures. The compaies are part of the fiacial sector ad their primary activity is the ivestmet i Romaia firms. The price series covers the period 04/0/200 05/06/2007, has a total of 564 observatios ad is adjusted for corporate evets. The missig data was replaced by the previous value of the series (or the ext value if data at the begiig of the series is missig). The origial series of prices was trasformed ito retur series with the help of the logarithmic formula: rt = log(p t /p ) t 2

22 II. Estimatio ad results. GARCH models Before the estimatio of the GARCH model, a aalysis of the data is made i order to verify if the returs are autocorrelated ad the volatility clusterig effect is preset i the series 6. The aalysis ca be made both visually by studyig the plot of the autocorrelatio fuctios (Appedix I, Figure ad Figure 2) ad statistically by usig a Ljug-Box test for radomess (Appedix I, Table ). From the visual aalysis we ca coclude that the series SIF, SIF2 ad SIF4 have a strog first order autocorrelatio ad the series SIF3 ad SIF5 display a weak at most of autocorrelatio, or eve a o-existet degree of autocorelatio. At the same time the visual aalysis of the squared returs suggests that a strog autocorrelatio is preset i all five series. The results of the Ljug-Box test cofirm the outcome of the visual aalysis. For the retur series, the ull hypothesis of the test (where data is radom) is rejected at a 5% sigificace level for all series except SIF3. Similar results are obtaied for the squared returs series where the ull hypothesis of the test is rejected at a 5% sigificace level for all equities. These results cofirm the assumptios that the retur series are autocorelated ad have a time-varyig volatility, thus a GARCH model should be appropriate for explaiig the data geeratig process of each series. I order to fid the best GARCH model for each series, a GJR-GARCH model is estimated at first ad the coefficiets that are ot statistically sigificat are removed, the the model is estimate agai i a simpler form. I additio to the verificatio of sigificace for each coefficiet, the Akaike criterio is also used for model selectio. The coefficiets are estimated with the maximum likelihood method ad a assumptio of t-distributed iovatios. The iitial model has the followig form: σ 2 t rt = c + φ r t + ε t = k + Gσ t + Aε t + LSgt ε t 6 The volatility clusterig effect ca be detected by calculatig the degree of autocorrelatio for the squared returs. 22

23 The results of the iitial estimatio (Appedix I, Table 2) show that the coefficiet of leverage is ot sigificat or has a wrog sig for all the five series. This implies that the volatility of the series is iflueced i a equal way by a egative or positive retur of the same magitude ad the assumptio that a egative retur has a higher impact does t seem to be supported by the data. Because the results do ot support the assumptio of a leverage effect, this parameter is removed ad the model becomes a GARCH(,). I additio, the coefficiet for the AR() parameter is ot sigificat for all the series, thus this parameter is also removed where it is foud to be irrelevat. The results of the fial estimatio (Appedix I, Table 3) show that the coefficiets are all sigificat at a 5% with the exceptio of the coefficiet of the AR() parameter of the SIF2 series which is sigificat oly at a 0% level. I order to evaluate the outcomes of the GARCH modellig, the residuals are first stadardized ad the the Ljug-Box test is applied to the stadardized residuals. The assumptio of the GARCH model is that ε t = z t σ t, where z t (stadardized residuals) is idepedet ad follows a Studet s t - distributio. The z t series is obtaied by dividig the residuals(ε t ) at the coditioal stadard deviatio (σ t ). The Ljug-Box Test applied to the stadardized residual (Appedix I, Table 4) shows that the ull hypothesis caot be rejected at a 5% sigificace level for SIF, SIF3, SIF4 ad at % sigificace level for SIF2 ad SIF5. I the case of the squared stadardised residuals the results are eve more relevat, the ull hypothesis beig accepted at a 5% sigificace level for all five series. These results prove that the GARCH models accurately describe the time series ad that the stadardized residuals fulfil the idepedece criteria that is ecessary i order to use the extreme value theory. 2. Extreme Value Theory (EVT) models Eve thought the estimatio of the GARCH model with the assumptio of t-distributed iovatios (stadardized residuals) ca explai a large degree of the excess kurtosis foud i the retur series, it still caot capture its asymmetry because the distributio is presumed to be symmetric. At the same time, the correct parameterizatio of the iovatio series is very importat because it is later used for the simulatio of the portfolio loss distributio. 23

24 I order to better describe the iovatio series resulted from the GARCH models, extreme value theory is used to estimate each tail of the distributio ad a kerel smoothig method is used for the iterior. The tail parameters ca be estimated usig oe of the methods described by the extreme value theory. However, for this study, the peak-over-threshold method is used maily because it eeds a smaller data sample compared to the block-maxima method. I order to idetify the tails of the iovatio series we sort the values of the series i ascedig order ad we cosider the first 0% of the values to be the lower tail ad the last 0% of the values to be the upper tail. By usig this method the threshold value is implicitly determied for the lower ad upper tail. The 0% fractio of the distributio is selected by takig ito cosideratio the simulatios performed by McNeil ad Frey (2000) ad Nyström ad Skoglud (200) that showed that the ML estimator is almost ivariat to the threshold value if the tail is cosidered betwee 5-3% of the sample data. The parameters of the GPD distributio that are estimated usig the maximumlikelihood method are displayed i (Appedix II, Table.). By studyig the results it ca be observed that the coefficiet that gives the heaviess of the tail (tail idex) is statistically differet for the upper compared with the lower tail, thus cofirmig the assumptio of a asymmetric distributio of iovatios. Furthermore the coefficiets of the tail idex for the lower tails are all statistically differet from zero, thus givig a idicatio that the lower tails are heavier the the tails of a ormal distributio. The estimated value of the tail idex for the upper tail is ot statistically differet from zero for either of the series, thus we ca draw the coclusio that the shape of these tails resembles the tail of a ormal distributio. By comparig the shape of the tails estimated usig the extreme value method with the shape of a Studet s t-distributio that has the same degrees of freedom as the parameter estimated i the GARCH model (Appedix II, Figure a,b ), it ca be observed that the extreme value method describes much better the distributios of the iovatios. I order to have a complete semi-parametrical distributio for each series of iovatios, a pseudo CDF (cumulative distributio fuctio) ad ICDF (iverse cumulative distributio fuctio) are built. The pseudo CDF fuctio receives a value, it idetifies where the value is situated i the estimated semi-parametrical distributio (i oe of the tails or i the cetre) by comparig it with the thresholds ad based o this iformatio it computes a correspodig cumulated probability. The pseudo ICDF fuctio is built o the same priciple, the cumulative probability that is give as a iput is mapped to oe area of the 24

25 semi-parametrical distributio ad a correspodig quatile is computed ad retured based o that iformatio. With the help of these pseudo fuctios a represetatio of each semiparametric distributio ca be build (Appedix II, Figure, c). 3. Copula models Oce the model for each time series is defied, the depedece structure of the portfolio ca be estimated by likig together the semi-parametrical distributios with copula methods. The Studet s t copula is selected for this task because i additio to a correlatio matrix it is also characterized by the degrees of freedom parameter, which defies the amout of tail depedece betwee the series. The copula is calibrated by usig the Caoical Maximum Likelihood (CML) method because this method allows a estimatio of the copula parameters without makig ay assumptio about the margial distributios. The CML method 7 ca be implemeted i two stages. T First we trasform the iitial data set X = (X t,, X t ) t= ito uiform variates usig the margial distributio fuctio, that is, for t=,.t, let u t = (u t,.., u t ) = [F (Xt),., F (X t )]. I this case, for equity i, X i represets the iovatio series ad Fi represets the pseudo-cumulative distributio. Secodly we estimate the vector of copula parameters α, via the followig relatio: Λ T t t α CML = argmax lc(u,.., u ; α) α where c is the copula desity fuctio, i this case the desity of the Studet s t copula. t= The actual estimatio of the copula parameters is doe i two steps; the first step maximizes the log-likelihood fuctio with respect to the liear correlatio matrix, give a fixed value for the degrees of freedom. The secod step uses the results from the first optimizatio i order to maximize the fuctio with respect to the degrees of freedom, thus maximizig the log-likelihood over all parameters. The fuctio that is maximized i the secod step is called the profile log-likelihood fuctio for the degrees of freedom. The estimated correlatio matrix (Appedix III, Table) shows a positive correlatio betwee all five series, while the relative small value of the degrees of freedom parameter (Appedix III, Table2) cofirms the presece of a strog tail depedece. The stadard error 7 See Mashal ad Zeevi, p 25. for more details. 25

26 of the degrees of freedom parameter ( ) was obtaied usig a simple bootstrap method. 4. Portfolio simulatio Oce the models for the margial distributios ad the depedece structure are estimated, we ca simulate the coditioal loss distributio of the portfolio for the ext period ad compute the risk measures of iterest. The first stage of the simulatio process represets the geeratio of depedet series by usig the depedece structure give by the t-copula, that is, for each series, for a horizo of h days, trials are geerated from a multivariate Studet s t distributio that has the same correlatio matrix ad degrees of freedom parameters as those estimated with the t-copula. The result of this step is a collectio of (o of equities x horizo) distributios that have the same depedece structure as the origial data. However these distributios were geerated usig a multivariate Studet s t distributio, so they must be trasformed i order to follow the semi-parametrical distributios used by the GARCH model. The trasformatio of each distributio is doe i two steps, first the distributio is shaped ito a uiform variate, by usig the cumulative distributio fuctio of the Studet s t distributio, ad secodly these uiform variates are coverted ito the semi-parametrical distributios by usig the pseudo-iverse cumulative distributio fuctio of the correspodig semi-parametrical distributio. A visual example of a simulatio for two correlated series ca be foud i (Appedix IV, Figure ). At the secod stage of the simulatio process, the semi-parametric distributios are give as a iput to the GARCH model that reitroduces the volatility ito the series ad gives as a output coditioal series of returs. At this stage we have a coditioal distributio of returs for each equity i the portfolio ad for each day of the horizo (h). These coditioal distributios ca be cumulated i order to build the loss distributio of the etire portfolio for a horizo of up to h days. 5. Measures of risk If we mark losses with a positive sig ad gais with a egative sig we ca estimate Value at Risk by takig the relevat quatile q α of the coditioal distributio. VaR α = qα 26

27 Also, Expected Shortfall is estimated by usig the followig formula: where represets the umber of trials. ES = E( X X > VaR) = ( X i= [ α ] ( ) ) /( [ a]) i Both measures of risk are applied to the idividual coditioal distributios ad the coditioal portfolio loss distributio (computed with a equal weight for each asset) ad the results are displayed i (Appedix IV, Table -4). The estimated risk is comparable betwee the equities, SIF3 displayig the highest level of risk while SIF5 the lowest. Furthermore, the high correlatio betwee the assets reduces the beefit of diversificatio to a miimum, thus makig the risk of the portfolio comparable to ay of the idividual equities. 6. Backtestig I order to evaluate the accuracy of the methodology used for the estimatio of risk, a backtest is applied for each idividual retur series ad also for the portfolio. The test implies the estimatio of Value-at-Risk for a umber of days for which we already kow the actual returs. By comparig the estimated Value-at-Risk with the actual returs we ca observe if the cofidece levels of the risk measure are ideed respected. The tests are performed with a day horizo, for the last 500 days of the series ad with a fixed data sample of 000 observatios. For each day the methodology is applied from the begiig ad all the parameters are reestimated. Plots of these tests ca be see i (Appedix IV, Figure 2) ad the umber of violatios for each series is displayed i (Appedix IV, Table 5). The backtestig results are ot very clear, firstly because the evaluatio at 90% ad 95% cofidece levels gave mixed results ad secodly because the accuracy of the risk measure at 99% cofidece level caot be test properly due to the small umber of back-testig days (500). For the idividual series, the results at a 90% cofidece level show that the risk is beig slightly uderestimated for two of the series (SIF, SIF2) ad slightly overestimated for the other three (SIF3, SIF4, SIF5). The situatio is differet at a 95% cofidece level, where we ca see a higher uderestimatio of the risk for the majority of the series. At a 99% cofidece level the risk appears to be overestimated for the majority of the series, perhaps due to the lack of sufficiet observatios leadig up to uclear coclusios. 27

28 At the portfolio level, the results show a uderestimatio of the risk at all three cofidece levels, although the degree of uderestimatio is rather small for the 90% ad 99% levels ad more sigificat for the 95% level. E. Coclusios. This paper aims at computig accurate estimates for the risk of a portfolio by costructig its coditioal loss distributio with a flexible methodology that separates the descriptio of the margial distributios from the depedece structure. The retur series for each of the equities was modelled usig GARCH methods i order to explai the autocorrelatio ad time-varyig volatility. The, the iovatio series resulted from the GARCH model is described as a semi-parametrical distributio with GPD tails ad a kerelsmoothed iterior that captures the asymmetry ad excess-kurtosis ofte foud i these series. The lik betwee the semi-parametrical distributios is the explaied by a Studet s t copula that gives the depedece structure of the etire portfolio. The estimated parameters of the margial distributios ad the depedece structure serve as a base for the simulatio of a coditioal portfolio distributio ad implicitly for the estimatio of the risk measures. By aalyzig the itermediary results of this methodology the followig coclusios ca be draw: - the GARCH models explai very well the autocorrelatio foud i the retur series ad the volatility clusterig effect - the distributios of the iovatios are asymmetric with heavy lower tails ad thi upper tails - the GPD descriptio for the tails of the iovatio series is more accurate compared to the descriptio give by the t-distributio estimated by the GARCH models. - the backtestig results for Value-at-Risk are ot coclusive but give a idicatio of a possible uderestimatio of the risk at 95% cofidece level Further research ca be doe i two mai directios; first this methodology could be applied for portfolios with differet risk factors i order to evaluate its accuracy for a larger collectio of assets. Secodly the methodology ca be improved by usig better measures of risk or more flexible tools for describig the data geeratig process for the returs of the portfolio. This implies for example, the estimatio of the GARCH models without ay 28

29 assumptio about the distributio of the iovatios, the use of differet copulas that might describe better the depedece structure of the assets or the use of spectral measures of risk that take ito accout the risk aversio of the risk maager. 29

30 F. Refereces. Alexader, C. (200), Market Models: A guide to Fiacial Data Aalysis 2. Artzer, Ph., F. Delbae, J.-M. Eber, ad D. Heath (998), Coheret Measures Of Risk, Uiversite Louis Pasteur, EidgeÄossische Techische Hochschule, Societe Geerale, Caregie Mello Uiversity, Pittsburgh 3. Bao, Y., T.H. Lee, ad B. Saltoglu (2004), Evaluatig Predictive Performace of Valueat-Risk Models i Emergig Markets: A Reality Check, UT Sa Atoio, UC Riverside, Marmara Uiversity 4. Bouyé, E., V. Durrlema, A. Nikeghbali, G. Riboulet, ad T. Rocalli (2000), Copulas for Fiace: A Readig Guide ad Some Applicatios, Fiacial Ecoometrics Research Cetre City Uiversity Busiess School Lodo 5. Clemete, A. ad C. Romao (2004a), Measurig ad optimizig portofolio credit risk: A Copula-Based Approach, Workig Paper. - Cetro Iterdipartimetale sul Diritto e l Ecoomia dei Mercati (2003b) Measurig portofolio value-at- risk by a Copula-Evt based approach 6. Cotter, J. ad K. Dowd (2005), "Extreme Spectral Risk Measures: A Applicatio to Futures Clearighouse Margi, Uiversity College Dubli, Nottigham Uiversity Busiess School 7. Daielsso, J. Ad C. G. de Vries (997), Value-at-Risk ad Extreme Returs, Lodo School of Ecoomics ad Istitute of Ecoomic Studies at Uiversity of Icelad, Tiberge Istitute ad Erasmus Uiversity 8. Demarta, S. ad A. J. McNeil (2004), The t Copula ad Related Copulas, Departmet of Mathematics Federal Istitute of Techology ETH Zetrum 9. Dias, A. ad P. Embrechts, Dyamic copula models for multivariate high-frequecy data i Fiace, Warwick Busiess School, - Fiace Group, Departmet of Mathematics, ETH Zurich 0. Diebold, F. X., T. Schuerma, ad J. D. Stroughair (998), Pitfalls ad Opportuities i the Use of Extreme Value Theory i Risk Maagemet, The Wharto Fiacial Istitutios Ceter. García, A. ad R. Geçay (2006), Risk-Cost Frotier ad Collateral Valuatio i Securities Settlemet Systems for Extreme Market Evets, Bak of Caada Workig Paper Hotta, L.K., E.C. Lucas, ad H.P. Palaro, Estimatio of VaR Usig Copula ad Extreme Value Theory, State Uiversity of Campias, Departmet of Statistics, Campias SP, Brazil. ESAMC, Campias SP 30

31 3. Joe, H. ad J. J. Xu, The Estimatio Method of Iferece Fuctios for Margis for Multivariate Models, Departmet of Statistics, Uiversity of British Columbia 4. Logi, F. M. (2000), From value at risk to stress testig: The extreme value approach, Joural of Bakig & Fiace 24, Mashal, R. ad A. Zeevi (2002), Beyod Correlatio: Extreme Co-movemets Betwee Fiacial Assets, Columbia Uiversity 6. McNeil, A.J. (996a), Estimatig the Tails of Loss Severity Distributios usig Extreme Value Theory, Departemet Mathematik ETH Zetrum 7. (998b), Calculatig Quatile Risk Measures for Fiacial Retur Series usig Extreme Value Theory, Departemet Mathematik ETH Zetrum 8. McNeil, A.J. ad R.Frey (2000), Estimatio of Tail-Related Risk Measures for Heteroscedastic Fiacial Time Series: a Extreme Value Approach, Departemet Mathematik ETH Zetrum 9. McNeil, A.J. ad T. Saladi (997), The Peaks over Thresholds Method for Estimatig High Quatiles of Loss Distributios, Departemet Mathematik ETH Zetrum 20. Nyström, K. ad J. Skoglud (2002a), A Framework for Sceariobased Risk Maagemet, Swedbak, Group Fiacial Risk Cotrol 2. (2002b), Uivariate Extreme Value Theory, GARCH ad Measures of Risk, Swedbak, Group Fiacial Risk Cotrol 22. Roseberg, J. V. ad T. Schuerma (2005), A Geeral Approach to Itegrated Risk Maagemet with Skewed, Fat-tailed Risks, Federal Reserve Bak of New York 3

32 Appedix I GARCH Figure. Correlograms for the retur series 32

33 Figure 2. Correlograms for the squared retur series 33

34 Table.Ljug-Box test results for the retur series ad the squared retur series. Table 2. Results for the first estimatio of the GARCH models (GJR-GARCH). 34

35 35

36 Table 3. Results for the fial estimatio of the GARCH models. 36

37 Table 4. Ljug-Box test results for the stadardized residuals ad squared residuals. 37

38 Appedix II EVT Table. Estimated GPD parameters for the tails of the stadardized residuals distributios. SIF Tail Tail Idex Std Error T-Stat Sigma Std Error T-Stat Lower Upper SIF2 Tail Tail Idex Std Error T-Stat Sigma Std Error T-Stat Lower Upper -8.E SIF3 Tail Tail Idex Std Error T-Stat Sigma Std Error T-Stat Lower Upper SIF4 Tail Tail Idex Std Error T-Stat Sigma Std Error T-Stat Lower Upper SIF5 Tail Tail Idex Std Error T-Stat Sigma Std Error T-Stat Lower Upper

39 Figure. a) CDF for the lower tail of the stadardized residuals distributio: GPD vs. Studet s t vs empirical b) CDF for the upper tail of the stadardized residuals distributio: GPD vs. Studet s t vs empirical b) Semi-parametric CDF with GPD tails ad kerel smoothed iterior (a) (b) (c) 39

40 (a) (b) (c) 40

41 (a) (b) (c) 4

42 (a) (b) (c) 42

43 (a) (b) (c) 43

44 Appedix III Copula Table. Estimated correlatio matrix SIF SIF2 SIF3 SIF4 SIF5 Correlatio Matrix SIF SIF2 SIF3 SIF4 SIF Table 2. Estimated degrees of freedom ad the equivalet stadard error. DoF Std Error Figure. The egative log-likelihood fuctio of the t-copula 44

45 Appedix IV Simulatio ad measures of risk Figure. a) Simulated semi-parametric series, SIF3 vs SIF5 uder the assumptio of idepedece (b) Simulated semi-parametric series, SIF3 vs SIF5 with the depedece structure give by the t-copula (a) (b) 45

46 Table. Estimated Value-at-Risk ad Expected Shortfall for a -day horizo idividual series SIF VaR ES 90% 95% 99% 90% 95% 99% -.86% -2.58% -5.40% -3.22% -4.30% -7.78% SIF2 VaR ES 90% 95% 99% 90% 95% 99% -2.02% -2.86% -5.68% -3.55% -4.73% -8.58% SIF3 VaR ES 90% 95% 99% 90% 95% 99% -2.50% -3.49% -6.39% -4.% -5.37% -8.63% SIF4 VaR ES 90% 95% 99% 90% 95% 99% -2.40% -3.09% -6.08% -3.80% -4.92% -8.33% SIF5 VaR ES 90% 95% 99% 90% 95% 99% -.83% -2.4% -4.3% -2.94% -3.78% -6.65% Table 2. Estimated Value-at-Risk ad Expected Shortfall for a 0-day horizo idividual series SIF VaR ES 90% 95% 99% 90% 95% 99% -5.57% -8.5% -5.75% -0.36% -3.87% % SIF2 VaR ES 90% 95% 99% 90% 95% 99% -6.85% -9.98% -6.88% -.47% -4.75% -23.9% SIF3 VaR ES 90% 95% 99% 90% 95% 99% -6.80% -0.2% -8.72% -2.9% -6.08% % SIF4 VaR ES 90% 95% 99% 90% 95% 99% -6.23% -9.52% -6.99% -.07% -4.37% % SIF5 VaR ES 90% 95% 99% 90% 95% 99% -5.75% -8.69% % -2.24% -7.45% % 46

47 Table 3. Estimated Value-at-Risk ad Expected Shortfall for a -day horizo portfolio PORTFOLIO VaR ES 90% 95% 99% 90% 95% 99% -.8% -2.52% -4.87% -3.03% -3.93% -6.40% Table 4. Estimated Value-at-Risk ad Expected Shortfall for a -day horizo portfolio PORTFOLIO VaR ES 90% 95% 99% 90% 95% 99% -5.03% -7.54% -5.74% -9.58% -3.03% -24.4% Table 5.Backtestig results actual vs expected umber of exceedaces for Value-at-Risk Backtestig Results VaR - 90% VaR - 95% VaR - 99% Expected SIF SIF SIF SIF SIF Portfolio

48 Figure 2. A display of the backtestig results for the idividual series ad for the portfolio. The exceedaces are marked with red. 48

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