Dynamic Copula Modelling for Value at Risk. Dean Fantazzini

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1 Dynamic Copula Modelling for Value a Risk Dean Fanazzini Absrac This paper proposes dynamic copula and marginals funcions o model he join disribuion of risk facor reurns affecing porfolios profi and loss disribuion over a specified holding period. By using copulas, we can separae he marginal disribuions from he dependence srucure and esimae porfolio Value-a-Risk, assuming for he risk facors a mulivariae disribuion ha can be differen from he condiional normal one. Moreover, we consider marginal funcions able o model higher momens han he second, as in he normal. This enables us o beer undersand why VaR esimaes are oo aggressive or oo conservaive. We apply his mehodology o esimae he 95%, 99% VaR by using Mone-Carlo simulaion, for porfolios made of he SP500 sock index, he Dax Index and he Nikkei5 Index. We use he iniial par of he sample o esimae he models, and he he remaining par o compare he ou-of-sample performances of he differen approaches, using various back-esing echniques. Keywords: Copulae, Value a Risk, Dynamic Modelling JEL classificaion: G11, G1, G3 Moscow School of Economics Moscow Sae Universiy. fanazzini@msemsu.ru Phone , Fax An earlier version of his paper has been presened a conferences in Konsanz, Venice, Dublin. For helpful commens and suggesions I d like o hank, Luc Bauwens, Elena M. DeGiuli, Mario Maggi, Winfried Pohlmeier, Eduardo Rossi, Giovanni Urga, and Bas Werker. Financial suppor is graefully acknowledged by he Research and Training Nework (RTN) Microsrucure of Financial Markes in Europe. Elecronic copy available a: hp://ssrn.com/absrac=154608

2 1 - Inroducion Following he increase in financial uncerainy in he 90s, here has been inensive research from financial insiuions, regulaors and academics o beer develop sophisicaed models for marke risk esimaion (see Jorion (1997), Crouhy, Galai, and Mark (001), Dowd (1998)). The mos well known risk measure is he Value-a-Risk ( VaR ), which is defined as he maximum loss which may be incurred by a porfolio, a a given ime horizon and a a given level of confidence. Jorion (000) provides an inroducion o Value a Risk as well as discussing is esimaion, while he websie comprehensively cies he Value a Risk lieraure as well as providing oher VaR resources. VaR is now recognized by boh official bodies and privae secor groups as an imporan marke risk measuremen ool: official bodies ha have embraced he concep of VaR include he Basle Commiee on Banking supervision (Basle Commiee on Banking Supervision (1998)) and he Bank for Inernaional Selemens (Fisher-Repor (1994)). Privae secor organizaions, such as he Group of-thiry (1993), which consiss of bankers and oher derivaives marke paricipans, have advocaed he use of Var for seing risk managemen sandards. However, we remark ha Value-a-Risk (VaR) as a risk measure is criicized for no being sub-addiive (see Embrechs (000) for an overview of he criicism). This means ha he risk of a porfolio can be larger han he sum of he sand-alone risks of is componens when measured by VaR (Arzner, Delbaen, Eber, and Heah (1997) and Arzner, Delbaen, Eber, and Heah (1999)). Hence, managing risk by VaR may fail o simulae diversificaion. Sill, he final Basle Capial Accord ha will come in force since 007 focus on VaR only (see he Basle Commiee on Banking Supervision (005)). Therefore, we explore he feaures of VaR in a mulivariae seing. In order o esimae VaR, we can use eiher non-parameric or parameric mehods. The hisorical simulaion mehodology belongs o he former family and, insead of making disribuional assumpions on reurns, i uses realized pas reurns o esimae he given percenile. The parameric mehods make specific disribuional assumpions on reurns and, herefore compues (analyically or by simulaion) he corresponding percenile. Crucial for he VaR is he disribuion funcion of he reurns on marke value. Usually, as allowed by he Basle Commiee, a normal disribuion is assumed for he marke reurn. However i is a known sylized fac ha he normal disribuion underesimaes he probabiliy in he ail and Elecronic copy available a: hp://ssrn.com/absrac=154608

3 hence he VaR (see McNeil, Frey, and Embrechs (005) for a general review). Nowadays, alernaive disribuions and mehods are proposed ha focus more on he ail behavior of he reurns. See, for example, Embrechs, Kluppelberg, and Mikosch (1997), Lucas and Klaassen (1998), Tsay (00) for a discussion. Popular alernaives in he financial lieraure include GARCH-ype models o allow for ime-varying volailiy and he Suden- disribuion, since i allows for more probabiliy mass in he ail han he normal disribuion. For a review on (G)ARCH models we refer o Bollerslev, Engle, and Nelson (1994). While univariae VaR esimaion has been widely invesigaed, he mulivariae case has been deal only by a smaller and more recen lieraure, regarding he forecasing of correlaions beween asses, oo: empirical works which deal wih his issue are hose of Engle and Sheppard (001), Gio and Lauren (003), Bauwens and Lauren (005) and Chong (004). When we use parameric mehods, VaR esimaion for a porfolio of asses can become very difficul due o he complexiy of join mulivariae modelling. Moreover, compuaional problems arise when increasing he number of asses. In order o overcome hese problems, we can resor o Copula heory: copula funcions allow o consruc flexible mulivariae disribuion wih differen margins and differen dependence srucure, wihou he consrains of he radiional join normal disribuion. Hull and Whie (1998) were among he firs o consider his kind of modelling, even hough hey did no refer explicily o copulas, bu raher o mapping observaions from he assumed disribuion of daily changes ino a sandard normal disribuion on a fracileo-fracile basis. More recenly, copulae have been explicily used for measuring porfolio Value a Risk by Bouye, Durrleman, Nikeghbali, Riboule, and Roncalli (001), Embrechs, Lindskog, and McNeil (003) and Cherubini, Vecchiao, and Luciano (004).Glasserman, Heidelberger, and Shahabuddin (00) exended his framework o a porfolio of derivaives by considering a paricular exension of he mulivariae disribuion. Ye, he applicaions made so far deal wih uncondiional disribuions, only. Wha we do in his work is o propose a Mone Carlo mehodology for measuring porfolio Value a Risk by performing simulaions from differen condiional mulivariae disribuions, in order o evaluae wha are he main deerminans when doing VaR forecass for a porfolio of asses. We compare differen disribuion assumpions for he marginals, as well as differen dynamic specificaions for heir momens, o undersand wheher he proper modelling of he laer is more imporan han he ype of disribuion. 3

4 In a similar fashion, we consider differen ype of copulas and differen condiional specificaions for heir parameers. We analyze hree porfolios composed of he Sandard and Poor 500 sock index, he Dax Index and he Nikkei5 index, wih daily daa aking ino consideraion he very volaile period beween January 1h 1994 ill Augus 1h 000. This period of ime winessed he fas raise of world financial markes followed by he he burs of he high-ech bubble. Thus, his sample is perfecly suied o highligh he differences among he considered models. We use he iniial par of he sample o esimae he 95%, 99% VaR, and he remaining par o compare he ou-of-sample performances of he differen models, using various backesing echniques. The res of he paper is organized as follows. In Secion we provide an ouline of mulivariae modelling, focusing he aenion o copula heory and skewed marginals. Secion 3 presens he necessary seps o perform Mone-Carlo simulaion for VaR forecasing wihin a flexible copula framework, while Secion 4 presens he empirical resuls. We conclude in Secion 5. - Condiional mulivariae modelling Copula heory provide an easy way o deal wih (oherwise) complex mulivariae modelling. The essenial idea of he copula approach is ha a join disribuion can be facored ino he marginals and a dependence funcion called a copula. The erm copula comes he Lain language and means link : he copula couples he marginal disribuions ogeher o form a join disribuion. The dependence relaionship is enirely deermined by he copula, while scaling and shape (mean, sandard deviaion, skewness, and kurosis) are enirely deermined by he marginals. Copulas can be useful for combining risks when he marginal disribuions are esimaed individually: marginals are iniially esimaed separaely and can hen be combined in a join densiy by a copula ha preserves he characerisics of he marginals. Copulas can also be used o obain more realisic mulivariae densiies han he radiional join normal. For example, he normal dependence relaion can be preserved using a normal copula, bu marginals can be enirely general (for example, a normal copula wih one T-suden marginal and one logisic marginal). In his case we have he so-called mea join disribuion funcions, whose heoreical background has been sudied recenly by Embrechs, Hoing, and Juri (003) and Fang and Fang (00). 4

5 In he following, we briefly describe he condiional marginal and copula funcions used o consruc mulivariae disribuions..1 Copula modelling In wha follows, he definiion of a copula funcion and some of is basic properies are given, while he reader ineresed in a more deailed reamen is referred o Nelsen (1999) and Joe (1997). As ime series analysis is ofen ineresed in random variables condiioned on some variables, we allow for condiioning variables represened by F 1 (for example, lagged reurns). Besides, we consider he bivariae case since i will be laer used in he empirical analysis. Definiion.1 (Condiional Copula): The condiional copula of (x,y) 1, where x 1 F and y 1 G, is he condiional join disribuion funcion of U F (x 1 ) and V G (y 1 ), given 1. I is easy o observe ha a wo-dimensional condiional copula is he condiional join disribuion of he probabiliy inegral ransforms of each of he variables X and Y wih respec o heir marginal disribuions, F and G. In order o undersand why he univariae margins and he mulivariae dependence srucure can be separaed, we have o consider he following heorem: Theorem.1 (Sklar s heorem for condiional disribuions): Le F be he condiional disribuion of x 1, given he condiioning se 1, G be he condiional disribuion of y 1, and H be he join condiional bivariae disribuion of (x, y 1 ). Assume ha F and G are coninuous in x and y. Then here exiss a unique condiional copula C such ha H (x, y 1 ) = C (F (x 1 ), G (y 1 ) 1 ) (.1) Conversely, if we le F and G be he condiional disribuions of he wo random variables X and Y, and C be a condiional copula, hen he funcion H defined as (.1) is a condiional bivariae disribuion funcion wih condiional marginal disribuions F and G (see Sklar (1959), Paon (006), for a proof ). 5

6 A copula is hus a funcion ha, when applied o univariae marginals, resuls in a proper mulivariae probabiliy densiy funcion (p.d.f.): since his p.d.f. embodies all he informaion abou he random vecor, i conains all he informaion abou he dependence srucure of is componens. Using copulas in his way splis he disribuion of a random vecor ino individual componens (he marginals), wih a dependence srucure among hem given by he copula, wihou losing any informaion. The Sklar s heorem for condiional disribuions implies ha he condiioning variable(s), 1, mus be he same for boh marginal disribuions and he copula: if we do no use he same condiioning variable for F, G and C, he funcion H will no be, in general, a join condiional disribuion funcion. The only case when H will be he join disribuion of (x, y 1 ) = (x, y w 1,w ), is when F (x w 1 ) = F (x w 1, w ) and G (y w ) = G (y w 1, w ), ha is when some variables affec he condiional disribuion of one variable bu no he oher. I is sraighforward o see ha i is possible o exrac he implied condiional copula from any bivariae condiional disribuion. By applying Sklar s heorem and using he relaion beween he disribuion and he densiy funcion, we can derive he bivariae copula densiy c (F (x 1 ), G (y 1 ) 1 ), associaed o a copula funcion C (F (x 1 ), G (y 1 ) 1 ): [ C( F( x F 1), G( y F 1) F 1)] F( x F 1) G( y F 1) h( x, y F 1 ) F( x F ), G ( y F ) x y 1 1 c ( F( x F ), G ( y F ) F ) f ( x F ) g ( y F ) h ( x, y F ) c( u, v F 1 ) = f x g y 1 ( F 1 ) ( F 1 ) (.) where u F (x 1 ) and v G (y 1 ). By using his procedure, we can derive he Normal and he T-copula: 1 The copula of he bivariae Normal disribuion is he Normalcopula, whose probabiliy densiy funcion in he bivariae case is he following 6

7 1 cu (, v; F 1 ) [( ( u)) ( ( v)) - ( u) ( v)] exp exp [( ( u)) ( ( v)) ] (.3) where is he condiional linear correlaion, given he condiioning se 1, and -1 is he inverse of he sandard univariae Normal disribuion. On he oher hand, he copula of he bivariae Suden s - disribuion is he Suden s T-copula, whose densiy funcion is cu (, v;, F ) [( ( u )) ( ( v )) - ( u ) ( v )] exp (1 ) ( ( u )) ( ( v )) exp (.4) where is he condiional linear correlaion, are he condiional degrees of freedom, while 1 denoes he inverse of he Suden s cumulaived isribuion funcion. Boh hese copulae belong o he class of Ellipical copulae 1. An alernaive o Ellipical copulae is given by Archimedean copulae: however, hey presen he serious limiaion o model only posiive dependence (or only parial negaive dependence), while heir mulivariae exension involve sric resricions on bivariae dependence parameers. This is why we do no consider hem here. 1 See Cherubini, Vecchiao, and Luciano (004) for more deails. 7

8 . Marginal modelling The marginal disribuions ha we used o build a join mulivariae disribuion are he following four: 1. Normal;. Skew-Normal; 3. T-suden; 4. Skew-T suden. Two well known deviaions from normaliy are fa ails and asymmery. One disribuion ha is used o allow for excess kurosis is he Sudens-T and i has been generalized oallow for skewness by Hansen (1994). Despie oher generalizaions have been proposed, we chose his one due o is simpliciy and is pas success in modelling economic variables (Paon (004), Paon (006), Jondeau and Rockinger (003)). Definiion. (Skew-T disribuion): Le y be a random variable which follows a condiional Skewed-T disribuion wih densiy funcion f(, ) and mean zero and variance one by consrucion, in order o be a suiable model for he sandardized residuals of a condiional mean and variance model. The condiional parameers, conrol he kurosis and skewness of he variable, respecively, while he densiy funcion is repored below (for more deails, see Hansen (1994)): 1 1 by a a bc 1 for y 1 b Skewed-T f( y; v, F 1 ) 1 1 by a a bc 1 for y 1 b where 1 c ( ) b 13 a a 4 c 1 8

9 This densiy is defined for < < and 1 < λ <+1. Moreover, his densiy encompasses a large se of convenional densiies: 1. if λ =0, he Skew- reduces o he radiional Suden-T disribuion.. If we have he skew-normal densiy. 3. If λ = 0 and we have he normal densiy. Similarly o Suden s T, given he resricion >, his disribuion is well defined and is second momen exiss, while skewness exiss if > 3 and kurosis is defined if > 4. The parameer λ conrols for skewness: If i is bigger han zero, we have posiive skewness, while if i is smaller han zero he disribuion is negaive skewed. Fig.1 displays differen densiies obained for various values of degrees of freedom and skewness parameer λ. The cumulaive disribuion funcion (c.d.f.), he inverse-c.d.f. and relaive proofs are repored in Appendix A. Figure 1: Densiy of Hansen s Skewed-T disribuion 9

10 .3 Esimaion: he inference for margins mehod (IFM) As we have seen, he join densiy funcion in he bivariae condiional case is: h ( x, y F ; ) f ( x F ; ) g ( y F ; ) c ( u, v F ; ) (.5) 1 h 1 f 1 g 1 c where u F (x 1 ; f ) and v G (y 1 ; g ), and θ h, f, g, c are he join densiy, marginals and copula parameers vecors, respecively, wih θ h, [ f, g, c ]. Maximum likelihood analysis implies, Lxy(θ h ) = Lx( f ) + Ly( g ) + Lc( f, g, c ), where Lxy( h ) log h (x, y 1 ; h ), Lx( f ) log f (x 1 ; f ), Ly( g ) log g (y 1 ; θg), and Lc( f, g, c ) log c(u, v 1 ; c ). I will no always be he case ha he parameer vecor of he join disribuion θ h decomposes so nealy ino n+1 componens, associaed wih he n margins (in his case n = ) and he copula. Examples of common models where such a decomposiion is possible are vecor auoregressions and mulivariae auoregressive condiional heeroscedasiciy (ARCH) models: in hese cases, he n margins parameer vecors are variaion free w.r. each oher, and can be esimaed separaely. Common models where he previous decomposiion is no possible are mulivariae ARMA models and mulivariae generalized ARCH models. When his decomposiion fail o hold, all he n marginal models mus be esimaed joinly. According o he IFM mehod, he parameers of he marginal disribuions are esimaed separaely from he parameers of he copula. In oher words, he esimaion process is divided ino he following wo seps: 1.Esimaing he parameers f and g of he marginal disribuion F and G using he ML mehod: ˆ arg max L( )arg max log f ( x ; ) f f f 1 ˆ arg max L( )arg max log g ( y ; ) (.6) g g g 1 T T if he parameers vecors are variaion free; oherwise, 10

11 ˆ, ˆ argmax[ L( ) L( )] argmax [log f ( x ; ) log g ( x ; )] (.7) f g f g f g 1 T.Esimaing he copula parameers θ c, given sep 1): T ˆ arg max[ L( )] arg max log[ c ( F ( x ; ˆ ), G ( x ; ˆ ); )] (.8) c c f g c 1 Like he ML esimaor i verifies he properies of asympoic normaliy, bu he covariance marix mus be modified (Joe and Xu (1996), Joe (1997), Paon (006)): T( ˆ ) N(0, V ( )) h where V(θ 0 ) = D 1 M(D 1 ) T is he Godambe Informaion Marix, while D = E[ g(θ) T / θ], M = E [g(θ) T g(θ)], and g(θ) = ( L x / θ f, L y / θ g, Lc/ θ c ) is he score funcion..3.1 In-sample resuls: he daase We consider hree imporan risk facors, such as he Sandard and Poor 500 index, he Dax Index and he Nikkei5 Index, wih daily daa aking ino consideraion he very volaile period beween January 1h 1994 ill Augus 1h 000. This period of ime winessed he fas raise of world financial markes followed by he he burs of he high-ech bubble. The descripive saisics of he (raw) log-reurns are presened in Table 1. The hree indexes generally exhibied negaive skewness (he Nikkei slighly posiive, insead) and excess kurosis, while he Jarque-Bera saisic indicaes ha neiher series is uncondiionally normal. Besides, he means and volailiies are very similar, as expeced. Table 1: SP500, DAX, NIKKEI5 Log-Reurns Descripive saisics. 11

12 .3. In sample resuls: marginals models We sared modelling each marginal ime series by a general AR(1)- Threshold GARCH(1,1) model for he coninuously compounded reurns y = 100[log(P ) log(p -1 )], given by: y y 1 1 iid (.9) h, f(0.1) (.10) h D h (.11) where D 1 = 1 if ε 1 < 0, and 0 oherwise. Good news ε 1 > 0 and bad news ε 1 < 0, have differen effecs on he condiional variance in his model: good news has an impac of α, while bad news has an impac of α + γ. If γ > 0 we say ha he leverage effec exiss, while if γ0 he news impac is asymmeric. We ried an EGARCH specificaion Nelson (1991), oo: however, since he diagnosic ess and he Schwarz crierion were similar o ha of he TGARCH, bu he numerical maximizaion of he log-likelihood funcion resuled much more difficul when working wih no normal disribuions, we resor o he TGARCH specificaion (his evidence confirms similar resuls by Angelidis, Benos, and Degiannakis (003)). We esimae he AR(1)-TGARCH(1,1) model assuming four differen densiy funcions f(0, 1) for η : he Normal, he Skew-Normal, he Suden s-t and he Skew-T as presened in secion.. When working wih he laer hree disribuions, we have o specify a dynamic model for he condiional skewness parameer and/or he condiional degrees of freedom, as well. We propose here a specificaion similar o Hansen (1994) and Rockinger and Jondeau (001): ( 1) (.1) ( ) (.13) 1 For he SP500 we used an AR(3)-Threshold GARCH(1,1) model, insead, since Likelihood Raio ess and diagnosic ess indicae his as he bes model. 1

13 where ( ) is a modified logisic ransformaion designed o keep he condiional skewness parameer λ in (-1,1) a all imes, while ( ) is a logisic ransformaion designed o keep he condiional degrees of freedom in (,30) a all imes (see Hansen 1994). We avoid an auoregressive specificaion, in so far as i may lead o spuriously significan parameers (see Rockinger and Jondeau (001), for a proof). Moreover, we ried differen specificaions, bu wih similar resuls and increased compuaional ime. This is why we resor o his simple modelling. The models esimaed using he enire daase available are repored below. Table : AR(1)-TGARCH(1,1) models wih differen disribuions assumpions. (*) The - parameer correspond o an AR(3) erm The bes specificaion is given by he Skew-T disribuion, as expeced, even hough all four disribuions passed Box-Pierce ess on he sandardized residuals (no repored). The esimaed parameers are raher similar across differen disribuions: however, Table shows ha when skewness and kurosis are no aken ino accoun, boh he parameers of he mean and of he variance equaions appear o be slighly overesimaed; his happens for he Normal disribuion, oo. These resuls confirm Newey and Seigerwald (1997), who showed ha Quasi-ML esimaors can give no consisen esimaes under cerain condiions. 13

14 The leverage effec is saisically significan in all he considered series, as well as he condiional skewness and degrees of freedom. However, he skewness parameers are no significan for he Nikkei55, showing ha he skewness for his series is no saisically differen from zero, as we ve previously seen in he descripive saisics in Table In-sample resuls: copula models Afer having esimaed he parameers of he marginal disribuions {F,G } in he firs sage, we proceeded o esimae he copula parameers in he second sage, as previously explained in paragraph.3. Since we ll consider hree porfolios for VaR evaluaion, composed of he SP500 and he Dax, he SP500 and he NIkkei5, and he Nikkei5 and he Dax, respecively, we fied he Normal (.3) and T-copula (.4) o hese asse pairs, by using he cumulaive disribuion funcion of he sandardized residuals ˆ esimaed from he marginal models: x x X X x y, CNormalcopula F ( ), G ( ); F 1 x y h h x x X X x y, CT copula F ( ), G ( );, F 1 (.14) x y h h where [ρ, υ ] are he condiional correlaion and condiional degrees of freedom, respecively, [μ, h ] he condiional means and variances, while {F,G } can be Normal / Skew-Normal / Suden s T / Skew T. When we consider a dynamic specificaion for a copula, i is imporan o remember ha he Sklar s heorem for condiional disribuions implies ha he condiioning variable(s) mus be he same for boh marginal disribuions and he copula. Following Paon (006) and Embrechs and Dias (003), we sugges his general evoluion equaion as saring poin for our dynamic copula specificaions: ( u 1 v 1 ) (.15) ( u v ) (.16)

15 where (x) (1 e x )/(1+e x ) is he modified logisic ransformaion, designed o keep he condiional correlaion ρ in ( 1,1) a all imes, while ( ) is a logisic ransformaion designed o keep he condiional degrees of freedom in (, 100) a all imes. The models esimaed using he enire daase available are presened in Table 3, while he condiional correlaion for he hree asse pairs, as well as he condiional degrees of freedom for he T-copula are repored in Figures -3 (for sake of space, we repor he case of Normal and Skew-T marginals, only). Table 3: Differen copula models wih differen marginals Figure : Condiional. Correlaion T-copula: SP500-DAX, SP500-NIKKEI, NIKKEI-DAX. 15

16 Figure 3: Condiional. D.o.F. T-copula: SP500-DAX, SP500-NIKKEI, NIKKEI-DAX. As Table 3 and figures -3 clearly show, he simple normal copula wih consan correlaion is he bes soluion, since he dynamic parameers are no significan and he degrees of freedom of he T-copula are always above 0: ha is he T-copula and he Normal copula are no more disinguishable. 3 - Value a risk applicaions 3.1 Inroducion Value a Risk (or VaR) is a concep developed in he field of risk managemen ha is defined as he maximum amoun of money ha one could expec o lose wih a given probabiliy over a specific period of ime. While VaR is widely used, i is, noneheless, a conroversial concep, primarily due o he diverse mehods used in obaining VaR, he widely divergen values so obained and he fear ha managemen will rely oo heavily on VaR wih lile regard for oher kinds of risks. The VaR concep embodies hree facors: 1. A given ime horizon. A risk manager migh be concerned abou possible losses over one day, one week, ec.. VaR is associaed wih a probabiliy. The saed VaR represens he possible loss over a given period of ime wih a given probabiliy. 3. The acual amoun of money invesed. If we call V(l) he change in he value of he asses in he financial posiion from o +l and F l (x) he cumulaive disribuion funcion ( cdf ) of V(l), he VaR is formally defined as follows: Definiion 3.1 (Value a Risk) We define he VaR of a long posiion over ime horizon l wih probabiliy p as he real VaR (p, l) such ha p = Pr[V(l) VaR (p, l)] = F l (VaR (p, l)). 16

17 where VaR is defined as a negaive value (loss). If he cdf is known, hen VaR is simply is p-h quanile imes he value of he financial posiion: however, he cdf is no known in pracice and mus be esimaed. We remark ha for any univariae cdf F l (x) and probabiliy p, he quaniy x p = inf{x F l (x) p} is called he p-h quanile of F l (x), i.e. he smalles real number x saisfying F l (x) p. Besides, he definiion of VaR for a long posiion coninuous o apply o a shor posiion if one uses he disribuion of V (l). Therefore is suffices o discuss mehods of VaR calculaions using a long posiion (for more deails see Jorion (000), Tsay (00)). 3. VaR esimaion: problem seup Our goal is o esimae he daily Value a Risk for a porfolio of wo asses (or risk facors, in a general framework), by using he ime series of daily reurns for each risk facor. A his sage, we have o find he mos appropriae join disribuion funcion which describes hese daa: This is done by choosing specific marginals for he single risk reurns and a copula o link hem ogeher ino a join disribuion funcion. Le x and y denoe he asses log-reurns a ime and be β (0, 1) he allocaion weigh, so ha he porfolio reurn is given by z = β x +(1 β)y. The condiional join disribuion funcion esimaed a ime 1 is given as in (.1): H (x, y 1 ) = C (F (x 1 ), G (y 1 ) 1 ), (3.1) while he relaive densiy funcion is given in (.). Hence, he cumulaive disribuion funcion for he porfolio reurn Z is given by: ( z) Pr( Z z ) Pr( X (1 ) Y z ) 1 1 z y c( F( x F 1), G( y F 1) F 1) f( x F 1) dxg( y F 1) dy (3.) 17

18 The one-sep-ahead VaR compued in 1 for he porfolio a a confidence level p is he soluion z * of he equaion ζ(z * )=p, imes he value of he financial posiion a -1. Apar for he mulivariae normal and mulivariae T case, when using copulas wih differen marginals he compuaion of he one-sep-ahead VaR is no sraighforward, since here are no analyic and easy-o-use formulae o swich from he condiional means and volailiies o he long and shor VaR of he porfolio. However, he VaR can be compued using a simple Mone Carlo simulaion as widely used in quaniaive finance and VaR applicaions, see Jorion (000), Gio and Lauren (003) and Bauwens and Lauren (005). Besides, Gio and Lauren (003) show ha a choice of 100,000 simulaions provides accurae esimaes of he quanile. Following his soluion, we generae a large number of one-day-ahead reurns {x, y } for he wo asses, by simulaing 100,000 random reurns wih he condiional disribuion funcion (3.1) and we revaluae he porfolio a ime. We hen deermine he Value a Risk a a given confidence level p, by simply aking he empirical quanile a p of he simulaed porfolio profi and loss disribuion. The deailed seps of he procedure for esimaing he 95%, 99% VaR over a one-day holding period are he following: 1. Le consider he porfolio z which conains one posiion for each of he wo risk facors (or asses), whose value a ime 1 is: P z, 1 = P x, 1 + P y, 1 where P x, 1, P y, 1 are he marke prices of he wo asses a ime 1.. We simulae j = 100,000 MC scenarios for each asse logreurns, {x j,, y j, }, over he ime horizon [ 1,], using he condiional join disribuional funcion (3.1). (a) Firs, we have o simulae a j random variae (u j,x, u j,y ) from he copula C ( ). See Cherubini, Vecchiao, and Luciano (004), for a discussion abou copula simulaion. i. If we are using a -dimensional condiional Normal copula (.3), we have o run he following algorihm: 18

19 Find he Cholesky decomposiion A of he correlaion marix, i.e. [1 ˆ, ˆ 1], where ˆ is he forecased condiional correlaion, esimaed wih he dynamic model described in secion.3.3; Simulae independen sandard normal random variaes z j = (z j,1, z j, ) ; Se he vecor b j = Az j ; Deermine he componens (u j,x, u j,y ) = ((b j,1 ), (b j, )), where ( ) is he sandard normal c.d.f. The vecor (u j,x, u j,y ) is a j random variae from he -dimensional Normal copula C Normal ( ; ˆ 1 ) ii. If we are using a -dimensional condiional T-copula (.4), we have o run he following algorihm: Find he Cholesky decomposiion A of he correlaion marix, i.e. [1 ˆ, ˆ 1], where ˆ is he forecased condiional correlaion, esimaed wih he dynamic model described in secion.3.3; Simulae independen sandard normal random variaes z j = (z j,1, z j, ) ; Simulae a random variae, s j, from a ˆ disribuion, independen of z j, where ˆ is he forecased degrees of freedom parameer, esimaed wih he dynamic model described in secion.3.3; Deermine he vecor b j = Az j ; ˆ Se c b j s j j Deermine he componens (u j,x, u j,y ) = ( ˆ (c j,1 ), ˆ (c j, )), where ˆ ( ) is he Suden s T c.d.f. wih degrees of freedom equal o υ. The vecor (u j,x, u j,y ) is a j random variae from he -dimensional T-copula C T cop ( ; ˆ, ˆ 1 ). (b) Second, we ge he sandardized asse log-reurns by using he inverse funcions of he esimaed marginals, which can be Normal / Skew-Normal / T-Suden / Skew-T, as described in secion. and Appendix A: 19

20 ( q, q )' ( Fˆ ( u ); Gˆ ( u )) 1 1 Qj jx, jx, jx, jx, (c) Third, we rescale he sandardized asses log-reurns by using he forecased means and variances, esimaed wih AR-GARCH models as described in secion.3.: { x, y } ˆ q hˆ, ˆ q hˆ j, j, x, j, x x, y, j, y y, (d) Finally, we repea his procedure for j = 100, 000 imes. 3. By using hese 100,000 scenarios, he porfolio is being revaluaed a ime, ha is: P z, j = P x, 1 exp(x j, ) + P y, 1 exp(y j, ), j = 1,, Porfolio Losses in each scenario j are hen compued: Loss j = P z, j P z,-1, j = 1,..., 100, The calculus of 95%, 99% VaR is very simple: a) We order he 100,000 Loss j in increasing order; b) 99% VaR is he 1000 h ordered scenario; c) 95% VaR is he 5000 h ordered scenario. As an example, Figure 4 shows a simulaed profi and loss disribuion for a given porfolio. 0

21 Figure 4: Profi & Loss disribuion for a given porfolio z. We wan o remark ha he radiional join normal disribuion is a special case nesed in he more general copula-marginals framework: looking a equaion (.), we can easily see ha a random vecor (X,Y) is mulivariae normal if he univariae margins F, G are Normal, and he dependence srucure among he margins is described by he Normal copula (.3). 4 - VaR evaluaion and resuls The goal of his work is o evaluae wha are he main deerminans when doing VaR forecass for a porfolio of asses. We saw in Secion ha four elemens have o be considered in a condiional mulivariae analysis: 1. The choice of he marginals disribuion;. The specificaion of he condiional momens of he marginals, ranging from he mean ill he kurosis; 3. The choice of he copula; 4. The specificaion of he condiional copula parameers The in-sample resuls showed ha an AR(1)-TGARCH(1,1) model wih a dynamic Skew-T disribuion resuled o be he bes choice for marginals specificaions, while a simple normal copula wih consan correlaion was he bes opion for he bivariae asse pairs. 1

22 In order o evaluae how imporan are he cied four elemens, we generae ou-of-sample VaR forecass for he considered hree porfolios (SP500-DAX, SP500-NIKKEI5, NIKKEI5-DAX), by using hese differen condiional mulivariae disribuions: Dynamic Normal copula + Normal marginals, wih he following specificaions: 1. Consan mean and Consan Variance;. AR(1) specificaion for he mean and Consan Variance; 3. AR(1) specificaion for he mean and GARCH(1,1) for he Variance; 4. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance; Dynamic Normal copula + Skew-Normal marginals, wih he following specificaions: 1. Consan mean, Consan Variance, Consan Skewness parameer;. AR(1) specificaion for he mean, Consan Variance, and Consan Skewness parameer; 3. AR(1) specificaion for he mean and GARCH(1,1) for he Variance, and Consan Skewness parameer; 4. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, and Consan Skewness parameer; 5. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, and Dynamic Skewness parameer given by (.1) Dynamic Normal copula + Suden s T marginals, wih he following specificaions: 1. Consan mean, Consan Variance and Consan Degrees of Freedom;. AR(1) specificaion for he mean, Consan Variance, and and Consan Degrees of Freedom; 3. AR(1) specificaion for he mean and GARCH(1,1) for he Variance, and Consan Degrees of Freedom; 4. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, and Consan Degrees of Freedom; 5. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, and Dynamic Degrees of Freedom given by (.13);

23 Dynamic Normal copula + Skew-T marginals, wih he following specificaions: 1. Consan mean, Consan Variance, Consan Skewness parameer, and Consan Degrees of Freedom;. AR(1) specificaion for he mean, Consan Variance, Consan Skewness parameer, and Consan Degrees of Freedom; 3. AR(1) specificaion for he mean and GARCH(1,1) for he Variance, Consan Skewness parameer, and Consan Degrees of Freedom; 4. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, Consan Skewness parameer, and Consan Degrees of Freedom; 5. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, Dynamic Skewness parameer given by (.1), and Consan Degrees of Freedom; 6. AR(1) specificaion for he mean and T-GARCH(1,1) for he Variance, Dynamic Skewness parameer given by (.1), and Dynamic Degrees of Freedom given by (.13); Consan Normal copula + Normal / Skew-Normal / Suden s T /Skew-T marginals, wih he previous specificaions; Dynamic T-copula + Normal / Skew-Normal / Suden s T /Skew- T marginals, wih he previous specificaions; Consan T-copula + Normal / Skew-Normal / Suden s T /Skew- T marginals, wih he previous specificaions. We generae porfolio Value a Risk forecass a he 95 % - 99 % confidence levels, following he procedure described in secion 3.. The prediced one-sep-ahead VaR forecass are hen compared wih he observed porfolio losses and boh resuls are recorded for laer assessmen using a back-esing procedure. The firs esimaion sample is given by he firs 700 observaion: a he j-h ieraion where j goes from 700 o 1699 (for a oal of 1000 observaions), he esimaion sample is augmened o include one more observaion, and his procedure is ieraed unil all days have been included in he esimaion sample. We finally assess he performance of he compeing models over hese 1000 observaions using he following back-esing echniques: Kupiec s uncondiional coverage es; 3

24 Chrisoffersen s condiional coverage es; Loss funcions o evaluae VaR forecass accuracy. 4.1 Uncondiional model evaluaion: Kupiec s es This es is based on binomial heory and ess he difference beween he observed and expeced number of VaR exceedances of he effecive porfolio losses. Since VaR is based on a confidence level p, when we observe N losses in excess of VaR ou of T observaions, hence we observe N/T proporion of excessive losses: he Kupiec s es answers he quesion wheher N/T is saisically significanly differen from p. Following binomial heory, he probabiliy of observing N failures ou of T observaions is (1 p) T N p N, so ha he es of he null hypohesis ha he expeced excepion frequency N/T = p is given by a likelihood raio es saisic: LR UC = ln[(1 p) T N p N ] + ln[(1 N/T) T N (N/T) N ] (4.1) which is disribued as χ (1) under he H 0. This es can rejec a model for boh high and low failures bu, as saed in Kupiec (1995), is power is generally poor. 4. Condiional model evaluaion: Chrisoffersen s es A more complee es was made by Chrisoffersen (1998), who developed a likelihood raio saisic o es he join assumpion of uncondiional coverage and independence of failures. Is main advanage over he previous saisic is ha i akes accoun of any condiionaliy in our forecas: for example, if volailiies are low in some period and high in ohers, he VaR forecas should respond o his clusering even. The Chrisoffersen procedure enables us o separae clusering effecs from disribuional assumpion effecs. His saisic is compued as: LR CC = ln[(1 p) T N p N ] + ln[(1 π 01 ) n00 π n01 01 (1 π 11 ) n10 π n11 11 ] (4.) where n ij is he number of observaions wih value i followed by j for i, j = 0,1 and 4

25 π ij = n ij / ( j n ij ) (4.3) are he corresponding probabiliies. Under he H 0, his es is disribued as a χ (). If he sequence of N losses is independen, hen he probabiliies o observe or no observe a VaR violaion in he nex period mus be equal, which can be wrien more formally as π 01 = π 11 = p. The main advanage of his es is ha i can rejec a VaR model ha generaes eiher oo many or oo few clusered violaions, alhough i needs several hundred observaions in order o be accurae. 4.3 Measures of accuracy: loss funcions The previous ess do no show any power in disinguishing among differen, bu close, alernaives. Moreover, as noed by he Basle Commiee on Banking Supervision (1998), he magniude as well as he number of excepions are a maer of regulaory concern. This concern can be readily incorporaed ino a so called loss funcion by inroducing a magniude erm. This was firs accomplished by Lopez (1998). The general form of hese loss funcions is: C 1 f ( L, VaR ) if L VaR g ( L, VaR ) if L VaR where f(x, y) and g(x, y) are arbirary funcions such ha f(x, y) g(x, y) for a given y, while L is he porfolio loss. Under very general condiions, accurae VaR esimaes will generae he lowes possibile score. Many loss funcions can be consruced. Lopez (1998) has proposed he following quadraic loss funcion: C 1 1 ( L - VaR ) if L VaR 0 if L VaR (4.4) Thus, as before, a score of one is imposed when an excepion occurs, bu now, an addiional erm based on is magniude is included. The numerical score increases wih he magniudo of he excepion and can provide addiional informaion on how he underlying VaR model forecass he lower ail of he underlying L +1 disribuion. Blanco and Ihle (1999) sugges an alernaive way o deal wih he size of excepions by focusing on he average size of he excepions: 5

26 C 1 L - VaR VaR if if L VaR 1 1 L VaR 1 1 (4.5) A sraegy ha can be implemened wih he previous approaches is he following: 1. Apply Kupiec s and Chrisoffersen ess a a firs sage o choose he bes models;. Then use he loss funcions o compare he coss of differen admissible choices. 4.3 VaR ou-of-sample resuls Tables 4-6 repor he real VaR exceedances N/T, he p-value p UC of Kupiec s Uncondiional Coverage es, and he p-value p CC of Chrisoffersen s Condiional Coverage es, for he he VaR forecass a 95% - 99% confidence levels, relaive o he hree differen considered porfolios (SP500-DAX, SP500-NIKKEI5, NIKKEI5-DAX). Table 4: VaR resuls: SP500-DAX porfolio. 6

27 Table 5: VaR resuls: SP500-NIKKEI porfolio. Table 6: VaR resuls: NIKKEI-DAX porfolio 7

28 The wo loss funcions discussed in Secion 4.3 are presened in Appendix B, Tables 10-1, insead. The resuls using he Normal copula wih consan correlaion, as well as he T-copula wih dynamic and consan parameers, delivered almos idenical resuls, hus confirming he in-sample resuls: a simple normal copula wih consan correlaion is sufficien o describe he dependence srucure of he considered risk-facors. For sake of space, we repor only he VaR resuls relaive o he AR(1)-TGARCH(1,1) specificaion wih a dynamic Skew-T disribuion. The complee se of resuls is available from he auhor. Table 7: VaR resuls: SP500-DAX porfolio (Dynamic Skew-T + oher copulas) Table 8: VaR resuls: SP500-NIKKEI porfolio (Dynamic Skew-T + oher copulas) Table 9: VaR resuls: NIKKEI-DAX porfolio (Dynamic Skew-T + oher copulas) The previous ables highlighs some ineresing resuls: 1. The GARCH specificaion for he variance is absoluely fundamenal o have good VaR forecass, whaever he marginal disribuion is; 8

29 . The asymmeric GARCH specificaion is imporan o ge precise VaR esimaes a he 95% confidence level. However, i can produce conservaive esimaes a he 99% level when dealing wih srongly lepokuric asses and he normal (or skew-normal) disribuion is used; 3. The AR specificaion of he mean is no relevan in all cases; 4. The GARCH specificaion seems o model mos of he lepokurosis presen in he daa. However, when he asses are srongly lepokuric, a Skew-T disribuion is he bes choice (similarly o poin.); 5. Using a Suden s T disribuion wih srongly skewed asses can produce very aggressive VaR forecass a he 95% level; 6. The Skew-T and Skew-Normal disribuions presen he mos precise VaR forecass, according o he ess and Loss funcions used; 7. Allowing for dynamics in he skewness and degrees of freedom parameers produces more conservaive VaR forecass in almos all cases; 8. The ype of copula as well as he dynamics in is parameers are no relevan: a simple normal copula wih consan correlaion resuled o be sufficien in all cases. These resuls seem o poin ou ha a Skew-T disribuion wih a T- GARCH(1,1) specificaion and consan skewness and degrees of freedom parameers for he marginals, ogeher wih a consan normal copula, should be a good compromise for precise VaR esimaes when dealing wih a porfolio of differen asses. This evidence is consisen wih oher resuls repored in Marshal and Zeevi (00) and Chen, Fan, and Paon (004) ha found he normal copula o be a plausible choice when dealing wih bivariae porfolios, while his is no he case for larger collecions of financial asses. Paricularly, Chen, Fan, and Paon (004) poin ou ha he Suden s copula is a beer approximaion o he rue copula when porfolio dimensionaliy increases, bu he possibiliy of misspecificaion sill exiss. 9

30 5 Conclusions We have described in his work a model for esimaing porfolio VaR by Mone Carlo simulaion and condiional mulivariae disribuions. Tradiional risk measuremen models assume ha he risk facors disribuion is mulivariae normal: unforunaely empirical evidence shows ha he rue marginal disribuions of financial asses usually presen fa ails and srong skewness. We herefore inroduced a model o generae scenarios for porfolio risk facors from differen condiional mulivariae disribuions, o find ou wha are he main deerminans when doing VaR forecass for a porfolio of asses: we used dynamic Gaussian or T-copulas o model asses dependence srucure, ogeher wih differen dynamic marginal specificaions. We esimaed he 95 %, 99% VaR for hree differen porfolios composed of he SP500 and he DAX, he SP500 and he NIKKEI5, and he NIKKEI5 and he DAX, respecively, using 100,000MC scenarios. We hen compared he accuracy of he differen VaR esimaes wih a backesing procedure over a ime window of 1000 daily observaions: we apply Kupiec s and Chrisoffersen s ess in he firs sage, and loss funcions in he second sage o compare he coss of differen admissible choices. The empirical analysis showed ha correc marginals specificaion is absoluely crucial for VaR forecasing, while copula specificaion plays a minor role. Paricularly, a T-GARCH(1,1) specificaion ogeher wih a Skew- T or Skew-normal disribuion gave he bes resuls. The use of symmeric disribuions, such as he Suden s T, wih skewed asses, produced aggressive VaR esimaes a he 95 % confidence level, insead. The analysis showed ha he choice of copula was no relevan for correc VaR esimaes, and a simple consan normal copula was sufficien in all cases. This evidence confirms previous resuls for bivariae porfolios bu wih a differen approach, since we used a VaR backesing procedure insead of goodness-of-fi ess for copulas as in Chen, Fan, and Paon (004). Possible exensions ha can be considered for fuure research include he passage from condiional bivariae o condiional mulivariae copula, following he suggesion of Engle (004) who recenly proposed he Rank- DCC model. Secondly, a soluion o he rade-off beween parsimonious modelling and unbiased esimaes may be given by a copula facor model. The direc consideraion of ransacion coss can surely help in deecing which facors are useful and which are no. Thirdly, explore he feaures of VaR wih porfolios composed of differen financial asses, such as bonds or exchange raes. 30

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33 McNeil, A., R. Frey, and P. Embrechs, 005. Quaniaive Risk Managemen: Conceps, Techniques and Tools. (Princeon Universiy Press). Nelsen, R., An Inroducion o Copulas. Lecure Noes in Saisics 139, (Springer, N.Y.). Nelson, D., Condiional Heeroskedasiciy in Asse Reurns: A New Approach, Journal of Economerics, 43, Newey, W., and D. Seigerwald, Asympoic Bias for Quasi Maximum Likelihood Esimaors in Condiional Heeroskedasiciy Models, Economerica, 3, Paon, A., 004. On he Ou-of-Sample Imporance of Skewness and Asymmeric Dependence for Asse Allocaion, Journal of Financial Economerics, (1), Paon, A., 006. Modelling Asymmeric Exchange Rae Dependence, Inernaional Economic Review, 47(). Rockinger, M., and E. Jondeau, 001. Condiional dependency of financial series: an applicaion of copulas. Discussion Paper NER 8, Banque de France. Sklar, A., Foncions de repariion `a n dimensions e leurs marges. Publ. Ins. Sais. Univ. Paris, 8, Tsay, R. S., 00. Analysis of Financial Time Series. (Wiley, New York). Appendices A. Skew-T Cdf and Inverse Cdf A.1 CDF Proposiion 3: Le X be a sandard Suden s T random variable, wih mean zero and variance (/(-)), and define he c.d.f. of X as F(; ). The c.d.f. of a Skewed- random variable y wih mean zero and variance 1, in erms of F(;) is he following: by a a (1 ) F ;, for y 1 b a Gy ( ;, ) (1 )/ fory b by a a (1 ) F ;, for y 1 b 33

34 where a, b, c where defined before for he densiy. Proof: a) Consider he case where y < a/b. Given he skewed- densiy we have: ( 1)/ 1 by a G( y;, ) bc 1 dy 1 We now change variable, subsiuing u = (by + a)/(1 ) and he relaive Jacobian: ( bya)/(1 ) ( 1)/ (( 1) / ) 1 u 1 du ( /) ( ) We change he variable again, subsiuing z u and is Jacobian. By recalling ha he sandard Suden s T Cdf is: ( 1)/ (( 1) / ) 1 x F () 1 dx ( /) We finally ge he Cdf: by a Gy ( ;, ) (1 ) F ; 1 b) When y = a/b (1 ) Gy ( ;, ) c) When y > a/b ( 1)/ 1 by a Gy ( ;, ) G( ab / ;, ) bc 1 dy 1 a/ b 34

35 By using analogous compuaions we have... (1 ) by a (1 ) Gy ( ;, ) (1 ) F ; 1 by a (1 ) F ; 1 A. Inverse CDF We express he inverse c.d.f. of he Skewed- disribuion in erms of he inverse disribuion of a Suden s T random variable, denoed as F 1 (u; ). If we indicae wih u = G(y;, ) an uniform variae wih suppor [0; 1], and we use he previous CDF, we ge 1 1 u 1 (1 ) F ; a, for 0 u b 1 1 G ( y;, ) 1 1 u 1 (1 ) F ; a, for u 1 b 1 The inverse Cdf can be used o generae random draws from he skewed disribuion wih he following algorihm: 1. Simulae n uniform draws from he Uniform(0, 1) disribuion;. Then use he previous formula o compue y = G 1 (u ;, ) 35

36 B. Loss funcions ables Table 10: Loss funcions: SP500-DAX porfolio. Table 11: Loss funcions: SP500-NIKKEI porfolio. 36

37 Table 1: Loss funcions: NIKKEI-DAX porfolio 37