Which Archimedean is he righ one? CPA Mario R. Melchiori Universidad Nacional del Lioral Sana Fe - Argenina Third Version Sepember 2003 Published in he YieldCurve.com e-journal (www.yieldcurve.com), Ocober 2003 BICA Coop. E.M.Lda. 25 de Mayo 774 Sano omé SANTA FE- Argenina - E-mail: mrmelchi@grupobica.com.ar The opinions expressed in his paper are hose of he auhor and do no necessarily reflec views shared by BICA Coop. E.M.Lda. or is saff.
CONTENTS ABSTRACT... 2 INTRODUCTION... 2 A NORMAL WORLD... 2 RISK MANAGEMENT APPLICATION... 4 ARCHIMEDEAN COPULAS... 4 SELECTING THE RIGHT MARGINAL DISTRIBUTION... 5 DEPENDENCE. KENDALL TAU... 8 ARCHIMEDEAN BIVARIATE COPULA... 8 Algorihm:... 9 VBA code ha Generaes random variaes from he 2-dimensional Gumbel copula... 9 VBA Code ha compues Inverse Cumulaive Disribuion Funcion using numerical roofinding... 0 WHICH IS THE ARCHIMEDEAN COPULA RIGHT ONE?... VBA Code ha compues Kendall using Tau nonparameric esimaion... 3 VBA Code ha compues he pseudo-observaions T {number of ( X X ) X } 2j X2 i < / ( n ) for i,2,..., n. = < such ha j < i and i j i =... 3 X X NUMERICAL EXAMPLE... 3 The Pricing Firs-a-defaul Algorihm:... 4 CONCLUSIONS... 5 REFERENCES... 6 APPENDIX A OTHERS COPULA S PARAMETERS... 7 Clayon s Parameers... 7 Frank s Parameers... 7 APPENDIX B KENDALL τ REVISITED... 7 APPENDIX C - USING SIMTOOLS FEATURES IN VBA PROGRAMS... 8 APPENDIX D TAIL DEPENDENCE.... 8
2 Which Archimedean he righ one? Mario R. Melchiori 2 Absrac This paper presens he concep of copula from a pracical sandpoin. Given he widened use of he mulinormal disribuion, we argue is inadequacy, while advocae for using he copula as an alernaive and beer approach. We examine wha he copulas are used for wihin of risk managemen. Then we expose a guide o choose boh he margins and he Archimedean copula ha beer fi o daa. In addiion, we provide an algorihm o simulae random bivariae from Archimedean copula. In order o cover he gap beween he heory and is pracical implemenaion VBA codes are provided. They are used in a numerical example ha illusraes he use of he copula in he pricing of a firs-a-defaul conrac. Two spreadshees accompany o paper, by presening sep by sep all pracical applicaions covered. Inroducion Keywords:, Kendall Tau, Dependence, and Credi Derivaives Since Li (2000) firs inroduced copulas ino defaul modeling, here has been increasing ineres in his approach. Unil ha momen, he copula concep was used frequenly in survival analysis and acuaries sciences. Following o Li (2000), a copula is a funcion ha links univariae marginals o heir full mulivariae disribuion. For muniform random variables 2 U, U,..., U m he join disribuion funcion C is defined as: Cu (, u,..., u, ρ ) = Pr U u, U u,..., U u 2 m 2 2 m m where ρ 3 is a dependence parameer, can also o be called a copula funcion. can be used o link marginal disribuions wih a join disribuion. For deerminae univariae marginal disribuion funcions F ( x), F2 ( x2),..., F m ( x m ), he funcion CF ( ( x ), F ( x ),..., F ( x ) ) = F ( x, x,..., x ) 2 2 m m 2 m which is defined using a copula funcion C, resuls in a mulivariae disribuion funcion wih univariae marginal disribuions specified by F ( x), F2 ( x2),..., F m ( x m ). Sklar (959) esablished he converse. He showed ha any join disribuion funcion F can be seen as a copula funcion. He proved ha if F ( x, x2,..., xm) is a join mulivariae disribuion funcion wih univariae marginal disribuion funcions, hen here exiss a copula funcion Cu (, u2,..., u m ) such ha F ( x, x,..., x ) = CF ( ( x ), F ( x ),..., F ( x ) ). 2 m 2 2 m m If eachf i is coninuous hen C is unique. Thus, copula funcions provide an unifying and flexible way o sudy join disribuions. Anoher imporan derivaion is ha he copula allows us o model he dependence srucure independenly from he marginal disribuions. In his paper, we will focus he bivariae copula funcion Cuv (, ) for uniform variables U and V, defined over he area { ( uv, ) 0 < u,0 < v } A normal world is commonly adoped boh in marke risk models and credi risk one, eiher explicily or implicily, when he models do use of he mulinormal disribuion. The commercial credi risk models KMV and CrediMerics use ha. From he copula s poin of view he mulinormal disribuion has normal marginal disribuion and Gaussian copula dependence. 2 I am graeful o Arcady Novosyolov, Carina Srada, Glyn Holon, Luciano Alloai and Moorad Choudhry for heir generous conribuion. All remaining errors are, of course, my own. I wan o hank o Mohamoud B Dualeh for encouraging me o wrie his paper. 3 As Embrechs e. al. (200) show, he correlaion is only a limied descripion of he dependence beween random variables, excep for he mulivariae normal disribuion where he correlaion fully describes he dependence srucure.
3 Hereafer, we will use he erm Normal for he univariae marginal disribuions and he erm Gaussian referring o he copula dependence. The advanage of using normal dependence srucure doesn arise, as should be suppose, from hisorical behavior of he financial nor credi marke, bu in is simpliciy, analyical manageabiliy and he easy esimaion he is only parameer, he correlaion marix. Empirical evidence suggess ha he use of mulinormal disribuion is inadequae 4. The non-normaliy of univariae and mulivariae equiy reurns is hisorically unmisakable. In oher words, here is clear evidence ha equiy reurns have uncondiional fa ails, o wi, he exreme evens are more probable han anicipaed by normal disribuion, no only in marginals bu also in higher dimensions. This is imporan boh for marke risk models as credi risk one, where equiy reurns are used as a proxy for asse reurns ha follow a mulivariae normal disribuion, and, herefore, defaul imes have a mulivariae normal dependence srucure as well. As Embrechs e. al. (200) show, here many pifalls o he normaliy assumpion. For us, he main snare is he small probabiliy of exreme join evens. In credi risk case, defauls are rare evens, so ha he ail dependence has a grea impac on he defaul srucure. Tail dependence can be measure. The ail dependence for wo random variables F and Y X and Y wih marginal disribuions X F measures he probabiliy ha Y will have a realizaion in he ail of is disribuion, condiioned ha X has had a realizaion in is own ail. Tail dependence relaes he amoun of dependence in he upper righ quadran ail or lower lef one of a bivariae disribuion, so we could have upper ail dependence, lower ail dependency or boh. Upper ail dependence exiss when here is a probabiliy ha posiive ouliers happen joinly. Upper ail dependence is defined as: where λ ( limpr Y F ( u) X F ( u) ) = (.) upper Y X u F denoes he inverse cumulaive disribuion funcion and u is an uniform variable defined over ( 0, ). ( Y X ) Since Pr Y F ( u) X F ( u) can be wrien as: Pr( X FX ( u) ) Pr Y FY ( u) + Pr X FX u, Y FY u ( ) ( ( ) ( ) ) Pr( X FX ( u) ) (.2) given ha: and ( Pr X F ( ) ) Pr( X u Y FY ( u) ) u = = (.3) ( Pr X F ( ) X u, Y FY ( u) ) Cuu (, ) = (.4) an alernaive and equivalen definiion (for coninuous random variables) of (.), is he following: λ upper = 2 u + Cuu (, ) lim u u (.5) Lower ail dependence is symmerically defined: ( Y X ) Since Pr Y F ( u) X F ( u) λ ( limpr Y F ( u) X F ( u) ) = (.6) Lower Y X u 0 can be wrien as: 4 See R. Mashal, M. Naldi, and A. Zeevi. The Dependence Srucure of Asse Reurns hp://www-.gsb.columbia.edu/faculy/azeevi/papers/equiy- Asse-Revised.pdf. Forhcoming, Risk.
4 Pr ( X FX ( u), Y FY ( u) ) Pr( X FX ( u) ) (.7) given (.3) and (.4) an alernaive and equivalen definiion (for coninuous random variables) of (.6), is he following: λ Lower Cuu (, ) lim u 0 u = (.8) The Gaussian copula wih correlaion ρ < does no have lower ail dependence nor upper one (, ) I is imporan o remark ha: he ail area dependency measure (, ) marginal disribuions. Lower upper λ λ. Lower upper λ λ depends on he copula and no on he Non-Gaussian copulas such as and Archimedean used as underlying dependence srucure wih anyone else marginal disribuion, have upper ail dependence, lower ail dependency or boh, so ha, hey could describe beer he realiy of he behavior of he financial and credi markes. See he appendix D for a non-parameric esimaion of he ail dependence 5. Risk Managemen Applicaion Unil here, we have seen wha he copula is and why he mulinormal disribuion is no an adequae assumpion. Now, we show wha he copula is used for wihin risk managemen. As i already was said, firs was used frequenly in survival analysis and acuaries sciences. In addiion, is employed in loss aggregaion, sress esing, defaul modeling and operaional risk. Hereafer, we concenrae he use of he copula in he defaul modeling scope, more concreely, in he Credi Derivaives one. Defaul risk has been exensively modeled a an individual level, bu lile is known abou defaul risk a a porfolio level where he defaul dependence is a meaningful aspec for considering. Furher, in recen years have appeared new financial insrumens, such as collaeralized deb obligaions (CDOs), n h -o-defaul baske, ha have coningen payoffs on he join defaul behavior of he underling securiies. In he case of n h -o-defaul baske, he join dependence is of vial imporance in is pricing, because he amoun of names are no large enough o ensure a correc diversificaion. Ahead, we give an example o illusrae he use of copula in he valuaion of firs-o-defaul conrac. The appropriae choice of he marginal disribuion is needed bu no enough o accuraely measure and price he risk exposure a a porfolio level, in addiion is criical o undersand and o model he defaul dependence o choose he fied join disribuion among he underling securiies. Archimedean copulas We will focus our aenion o one special class of copula ermed Archimedean one. An Archimedean copula can be wrien in he following way: ( ) ϕ Cu,..., u = ϕ ( u ) +... + ϕ ( u ) (.9) n for all 0 u,..., u and where ϕ is a funcion ermed generaor, saisfying: n 0; ϕ ( ) = ' for all ( 0, ), ϕ ( ) < 0, his is o say ϕ is decreasing; for all ( 0, ), ϕ ( ) 0, his is o say ϕ is convex. n 5 For a formal calculaion of (.) see EMBRECHTS, P., A. J. MCNEIL and D. STRAUMANN (999): Correlaion and Dependence in Risk Managemen: Properies and Pifalls - hp://www.mah.ehz.ch/~srauman/preprins/pifalls.pdf -
5 Examples of bivariae Archimedean copulas are he following: Produc or Independen copula: ( ) ln ; C ( uv, ) uv. (.0) ϕ = = Clayon copula 6 ϕ ( ) =, > 0; ( + ) Gumbel copula 7 Frank copula 8 ϕ Cu v (.) ( lnu) + ( lnv) = = (.2) ( ) ( ln ), ; Cuv (, ) e ( ) ( ) ( ) u v e e e ( ) = ln, ; Cuv (, ) = ln + e e ϕ R (.3) The mehod described ahead is able o selec he Archimedean copula, which fis beer real daa. An Archimedean copula has he analyical represenaion given by equaion (.9). So, in order o selec he copula, i is sufficien o idenify he generaor ( ) ϕ. Selecing he righ marginal disribuion Suppose you have wo hisorical ime series compound by 000 observed daa over a period of ime, like his: 9 Series Series 2 0.85667-0.609474 2.22406 0.974876 3 0.359444.088642 4 0.777068 0.6506 5 0.734274 0.962609......... 996-0.66260 -.240644 997-0.567470 -.96790 998 0.84934.45670 999-0.84523-0.757466,000 0.58057 0.688 Firs, i is necessary o deermine in wha manner he series are marginally disribued. For doing his, we using he char, oher daa can o reques more sophisicaed approach 0. Commercial simulaion sofware such as Crysal Ball or @Risk 2 supply ools for fiing hisorical daa o deerminae probabiliy disribuion. 6 Clayon (978), Cook-Johnson (98), Oakes (982). 7 Gumbel (960), Hougaard (986). 8 Frank (979). 9 In he conex of his paper hese series can be considered as equiy reurns ha are used as a proxy for asse reurns. Firs, we invesigae he marginal disribuion of each series and hen we inquire which is he dependence among hem. 0 A nex paper will inroduce some of he approach such as Maximum likelihood Esimaion Decisioneering, Inc. - hp://www.decisioneering.com/ - offers he Crysal Ball line of spreadshee modeling sofware for ime-series forecasing, risk analysis, and opimizaion using Mone Carlo simulaion. 2 Palisade Corporaion - hp://www.palisade.com/ - develops applicaions for risk and decision analysis using Mone Carlo simulaion and opimizaion, including @RISK. All are add-ins o Excel
6 Which Archimedean is he righ one? Series Series 2 Average 0.035 0.073 Sd Dev.02.008 Sd Err 0.032 0.032 Max 3.234 2.662 Min -3.247-2.788 Quanile 95% -.675 -.658 Series Bins Frequency -3.00 5-2.50 5-2.00 7 -.50 47 -.00 79-0.50 22 0.00 20 0.50 205.00 50.50 93 2.00 53 2.50 7 3.00 5 3.50 Series 2 Bins Frequency -3.50 - -3.00 - -2.50 3-2.00 4 -.50 53 -.00 69-0.50 44 0.00 86 0.50 20.00 34.50 06 2.00 45 2.50 32 3.00 4 3.50 -
7 Hisogram 250 20.0% Frequency 200 50 00-50 (5) (4) (3) (2) () 0 Bins 2 3 4 5 5.0% 0.0% 5.0% 0.0% Series : Hisogram of he hisorical realizaion. Frequency Normal Sandard Series 2: Hisogram of he hisorical realizaion. Daa and plos show ha he Normal Sandard Probabiliy Disribuion is a fied elecion in his case. Knowing he marginal disribuion, we are able o separae marginal behavior and dependence srucure. The dependence srucure is fully described by he join disribuion of uniform variaes obained from he marginal disribuions, Normal Sandard Disribuions in our case. This poin is of fundamenal imporance and ofen cause considerably roubles. Remember, dependence srucure doesn derive from he marginal disribuions, Normal Sandard in his example, bu from he uniform variaes obained from he marginal disribuions. We jus need o know marginal disribuions so ha o recognize he cumulaive disribuion funcions (CDF) ha allows us o compue he uniform variae. For example: Series Series 2 0.85667-0.609474 Φ ( 0.85667 ) = 0.80472 (.4) Φ ( -0.609474 ) = 0.2705 (.5) where Φ denoes he normal cumulaive disribuion funcion. In Excel language: = NormSDis( 0.85667) = 0.80472 (.6) =NormSDis( -0.609474 ) = 0.2705 (.7)
8 he dependence srucure refers o he relaionship beween 0.080472 and 0.2705. Now, wha if he marginal disribuion is Lognormal? Jus mus use he correc CDF (LogNormDis funcion in Excel). Dependence. Kendall Tau τ For invesigaing more deeply he dependence we need a measure for gauging i. I is known as Kendall τ (Tau). I is a rank correlaion measure, i is invarian under sricly increasing ransformaions of he underlying random variables. Linear correlaion (or Pearson s correlaion ( ρ ) ) is mos frequenly used in pracice as a measure of dependence, bu i lacks his propery. If we call c and d respecively he numbers of pairs of variables, which are concordan and discordan, hen Kendall s Tau wries : c d τ = = pc pd (.8) 3 c + d where p and p are respecively he probabiliies of concordance and discordance. c d Le V = X Y be a vecor of wo random variables a ime. In our = -0.66260 -.240644 concordan if ( X ) ( ) > 0 Xs Y Y s. Conversely, if we have ( X ) ( ) < 0 Xs Y Y s, V, for example. Then, wo disinc observaions V and case 996 996 996 V and V s are V are discordan s (i.e. : negaively dependen). Calculaing he lineal correlaion from boh marginal and uniform disribuion can see he propery of invariabiliy under sricly increasing ransformaions of he underlying random variables: Lineal Correlaion Uniform variables U(0,).000 0.655 0.655.000 Marginal Disribuions.000 0.629 0.629.000 They are differen. Correlaion s Pearson is varian under sricly increasing ransformaions of he underlying random variables. The fundamenal reason why correlaion fails as an invarian measure of dependency is due o he fac ha he Pearson Correlaion coefficien depends no only on he copula bu also on he marginal disribuions. Thus he measure is affeced by changes of scale in he marginal variables. Now we compue he Kendall Tau dependence: τ Uniform variables U(0,).000 0.458 0.458.000 Marginal Disribuions,000 0,458 0,458,000 They are alike. Kendall Tau is invarian under sricly increasing ransformaions of he underlying random variables. Archimedean Bivariae The following algorihm generaes random variaes ( uv, ) wih generaorϕ : T whose join disribuion is an Archimedean copula C n = 2 i< j 3 The formula τn sign ( X i _ Xj) ( X2i _ X2j) (.32) can be used for esimaing τ
9 Algorihm:. Simulae wo independen U ( 0,) random variaes sand q. 2. Se K ( q) =, where K is he disribuion funcion Cuv (, ). 3. Se u = ϕ ( s. ϕ ( ) ) and v ϕ ( ( s) ϕ ( ) ) For each Archimedean copula we need, o wi: =. A. Kendall τ (.32) B. Thea C. Generaor ϕ ( ) D. Generaor s firs derivae ϕ '( ) E. Generaor s Inverse ϕ ( ) ϕ = = ϕ F. The disribuion funcion of Cuv (, ) K G. Disribuion funcion inverse K ( ) '( ) ( When i has no a closed form as in case of Gumbel, Frank and Clayon Archimedean copula, i can be obained hrough he equaion numerical roo finding). For doing his, we need he firs derivae regard o of For he Gumbel copula, we have: Table B. (*) C. D. E. F. G. (**) = τ (.9) (*). Only posiive dependence. ( ln) (.20) ( ln) (.2) (**) There is no a closed form for he inverse disribuion funcion So ha: e (.22) ( ln) (.23) ϕ ( ) q ϕ' ( ) ϕ ( ). ϕ' ( ) ln( ) + (.24) K Gumbel, so G. will be used for obaining i by numerical roo finding. by u ( ( ( ) ) ) s ln = (.25) e v ( ( ) ( ( ) ) ) s ln = (.26) e VBA code ha generaes random variaes from he 2-dimensional Gumbel copula Funcion Gumbel(ByVal Thea As Double, Opional Random, Opional Random2) As Varian Generaes random variaes from he 2-dimensional Gumbel copula Dim As Double, s As Double, q As Double, u() As Double Applicaion.Volaile ReDim u( To 2)
0 Simulae wo independen U ( 0,) random variaes sandq. If IsMissing(Random) Then s = Rnd Else s = Random End If If IsMissing(Random2) Then q = Rnd Else q = Random2 End If Se K ( q) =, where K is he disribuion funcion of Cuv (, ). Because Gumbel has no a closed form KCg_Inv is obained hrough by numerical roo finding = KCg_Inv(Thea, q) Se u = ϕ ( s. ϕ ( ) ) and v ϕ ( ( s) ϕ ( ) ) =. u() = Exp(-(s * (-Log()) ^ Thea) ^ ( / Thea)) u(2) = Exp(-(( - s) * (-Log()) ^ Thea) ^ ( / Thea)) The vecor u(2) is a pair of pseudo random numbers ha are uniformly disribued on [0,] x [0,] and ha has a Gumbel copula as a join disribuion funcion. Gumbel = u End Funcion VBA Code ha compues Inverse Cumulaive Disribuion Funcion using numerical roo finding Funcion KCg_Inv(ByVal Thea As Double, ByVal q As Double, Opional olerance As Single = 0.000000000) As Double Because Gumbel has no a closed form K ( q) = is obained hrough by numerical roo finding Dim As Double, zero As Double, KCg As Double, dela As Double, diff As Double = olerancia zero = 0 Do While True The disribuion funcion of Cuv (, ) K ( ) '( ) ϕ = = ϕ.gumbel equal o ( ln) (.23) KCg = - ( * Log() / Thea) q Derivae of he disribuion funcion of Cuv (, ) K (.24) ( ) '( ) ϕ = = ϕ.gumbel equal o ln( ) + dela = -(Log() / Thea) - ( / Thea) + diff = KCg - zero If Abs(diff) < olerance Then Exi Do = + (-diff / dela) Loop KCg_Inv = Exi Funcion End Funcion
The vecor u(2) is a pair of pseudo random numbers ha are uniformly disribued on [0,] x [0,] and i has a Gumbel copula as a join disribuion funcion. Then ake he marginal disribuion funcions, in his case, normal sandard, we pu u = Φ ( r ) (.27) u=normsdis( r ) hen we have: v = Φ ( r 2 ) (.28) v=normsdis( r 2) r ( u) = Φ (.29) r = NormSInv( u) r2 ( v) = Φ (.30) r 2 = NormSInv( v) are pseudo random numbers wih disribuion funcion Φ ( Normal Sandard ) and join disribuion funcion Gumbel. Which is he Archimedean copula righ one? The disribuion funcion of an Archimedean copula, as i already had been exposed in F. is represened for he following formula: To idenifyϕ, we: ϕ = = ϕ Cuv (, ) K ( ) '( ) (.3). Esimae Kendall s correlaion coefficien using he usual nonparameric esimae: τ n n = sign ( X i _ Xj) ( X2i _ X2j) (.32) 2 i< j 2. Consruc a nonparameric esimae of, he following way: i. Firs, define he pseudo-observaions T {number of ( X X ) X j i K = < such ha i j i < X and X } 2 < X2 / ( n ) for i =,2,..., n. ii. Second, consruc he esimae of j i K as K ( ) n = proporion of i 3. Now consruc a parameric esimae of Kusing he relaionship. (.3) T s. For example, choose a generaorϕ, for his refers o Table and Appendix A, and use he esimae τ n o calculae an esimae de, say n. Use n say K n ( ). o esimae ϕ ( x), say ϕ ( ).Finally, use ϕ ( ) o esimae K ( ), In order o selec he Archimedean copula which fis beer he daa, Frees and Valdez (998) propose o use a Q-Q plo beween 2.ii) and 3) or by minimizing a disance such as K ( ) K ( ) dk ( ) Boh approach are presened below: x n x n 2 n n n.
2 The graphical approach shows ha he Gumbel copula is he beer fi. The nonparameric approach arrives o same resul. 4 Gumbel Clayon Frank τ 0.45835 0.45835 0.45835.84623.69245 5.02757 au au Frank K ( ) K ( ) ( ) n n dk n 2 Densiy Cumulaive Sample Gumbel Clayon Frank Gumbel Clayon Frank 0.0000 7 7 0.00700 0.00007 0.00002 0.000 0.000 0.000 0.000 0.0500 23 30 0.3000 0.35 0.07937 0.353 0.000 0.003 0.000 0.000 90 220 0.22000 0.22474 0.5790 0.298 0.000 0.004 0.000 0.500 80 300 0.30000 0.3045 0.23507 0.29088 0.000 0.004 0.000 0.2000 68 368 0.36800 0.37436 0.3043 0.35478 0.000 0.003 0.000 0.2500 8 449 0.44900 0.43773 0.38359 0.448 0.000 0.004 0.00 0.3000 54 503 0.50300 0.49565 0.4547 0.47047 0.000 0.002 0.00 0.3500 56 559 0.55900 0.54903 0.5283 0.52443 0.000 0.00 0.00 0.4000 47 606 0.60600 0.59853 0.58623 0.57650 0.000 0.000 0.00 0.4500 46 652 0.65200 0.64464 0.64707 0.62692 0.000 0.000 0.00 0.5000 4 693 0.69300 0.68773 0.70403 0.67579 0.000 0.000 0.000 0.5500 35 728 0.72800 0.728 0.75684 0.72305 0.000 0.00 0.000 0.6000 43 77 0.7700 0.76602 0.8059 0.76856 0.000 0.00 0.000 0.6500 28 799 0.79900 0.8067 0.8488 0.8204 0.000 0.002 0.000 0.7000 23 822 0.82200 0.83524 0.88745 0.85308 0.000 0.004 0.00 0.7500 3 853 0.85300 0.86687 0.92082 0.893 0.000 0.005 0.00 0.8000 30 883 0.88300 0.89670 0.94868 0.92542 0.000 0.004 0.002 0.8500 33 96 0.9600 0.92483 0.97078 0.95495 0.000 0.003 0.002 0.9000 36 952 0.95200 0.9537 0.98685 0.97843 0.000 0.00 0.00 0.9500 26 978 0.97800 0.97640 0.99667 0.9947 0.000 0.000 0.000.00000 22,000.00000.00000.00000.00000 0.00 0.045 0.02 Min K ( ) K ( ) ( ) n n dkn 2 = 0.00 for Gumbel, 0.045 for Clayon and 0.2 for Frank. Below wo VBA code necessary for performing nonparameric esimaion ha allow us o reply he quesions Which is he copula righ one? 4 An aached Excel shee develops he nonparameric mehod horoughly. Over here we only presen is resul.
3 VBA Code ha compues Kendall using Tau nonparameric esimaion Funcion K_au(ByVal X As Range, ByVal X2 As Range) As Double 'Esimae Kendall's correlaion coefficien using he usual nonparameric esimae τ n n = sign ( X i _ Xj) ( X2i _ X2j) (.32) 2 i< j Dim i As Long, j As Long, s As Long, n As Long n = X.Rows.Coun For i = To n For j = i To n If j > i Then s = s + Sgn((X.Cells(i, ) - X.Cells(j, )) * (X2.Cells(i, ) - X2.Cells(j, ))) End If Nex Nex K_au = (Applicaion.WorksheeFuncion.Combin(n, 2) ^ -) * s End Funcion VBA Code ha compues he pseudo-observaions T {number of ( X X ) X < X and X } 2 < X2 / ( n ) for i =,2,..., n. j i j i = < such ha i j i Funcion Ts(ByVal X As Range, ByVal X2 As Range, i As Long) As Double Dim j As Long, s As Long, n As Long n = X.Rows.Coun For j = To n If X.Cells(j, ) < X.Cells(i, ) And X2.Cells(j, ) < X2.Cells(i, ) Then Ts = Ts + End If Nex Ts = Ts / (n - ) End Funcion Numerical Example This example shows how value a firs-o-defaul swap. For doing his, we use Li model (Li (2000)). Under his model, defauls are assumed o occur for individual asses according o Poisson process wih a deerminisic inensiy called hazard rae h. This means ha defaul imes ( T ) are exponenially disribued wih mean equal o h. Li relaes he defaul imes using a Gaussian (Normal) copula, we employ Gumbel copula, oo. We assume ha: we have a porfolio of wo credis ( n = 2). he conrac is a wo-year ransacion ( = 2), which pays one dollar if he firs defaul happens during he firs wo years. h =. Each credi has a, consan for erm of de conrac, hazard rae of 0.0 A consan ineres rae of r = 0.0
4 The Pricing Firs-a-defaul Algorihm: For each Mone Carlo rial we do he following: Draw uniform bivariaes from chosen copula ( Gaussian, Gumbel ec.) Map uniform o defaul imes ( T ) using he inverse cumulaive exponenial disribuion funcion given a fixed h. Compue minimum defaul ime. If i is less han n, he presen value of he conrac is Then we average many rials and compue he expeced value of he conrac.. rt. e. We examine he impac of he asse correlaion on he value of he credi derivaive using independence, perfecly correlaed and using he following lineal correlaion marix: Lineal Correlaion.000 0.629 0.629.000 Our simulaion of 30,000 rials produces he following resuls: s -a-defaul Swap Price Independence 0.302 Perfecly Correlaed 0.65 Normal 0.249 Gumbel 0.255 When we assume independence or perfec correlaion below analyical soluion is possible: where, in he independence case: in he perfecly correlaed case: ht r + h T ( T ) ( r + e h ) ht ht hn. (.33) = (.34) = h (.35) The resul of he analyical soluion is following presened: Analyical Soluion Price Independence 0.30 Perfecly Correlaed 0.65
5 Conclusions There is clear evidence ha equiy reurns have uncondiional fa ails, o wi, he exreme evens are more probable han anicipaed by normal disribuion, no only in marginal bu also in higher dimensions. This is imporan boh for marke risk models as credi risk one, where equiy reurns are used as a proxy for asse reurns ha follow a mulivariae normal disribuion, and, herefore, defaul imes have a mulivariae normal dependence srucure as well. Oher han normal disribuion should be used boh in marginal as join disribuions. To overcome hese pifalls, he concep of copula, is basic properies and a special class of copula called Archimedean are inroduced. Then we expose a guide o choose boh he margins and he Archimedean copula ha beer fi o daa. In addiion, we provide an algorihm o simulae random bivariae from Archimedean copula. In order o cover he gap beween he heory and is pracical implemenaion VBA codes are provided. Finally we show a numerical example ha illusraes he use of he copula by pricing a firs-o-defaul conrac. For simpliciy s sake, and given ha he join disribuion is he major opic of his paper, when we value a firs-o-defaul conrac, we obviae in marginal disribuion, o use a differen disribuion o normal one, bu we employ Archimedean copula o model dependence srucure. This paper is accompanied by wo spreadshees ha presen sep by sep all pracical applicaions covered. The spreadshees are available on Coningency Analysis - hp://www.riskglossary.com/papers/.zip and hp://www.riskglossary.com/papers/firs-a-defaul.zip -.
6 References EMBRECHTS, P., A. J. MCNEIL and D. STRAUMANN (999): Correlaion and Dependence in Risk Managemen: Properies and Pifalls. hp://www.mah.ehz.ch/~srauman/preprins/pifalls.pdf FREES, E. W. and VALDEZ, E. A. (998): Undersanding relaionships using copulas, Norh American Acuarial Journal, 2, pp. -25. HOLTON, Glyn A. (2003): Value-a-Risk: Theory and Pracice. Academic Press. hp://www.value-a-risk.ne/ LI, D. (2000): On Defaul Correlaion: A Funcion Approach, working paper, RiskMerics Group, New York. hp://www.defaulrisk.com/pdf files/on Defaul Correlaion- A Funcion Approach.pdF LINDSKOG, F. (2000): Modeling Dependence wih s, ETH Zurich. hp://www.mah.ehz.ch/~mcneil/fp/dependencewihs.pdf MASHAL, R. and NALDI, M. Pricing Muliname Credi Derivaives: Heavy Tailed Hybrid Approach. 200. Working Paper, Columbia Business School hp://www.columbia.edu/~rm586/pub/mashal_naldi_hybrid.pdf MENEGUZZO, D. and VECCHIATO, W. (2002): Sensiiviy in Collaeralized Deb Obligaions and Baske Defaul Swaps Pricing and Risk Monioring. MYERSON, Roger: VBA code for Simools.xla (3.3) Copyrigh 996-2000. hp://home.uchicago.edu/~rmyerson/addins.hm NELSON, Roger (999): An Inroducion o s. Springer Verlag. hp://www.riskbook.com/iles/nelsen_r_(999).hm ROMANO, C. (2002): Applying Funcion o Risk Managemen, Universiy of Rome, La Sapienza, Working Paper. WANG, S. S. (999): Aggregaion of Correlaed Risk Porfolios: Models & Algorihms, CAS Commiee on Theory of Risk, Working Paper. hp://www.casac.org/pubs/proceed/proceed98/980848.pdf
7 Appendix A Ohers s Parameers Clayon s Parameers B.(*) C. D. E. F. G. (**) 2τ. = ( ) + + ( ) ( + ) (.37) (.38) τ + + (.39) (.36) (*) > 0. Only posiive dependence. (**) There is no a closed form for he inverse disribuion funcion K Clayon, so G. will be used for obaining i by numerical roo finding. (.40) (.4) Frank s Parameers C. D. E. F. G. (**) e ln e (.42) (*) < e (.43) ( e e + ) ln (.44) <. Posiive and negaive dependence. (**) There is no a closed form for he inverse disribuion funcion e ln e (.45) ( e ) e e ln + ( ) e (.46) K Frank, so G. will be used for obaining i by numerical roo finding. B. Frank copula has no close form ha allows us o calculae hea parameer. We use numerical roo finding for calculaing i. Press buon in he aached Excel shee for performing he following VBA code: au au Frank Privae Sub CommandBuon_Click() Dim kau As Double Inpu waned Kendall Tau τ kau = InpuBox("Kendal Tau: ", "Inpu") In ChangingCell "G4" we use formula (.50) See Appendix B Range("G3").GoalSeek Goal:=kau, ChangingCell:=Range("G4") End Sub Appendix B Kendall τ revisied The Kendall Tau can be calculaed boh using formula (.32) or he following one: Le X and Y be random variables wih an Archimedean copula C generaed by ϕ, Kendall s Tau of X and Y is given by: τ ϕ ( ) = + 4 d (.47) ϕ' ( ) 0 when Cis Gumbel (.47) is given by:
8 τ ( ln) ln =+4 d = + 4 d = (.48) 0 0 ( ln) when Cis Clayon (.47) is given by: when Cis Frank (.47) is given by: + τ =+4 d = + 4 d. = + 2 (.49) 0 0 where D ( ) k x is he Debye funcion, given by: τ 4 ( D ( ) ) = (.50) x k k Dk ( x) = d k x e (.5) 0 We use (.50) for calculaing Frank copula's τ. The inegral f ( xdx ) is calculaed using Riemann sums a mehod. I approximaes he inegral by dividing he inerval ab, ino msubinervals and approximaing f wih a consan funcion on each subinerval. Riemann sum approximaes our definie inegral wih: b b a m k ( ) ( ) f xdx f x x k= 5 (.52) Appendix C - Using Simools feaures in VBA programs. For performing Mone Carlo Simulaion we use a freeware called Simools. For is righ working, i is necessary o aach Simools.xla as a reference in VBA module, by applying he Tools: References menu command in he Visual Basic Edior and checking Simools.xla as an available reference. More informaion abou Simools click hp://home.uchicago.edu/~rmyerson/addins.pdf. Appendix D Tail dependence. An example can be useful o visualize he issue. We assume: Kendall Tau τ = 0.45835. ρ = 0.65937. 6 Gumbel copula (.2) and Gaussian one 7 5 We use a VBA code o figure ou his inegral. For is righ working, i is necessary o conain Microsof Scrip Conrol.0 as a reference in VBA module. If Microsof Scrip Conrol.0 is missing you could download from hp://www.microsof.com/downloads/deails.aspx?familyid=d7e3492-2595- 49E6-8C02-426FEC693AC&displaylang=en π 6 We use he following relaionship: ρij = sin τij 2 2 arcsin, τ = ( ρ ) ij π ij
9 Use (.9) o calculae. τ 0.45835 Upper Tail Dependence Gumbel.84623 u 0.99000 0.99250 0.99500 0.99750 0.99900 0.99950 0.99990 0.99995 CGumbel ( uu, ) 0.98548 0.9890 0.99273 0.99636 0.99854 0.99927 0.99985 0.99993 λ upper (.5) 0.548 0.547 0.546 0.545 0.545 0.545 0.544 0.544 Gaussian ρ 0.65937 u 0.99000 0.99250 0.99500 0.99750 0.99900 0.99950 0.99990 0.99995 CGaussian ( uu, ) 0.98232 0.9866 0.99097 0.9954 0.9983 0.99905 0.9998 0.99990 λ upper (.5) 0.232 0.25 0.94 0.63 0.29 0.09 0.074 0.063 τ 0.45835 Lower Tail Dependence Gumbel.84623 u 0 0.0000 0.00750 0.00500 0.00250 0.0000 0.00050 0.0000 0.00005 CGumbel ( uu, ) 0.0023 0.0008 0.00045 0.0006 0.00004 0.00002 0.00000 0.00000 λ lower (.8) 0.23 0.08 0.089 0.065 0.043 0.03 0.05 0.0 Gaussian ρ 0.65937 u 0 0.0000 0.00750 0.00500 0.00250 0.0000 0.00050 0.0000 0.00005 CGaussian ( uu, ) 0.00232 0.006 0.00097 0.0004 0.0003 0.00005 0.0000 0.00000 λ lower (.8) 0.232 0.25 0.94 0.63 0.29 0.09 0.074 0.063 In he Gumbel copula s case when u he ail upper dependence changes slighly. In Gaussian copula s case he upper ail dependence ends o zero. When u 0 he lower ail dependence ends o zero for Gaussian copula and Gumbel one. So ha, our example suggess ha Gumbel copula has upper ail dependence bu does no has lower ail one, whereas Gaussian copula does has neiher. in his case: The formulae for calculaing he upper ail dependence from Gumbel copula is: (.53) 2 2 ( ) 7 C ( uv,; ρ ) ( u), ( v) Gaussian funcion wih linear correlaion ρ, and Φ = NormSInv( u). For = Φ Φ Φ, where Φ denoes he join disribuion funcion of he bivariae sandard normal disribuion Φ denoes he inverse of he disribuion funcion of he univariae sandard normal disribuion. In Excel language Φ a VBA code is available on hp://my.dreamwiz.com/sjoo/source/bivariae_normal_disribuion.x
20.84623 2 2 = 0.5444 (.54)