The t copula with Multiple Parameters of Degrees of Freedom: Bivariate Characteristics and Application to Risk Management

Transcription

1 The copula wh Mulple Parameers of Degrees of Freedom: Bvarae Characerscs ad Applcao o Rsk Maageme Ths s a prepr of a arcle publshed Quaave Face November 9 DOI: 8/ wwwadfcouk/jourals/rquf Xaol Luo CSIRO Mahemacal ad Iformao Sceces Sydey Locked bag 7 Norh Ryde NSW 67 Ausrala e-mal: XaolLuo@csroau Pavel V Shevcheko CSIRO Mahemacal ad Iformao Sceces Sydey Locked bag 7 Norh Ryde NSW 67 Ausrala e-mal: PavelShevcheko@csroau Submed verso: 8 Ocober 7 Fal verso: 8 May 9

2 Absrac The copula s ofe used rsk maageme as allows for modellg al depedece bewee rsks ad s smple o smulae ad calbrae However he use of a sadard copula s ofe crczed due o s resrco of havg a sgle parameer for he degrees of freedom (dof) ha may lm s capably o model he al depedece srucure a mulvarae case To overcome hs problem grouped copula was proposed recely where rsks are grouped a pror such a way ha each group has a sadard copula wh s specfc dof parameer I hs paper we propose he use of a grouped copula where each group cosss of oe rsk facor oly so ha a pror groupg s o requred The copula characerscs he bvarae case are suded We expla smulao ad calbrao procedures cludg a smulao sudy o fe sample properes of he maxmum lkelhood esmaors ad Kedall s au approxmao Ths ew copula ca be sgfcaly dffere from he sadard copula erms of rsk measures such as al depedece value a rsk ad expeced shorfall Keywords: grouped copula al depedece rsk maageme

3 Iroduco Approprae modelg of depedeces bewee dffere facal markes ad rsk facors s a mpora ad challegg aspec of quaave rsk maageme Copula fucos have become popular ad flexble models hs feld The use of copula fucos eables he specfcao of he margal dsrbuos o be decoupled from he depedece srucure of varables whch ur helps wh he ask of modelg facal rsks uder a more realsc o-gaussa assumpo The cocep of copulas was frs roduced by Sklar 959 bu oly a decade ago became popular applcao o facal rsk maageme For a comprehesve revew of copula facal rsk maageme see McNel e al (5) Modelg depedeces he case of more ha wo depede rsks s a challegg ask cosdered by may researchers We would lke o meo wo flexble approaches: par copula cascade ad esed Archmeda copulas Buldg o he poeerg work of Bedford ad Cooke ( ) Aas e al (7) showed how he mulvarae depedece ca be modeled usg a cascade of par-copulas acg o wo varables a a me I he mos geeral form hs par-copula produces may possble cosrucos ad model seleco becomes crcal ad very challegg Nesed Archmedea copula see Joe (997) ad McNel (8) s aoher flexble way o model mulvarae depedece These esed copulas have bvarae Archmeda margals ad allow for dffere levels of posve depedece dffere bvarae margals however hey requre cosras o he copula parameers I pracce oe of he mos popular copula modelg mulvarae facal daa s perhaps he copula mpled by he mulvarae dsrbuo see Embrechs e al (); Fag e al (); Demara ad McNel (5) Ths s due o s smplcy erms of smulao ad calbrao combed wh s ably o model al depedece whch s ofe observed facal reurs daa Ths sylzed fac ca o be adequaely descrbed by he commoly used Gaussa copula Rece papers by Mashal e al (3) ad Breyma e al (3) have demosraed ha he emprcal f of he copula s superor mos cases whe compared o he Gaussa copula However s somemes crczed due o he resrco of havg oly oe parameer for he degrees of freedom (dof) ha may lm s ably o model al depedece a mulvarae case To overcome hs problem Daul e al (3) proposed o use grouped copula where rsks are grouped o classes ad each class has s ow copula wh a specfc dof Ths however requres a a pror choce of classes I s o always obvous how he rsk facors should be dvded o sub-groups A adequae choce of groupg requres subsaal addoal effor (cosderao of may possble combaos) f here s o aural groupg for example by secor or class of asses I hs paper o overcome he problem wh a pror choce of groups he grouped copula we propose he use a grouped copula wh each group havg oly oe member hereafer referred o as he copula wh mulple dof parameers For coveece deoe hs copula as copula where defes he vecor of dof parameers Though he paper by Daul e al (3) does o explcly specfy he sze of each group mplcly assumes ha each group cosss of wo or more rsk facors whch s a ecessary codo for he fg procedure suggesed her paper Whle he dof parameers of he grouped copula ca be esmaed margally by fg each group separaely by a sadard copula as suggesed by Daul e al (3) he dof parameers of he copula ca oly be esmaed joly I s worh og ha he copula s o a mea- dsrbuo cosdered by Embrechs e al () Fag e al () Alhough our ma movao for sudyg he copula s modelg mulvarae cases for smplcy hs paper wll cosder bvarae examples oly Applcao he geeral mulvarae case cludg model seleco wll be cosdered furher research Eve he bvarae case wll be demosraed ha here could be a sgfca mpac o porfolo 3

4 rsk measures (such as Value a Rsk ad Expeced Shorfall) f he sadard copula s used whe he rue copula s copula Furhermore he f of he copula o some FX daa s deed superor whe compared wh he sadard copula The orgasao of hs paper s as follows Seco descrbes he model ad oaos for copulas Explc represeaos ad calbrao of he copula are dscussed Seco 3 ad Seco 4 respecvely Seco 5 preses mpora characerscs of he bvarae copula cludg al depedece ad local asymmery Examples of calbraos ad applcaos o rsk quafcao are provded Seco 6 Cocludg remarks are gve he fal Seco Model I s well kow from Sklar s heorem see eg Joe (997) ha ay jo dsrbuo fuco F wh couous (srcly creasg) margs F F F has a uque copula C( u ) F( F ( u ) F ( u ) F ( u )) () The copulas are mos easly descrbed ad udersood by a sochasc represeao We roduce oao ad defos as follows: Defo Z ( Z Z ) s a radom vecor from he mulvarae Normal dsrbuo Φ (z Σ ) wh zero mea vecor u varaces ad posve defe correlao marx Σ ; U ( U U U ) s defed o [ ] doma; S s a radom varable from he uform () dsrbuo depede from Z ; / W χ ( S) where χ () s he verse cdf of he Ch-square dsrbuo wh dof For coveece of furher oao he dsrbuo ad s verse for a radom varable W are deoed as G ad G respecvely W s depede from Z ; () s he sadard uvarae dsrbuo wh dof ad () s s verse The we have he followg represeaos: Sadard copula The radom vecor s dsrbued from mulvarae dsrbuo ad radom vecor X W Z () U ( ( X ) ( X )) (3) s dsrbued from he sadard copula A radom vecor obaed by rasformg he above U margally e F ( U) F ( U ) where F () are some uvarae couous dsrbuos s sad o be dsrbued from mea- dsrbuo (Embrechs e al ; Fag e al ) 4

5 Grouped copula Paro { } o m o-overlappg sub-groups of szes m The he copula of he dsrbuo of he radom vecor X W Z WZ ) W Z + W Z + WmZ (4) ( where W G ( S ) k m s he grouped copula e k k U ( ) ( ) ( ) ( ) ( )) X X X + X + X (5) ( s a radom vecor from he grouped copula Here he copula of each group s a sadard copula wh s ow dof parameer copula wh mulple dof ( copula) Cosder he grouped copula whch each group has a sgle member I hs case he copula of he radom vecor X ( W Z WZ W Z ) (6) s sad o have a copula wh mulple dof whch we deoe as copula e U ( X ) ( X ) ( X )) (7) ( s a radom vecor dsrbued accordg o hs copula Remark W ( G ( S) G ( S) G ( S) ) e W W are perfecly depede Also oe ha he dsrbuo of a radom vecor X gve by (6) s o a mea- dsrbuo Gve he above sochasc represeao smulao of he copula s sraghforward: Smulae radom varables Z Z from he mulvarae Normal dsrbuo Φ (z Σ ) Smulae a sgle radom umber S (depede from Z ) from he sadard uform dsrbuo () ad calculae Wk G ( S) k k Calculae vecors X ( W Z W Z ) ad U ( ( X ) ( X )) The laer radom vecor U s a sample from he copula I he case of sadard copula ad he case of grouped copula he correspodg subses have he same dof parameer Noe ha he sadard copula ad grouped copula are specal cases of copula (see Appedx A) m 3 Explc preseao of he copula wh mulple dof parameers From he sochasc represeao (6-7) s easy o show see Appedx A ha he copula dsrbuo wh ) has he followg explc egral expresso ( ad s desy s Φ Σ Σ C ( u) ( z ( u s) z ( u s)) ds (8) Σ C ( u) ( ( ) ( )) [ ( )] ϕ Σ z u s z u s wk s ds u u k k Σ c ( u ) f ( xk ) (9) k 5

6 Here he followg defos are used: Defo 3 z ( u s) ( u ) / w ( s) k k k w ( s) G ( ) ; k k k ; k s k / ϕ ( ) exp( Σ z z z Σ z) /[(π ) de Σ] s he mulvarae Normal desy wh zero meas ad u varaces; x ( u ) k k k ; k + ( x) ( + x / ) Γ( ( + )) /[ Γ( ) π ] f s he uvarae desy Noe he mulvarae desy (9) volves a oe-dmesoal egrao whch makes fg of hs copula more compuaoally demadg ha fg sadard copula bu sll praccal If all he dof parameers are equal e he s easy o show (see Appedx A) ha he copula defed (8) becomes he sadard copula: C Σ ( u ) u ( ) ( + ) Γ( ( + )) + z Σ z / ( π ) de Σ Γ( ) ( u ) dz () 4 Calbrao Cosder rsks modeled by a radom vecor Y ( Y Y ) ad assume ha s daa () ( K ) Y Y are d To esmae a paramerc copula usg observed daa y j K oe ca follow he procedure descrbed McNel e al (5) The frs sep s o projec daa o ( j) [ ] doma o oba ˆ ( j) u F ( y ) usg esmaed margal dsrbuos F ˆ () Ths ca be doe by modelg margs usg paramerc dsrbuos or o-paramercally usg emprcal dsrbuos (or combao of hese mehods eg emprcal dsrbuo for he body ad Geeralzed Pareo dsrbuo for he al of a margal dsrbuo) Gve sample ( j) u cosruced usg he orgal daa (or smulaed drecly as our smulao expermes Seco 6) he copula parameers ca be esmaed usg eg he maxmum lkelhood (ML) mehod as dscussed below Noe f boh margs ad copula are modeled by paramerc dsrbuos he a beer ferece ca be obaed by esmag marg ad copula parameers joly hough mgh be more dffcul echcally ( j) 4 Maxmum lkelhood Icludg he correlao coeffces Σ j he copula has M ( +) / parameers: ( ) / off-dagoal correlao coeffces plus dof parameers Le θ be he vecor of hese M parameers Deoe he desy of he copula evaluaed a ( j) u as ( j) c ( u ) whch ca be evaluaed usg (9) The he Maxmum Lkelhood Esmaors (MLEs) θ θˆ are he values of θ ha maxmze he log-lkelhood fuco 6

7 l( θ u () u + ( K ) K j ) l j ( j) ( + ) l[ + ( x K c ( u θ ) ( j) ) l / ] + K ( j) ( j) ( x / w ( s) x / w ( s) ) K ϕ Σ j [ w ( s)] ( l( π ) + l[ Γ( ) / Γ( ( + ))] ) ds () ( j) ( j) where x ( u ) j K Evaluao of l (θ ) volves oedmesoal egrao K mes ad he umercal opmzao procedure s more compuaoal demadg whe compared o he case of a sadard copula However usg avalable fas ad accurae algorhms for he oe-dmesoal egrao makes he fg sll praccal I hs work we have used a globally adapve egrao scheme documeed Pesses e al (983) The maxmzao of () s subjec o cosras o sasfy he requreme ha he correlao marx Σ s posve defe To sasfy hs cosra s covee o maxmze () respec o he coeffces of he Cholesky lower ragular marx A Σ AA The he followg smpler equaly cosras o he elemes of A should be mposed j A j Noe ha A Oe ca le A A j o reduce he umber of parameers j o ( ) / ukows for he off-dagoal elemes of he Cholesky lower ragular marx subjec o equaly cosras A j < j Asympoc properes Ofe MLEs have useful asympoc properes gve by he followg heorem (precse regulary codos requred ad proofs ca be foud may exbooks see eg Lehma (983) Theorem 64) Theorem 4 If X XK are d each wh a desy f ( x θ) ad correspodg MLE θˆ he as he sample sze K creases: a) uder mld regulary codos θˆ s a cosse esmaor of he rue parameer θ e θˆ coverges o θ probably as K creases; b) uder sroger regulary codos K ( θ ˆ θ) s asympocally Normal wh zero mea ad covarace marx I ( θ) where I j ( θ ) E[ l f ( X θ) / θ θ j ] s he Fsher formao marx If I (θ) ca o be foud closed form he (for a gve realzao x x K ) ypcally s esmaed by he observed formao marx Iˆ K j ( θ) l f ( xk θ) K θ θ j k () ha coverges o he Fsher formao marx by he law of large umbers ad may eve lead o more accurae ferece as suggesed by Efro ad Hkley (978) Boh I ˆ j ( θ) ad I j ( θ) deped o ukow rue parameer θ so fally he covaraces cov( ˆ θ ˆ ) bewee MLEs θ j 7

8 are esmaed as ˆ [ I θ j (ˆ)] / K ha wll be used some umercal examples below Also we calculae d dervaves () umercally usg fe dfferece mehod Remarks: The requred regulary codos for he above asympoc heorem are codos o esure ha he desy s smooh wh regard o parameers ad here s ohg uusual abou he desy see Lehma (983) These clude ha: he rue parameer s a eror po of he parameer space; he desy suppor does o deped o he parameers; he desy dffereao wh respec o he parameer ad he egrao over x ca be swapped; hrd dervaves wh respec o he parameers are bouded; ad few ohers Though he requred codos are mld hey are ofe dffcul o be proved especally whe he desy has o closed form as he case of copula Here we jus assume ha hese codos are sasfed Wheher a sample sze s large eough o use he asympoc resuls s aoher dffcul queso ha should be addressed real applcaos 4 Kedall s au approxmao I pracce o smplfy he calbrao procedure for copula correlao marx coeffces are ofe esmaed par-wse usg Kedall s au rak correlao coeffces τ Y Y ) va he formula (see eg McNel e al (5)) ( πτ ( Y Y )) ( j Σ s (3) j j The he secod sage he dof parameers are esmaed I he case of grouped copula Daul e al (3) esmaed he dof for each rsk group margally (e usg daa from he group o esmae s dof) I he case of he copula wh mulple dof parameers should be esmaed joly Remark Srcly speakg formula (3) s vald for ellpcal dsrbuos ad he bvarae case oly Though (3) s also que accurae he mulvarae case may lead o a cosse correlao marx ad furher adjusmes wll be requred (replacg egave egevalues by small posve values) for deals see McNel e al (5) Also was oed Daul e al (3) ha formula (3) s sll hghly accurae eve whe s appled o fd he correlao coeffces bewee rsks from he dffere groups (wh possbly o-ellpcal dsrbuos) so ca be used wh he same success he case of copula Though he qualy of hs approxmao for fe samples for eher sadard or grouped copulas was o suded I Seco 6 we show resuls of he full jo MLE calbrao (fg correlao ad dof parameers joly) as well as usg Kedall s au approxmao (3) for he correlao parameers ad compare he resuls I he cases suded below we observed ha he bas roduced by Kedall s au approxmao s small 8

9 5 Bvarae copula characerscs The bvarae copula wh wo dof parameers s ( z ( u s) z ( u s ) C Φ ( u u) ) ds (4) where z ( ) u s ( u ) / w ( s) ad Φ ( x ) x s he bvarae Normal dsrbuo wh zero meas u varaces ad correlao coeffce see (8) Accurae ad fas algorhms for evaluag he bvarae Normal dsrbuo are avalable ad hus he above copula dsrbuo fuco s effecvely a oe-dmesoal egrao ha ca be compued accuraely ad effcely usg umercal egrao Fgure shows he CDF surface for 7 8( u ) ad Fgure shows s desy surface We ow proceed o dscuss several C u releva properes of he bvarae C ( u ) copula u 5 Radal symmery Defo 5 A radom vecor X (or s df) s radally symmerc abou a f X a d a X Smlar o he sadard copula wh a sgle dof parameer he desy c ( u u) s radally symmerc abou he cere ( 55) e s desy sasfes c u u ) c ( u u ) U U (5) ( Obvously hs radal symmery meas ha Pr( U < 5 U < 5) Pr( U > 5 U > 5) ad Pr( U > 5 > U) Pr( U > 5 > U ) whch also mples Pr( U < U ) Pr( U > U ) ad Pr( U + U < ) Pr( U + U > ) d 5 Exchageably d ( Π() Π( ) Defo 5 A radom vecor X s exchageable f X X ) ( X X ) for ay permuao ( Π () Π( )) of ( ) The C ( u u) copula s o exchageable f e he desy c ( u u ) c ( u u ) f (6) Ths wll cause some local asymmery he dsrbuo as dscussed below 53 Symmery relaed o he correlao coeffce The copula desy u u ) sasfes c ( ( u u) c ( u u c ) ad c u u ) c ( u u ) ( 9

10 The above symmery s appare from he explc desy expresso (9) whe chagg u o u whle u s fxed he sg of z s chaged so a chage of sg wll cacel ou hs chage he bvarae Gaussa desy he egrad of (9) The sg chage x wll have o effec because he uvarae -dsrbuo s symmerc abou zero Ths symmery propery relaed o s very useful because usg hs symmery argume mos of he characerscs ad resuls preseed below for a posve correlao coeffce apply equally o he cases wh a egave 54 Asymmery wh respec o he axs U U To sudy some local asymmeres dvde he [ ] doma alog axes u u ad u + u addo o axes u 5 ad u 5 These four axes dvde he doma o egh equal area regos as show Fgure 3 umbered from o 8 clockwse sarg from he op-lef corer U U Fgure CDF surface for he bvarae copula 7 u ) C 8( u

11 U U Fgure PDF surface for he bvarae copula 7 u ) C 8( u Fgure 3 Paro of bvarae uform doma Deoe he probably of he varable ( ) rego ( 8 ) as Pr e: Pr Pr( U + U < U > 5) ; Pr Pr( U + U > U < 5) ; Pr3 Pr( U > U > 5) ; Pr4 Pr( U > U > 5) ; Pr5 Pr( U + U > U < 5 ) ; Pr6 Pr( U + U < U > 5) ; Pr7 Pr( U < U < 5) ; Pr8 Pr( U < U < 5) The he radal symmery esures ha whch also mples Pr Pr 5 Pr6 Pr Pr 3 Pr7 ad Pr 4 Pr8 + Pr Pr5 6 ad Pr 3 Pr4 Pr7 + Pr8 Pr + Pr + Radal symmery plus exchageably esures ha

12 Pr Pr5 6 ad Pr 3 Pr4 Pr7 Pr8 Pr Pr whch s he case for he sadard bvarae copula wh a sgle dof parameer I he case of copula (4) wh wo dof parameers he lack of exchageably mples asymmery such ha Pr Pr 4 Pr3 Pr Pr5 Pr6 ad Pr7 Pr8 bu s radal symmery mples Pr / Pr Pr5/ Pr6 ad 3/ Pr4 Pr7/ Pr8 Pr 7 Usg mllo Moe Carlo (MC) samples from he copula C 8( u u ) we fd Pr/ Pr 78 ad Pr3/ Pr4 37 (MC umercal sadard errors are of he order of ) Furhermore hs asymmery s more proouced he al of he dsrbuo Ths ehaced asymmery he al also occurs whe parameer s egave however ow he rao Pr 3/ Pr 4 s less ha oe for egave whle s larger ha oe for posve I fac due o he symmery c ( u u) c ( u u) we have Pr ( ) / Pr ( ) Pr ( ) / Pr ( ) 3 4 Thus he al he deparure of he asymmery rao from oe a measure of hs asymmery s more proouced for boh egave ad posve Regos 3 ad 4 defe a doma where boh margal varables are above he 5 quale Deoe PrA Pr( U > U > q) ad PrB Pr( U > U > q) where 5 < q < s a hgh quale level The we ca quafy he asymmery for he upper al rego by he rao ξ q : Pr( U > U > q) ξ q (7) Pr( U > U > q) The wo probables are gve by ad x Pr( U > U > q) ds dy ϕ ( y y) dy (8) x ( q s) x ( q s) x Pr( U > U > q) ds dy ϕ ( y y ) dy (9) x ( q s) x ( q s)

13 where ϕ x x ) s he bvarae Normal desy wh zero meas u varaces ad ( correlao ad x ( q s) ( q) / G ( s) Drec umercal egrao of (8) ad (9) s possble bu for smplcy ad effcecy we have used MC smulao from he copula o calculae he resuls below 7 Usg mllo MC samples we fd ξ for he C 8( u u ) copula e he upper al rego ( U > 99 U > 99) he rao PrA /(PrB + PrA) 6 I geeral ca be umercally verfed ha Pr( U > U > q) ξ > f q > 5 < Pr( U > U > q) > q () Table lss values of ξ 99 a wh varous values for parameers ad All values ξ Table are larger ha cofrmg ha he upper al U > q U > ) he of 99 ( q ( U probably of U > U s sgfcaly larger ha ha of U < U f U ) are from copula C wh > ad > The asymmery rao ξ q frs rapdly creases wh ad he becomes fla before slowly decreasg Ths asymmery s a feaure ha he sadard copula (wh a sgle dof parameer) does o have Table Asymmery rao ξ 99 for he C ( u u ) copula esmaed by mllo MC smulaos ξ q Due o radal symmery here s a smlar asymmery for he lower al Pr( U < U < q) η ξ > f < 5 < Pr( < < ) > q q q () U U q The above asymmery ca be see from he copula desy surfaces To reveal he asymmery 7 clearly Fgure 4 shows he dfferece bewee he desy of C ad he desy of 8 7 C 8 Noe he sadard bvarae copula s symmerc respec o he axs U U Observg he sg of he dfferece ear he upper al U > q U > q) Fgure 4 s clear he desy of C s larger ha ha of 7 8 ( q 7 8 C he upper ragle rego 3

14 ( U > U > q) bu s he oppose he lower ragle rego ( U > U > q) cofrmg he equaly () U U 7 7 Fgure 4 The dfferece bewee deses of C ad C Tal depedece Defo 53 The lmg lower ad upper al depedece coeffces (TDC) of wo rvs X ad X are defed hrough he copula as C( q q) λ L lm () q + q ad C ( q q) Pr( U > q U > q) λ U lm lm (3) q q q q respecvely see eg McNel e al (5) Here C q q) Pr( U > q U > ) ( q Noe ha for copulas wh radal symmery λ L λ U λ whch s he case for he copulas dscussed hs paper Also for he sadard bvarae copula wh a sgle dof parameer he al depedece coeffce s see McNel e al (5) λl ( ) + ( ( + )( ) /( + )) (4) By defo () he case of copula wh wo dof parameers he lower TDC s ( ( q) / w ( s) ( q) / w ( s ) ) lm Φ + ) q q λ L( ds (5) 4

15 Takg he lm aalycally (see Appedx B) gves λ ( ) Ω( ) + Ω( ) L where Ω( ) B / / g + ( ) F N Γ[( + ) / ] [( ) / ] Γ + / ( ( B ) / ) / g ( ) / e Γ( / ) d Here g () s he Ch-square desy wh dof ad F N () s he sadard Normal dsrbuo I ca be show (see Appedx B) ha he case of (6) s decal o (4) recoverg he TDC for he sadard copula as expeced The correcess of (6) was checked by comparg wh he umercal lmg value of (5) ad wh (4) whe for a wde rage of parameers Radal symmery assures ha λl ( ) λu ( ) Defo 54The TDC for he orh-wes quadra λ NW ad for he souh-eas quadra λ SE are defed as λ NW Pr( U < q U > q) Pr( U > q U < q) lm λse lm + q q q q + (6) Usg he symmery propery c ( u u) c ( u ) for copula we easly oba λ u ( ) λse ( ) λl( ) NW Smlar argume also apples o he sadard copula e λ ( ) λ ( ) λ ( ) NW SE L Numercal examples Fgure 5 shows λ L ( ) as a fuco of he wo dof parameers for 7 wh boh dof parameers ragg from o Fgure 6 shows λ NW ( ) for he same Some seleced umercal values (rouded o 3 decmal dgs) are gve Table As expeced he hghes al depedece occurs a ( ) ad he lowes a ( ) Of course he al depedece s symmerc f ad are swapped Table Lower al depedece coeffce λ L ( ) of he copula wh wo dof parameers ad 7 as a fuco of \

16 7 Fgure 5 Lower al depedece λ L ) for C copula ( 7 Fgure 6 Tal depedece λ NW ) for C copula he orh-wes (or souh-eas) quadra ( I s eresg o oe Table ha λ L ( ) does o ecessarly always decrease mooocally wh creasg ( ) whe ( ) s fxed Neverheless we ca sll observe from Table ha here s a cosse local moooc paer he values lsed Table Sarg from ay dagoal eleme ( bold face Table ) he value of λ L ( ) akes maxmum value compared wh all he elemes below or o he rgh of Furhermore he elemes he colum below or he row o he rgh of he dagoal eleme mooocally decrease wh creasg dof Tha s he followg moooc relaoshp holds λ L ( a ) < λl( b ) f a > b > (colum below he dagoal) λ L ( a) < λl( b) f a > b > (row o he rgh of he dagoal) 6

17 I oher words f we look a he lower ragular marx for he al depedece coeffces Table he value o he dagoal s he larges each colum ad decreases mooocally as he poso moves dow he colum The same ca be sad for rows he upper ragle he dagoal value s he maxmum each row he upper ragle 6 Examples of calbrao ad applcao o rsk quafcao Dffere copulas may o ecessarly lead o a sgfca dffereces erms of rsk measures such as Value a Rsk (VaR) ad Expeced Shorfall (ES) If hs s he case he use of overly complcaed copulas may o be jusfed ad he smpler oes are preferred for rsk maageme purposes Examples below are lmed o bvarae case oly I s geerally expeced ha he mpac of he copula choce o predcg rsk measures would o be very sgfca bvarae case O he oher had a small mpac bvarae case does o ecessarly mea a small mpac hgh dmesoal cases ad vce versa Neverheless he followg smulao expermes we demosrae ha he bvarae case f he porfolo of wo asses has asymmerc weghg facors he mpac of copula wh mulple dofs ca be sgfca ad he wrog choce of a sadard copula could lead o a subsaal uder-esmao of rsk measures Whle he MLE procedure descrbed Seco 4 s commoly pracced amog facal pracoers for he sadard copula poeal pfalls exs a leas for small sample szes I s mpora o sudy fg dffere copulas whe he sample sze s small (whch s he case for may rsk maageme problems) We wll carry ou a smulao sudy o he fe sample behavour of he MLEs for he copula ad sadard copula quafyg he bas ad he error of he MLEs a varous sample szes for a se of parameers I aoher example we cosder real foreg exchage daly daa o show ha he copula wh mulple dofs ca f beer ha he Gaussa or he sadard copulas I geeral he mpac of parameer uceray due o fe sample sze ca be sgfca ad always warras grea aeo as we have show recely a dffere sudy he operaoal rsk coex (Luo e al (7)) where a much smaller sample sze (<) was used o demosrae he mpac of uceray More comprehesve aalyss of copulas uder parameer uceray s beyod he scope of he curre paper ad s a ope feld for research 6 Smulao sudy of model rsk To demosrae he mpac of he copula model choce e model error we frs use smulaed daa wh a very large sample sze so ha fg errors due o fe sample sze ca be gored Cosder a porfolo of wo asses wh lear-reurs represeed by radom varables X ad Y ad weghg facors w X ad w Y respecvely The lear-reur of he porfolo s smply Z wx X + wy Y Le w X ad w Y he he porfolo lear-reur s smply Z X Y Ths porfolo reflecs a commoly ecouered hedge poso real facal world where oe sars wh a zero e capal ad wshes o maage he rsks volved wh hs al zero-sum porfolo I s acpaed ha he asymmery of he copula respec o he U U axs wll have a larger mpac o rsk measure predcos of a asymmerc porfolo I he bvarae case a asymmerc porfolo cosss of a log poso oe asse ad a shor poso aoher a ypcal spread poso facal markes 7

18 Copula models Assume ha he rue model for he depedece bewee X ad Y s copula wh 9 ( 9) deoed as C We smulae 5 samples of ( X Y ) from hs copula ad he sadard Normal margs ad he f he Gaussa copula ad he sadard copula ( hs fg he rue margs are used) For he Gaussa copula he correlao coeffce s esmaed by he sample lear correlao For he copula he correlao coeffce s esmaed usg (3) wh Kedall s au esmaed from he sample The dof parameer of he copula s he esmaed by he ML mehod Table 3 shows he calbrao resuls for he Gaussa copula ad he copula The sadard devao of ˆ due o fe sample sze esmaed usg () was less ha % Re-samplg sudy cofrms hs MLE error esmae ad also shows ha he error for ˆ s much smaller Table 3 Gaussa ad copula parameer esmaes usg 5 samples from he copula wh ad 9 Copula Model Correlao coeffce dof Gaussa copula ˆ 868 N/A copula ˆ 885 ˆ 7 84 Gaussa margs Assume ha margally boh X ad Y are dsrbued from he sadard Normal dsrbuo wh her depedece modelled by he copula models descrbed above The mllo Moe Carlo (MC) smulaos were performed for each of he hree copula models (he Gaussa ad copulas) wh he rue sadard Normal margs usg he parameers gve Table 3 o calculae he 99 VaR ad he 99 ES of he porfolo deoed as Qˆ ad Ψˆ respecvely I Table 4a Q ad Ψ are he rue values calculaed from he correc model e he copula ad sadard Normal margs The MC umercal sadard errors (due o fe umber of MC smulaos) for he calculaed VaR ad ES are very small (of he order of %) Relave perceage dffereces of hese rsk measures uder correc copula models agas he values from he copula are also gve Table 4a Table 4a Porfolo 99 VaRQˆ ad 99ES Ψˆ he case of sadard Normal margs Copula model Qˆ ( Q ˆ Q) / Q Ψˆ ( Ψ ˆ Ψ) / Ψ Gaussa copula 85-39% 475-5% copula -% 47-55% copula 337 % 74 % From Table 4a oe ca see ha he relave dfferece he 99 VaR bewee he copula ad he sadard copula cases s more ha % ad he correspodg dfferece he 99 ES s more ha 5% I s eresg o oe ha he sadard copula uder-esmaes he 99 quale (by 6%) eve relave o he Gaussa copula For he ES he copula gves a predco very close o ha of he Gaussa copula Boh Gaussa ad copulas uderesmae he ES cosderably comparso wh he copula I erms of boh he predced 8

19 values for he 99 VaR ad ES he dfferece bewee he copula ad he copula models s much larger ha he dfferece bewee he copula ad he Gaussa copula Ieresgly hs example he al depedece coeffce of he copula s 48 whle s 4 for he copula Despe havg a larger TDC he copula sll uderesmaes he 99 VaR ad ES For hs asymmerc porfolo appears he asymmerc propery of he copula play a more sgfca role ha he al depedece Addoally f he dfferece bewee ad he copula creases (e he asymmery creases) he uderesmao of rsk by he Gaussa ad copulas creases oo For example f we le 5 ad repea he above experme he uder-esmao of he porfolo 99 ES by boh he Gaussa ad he copulas s approxmaely % -margs Boh he quale ad expeced shorfall deped o margs as well as o he copula hus s eresg o explore he copula model mpac uder dffere margs ( calculaos he rue margs are used so ha mpac s sll due o correc copula model oly) Isead of Normal margs used he above example assume he sadard dsrbuo margs wh he same dof parameer e X () ad Y () Table 4b shows resuls for he 99 VaR ad ES usg -margs for dffere values No surprsgly he dfferece of 99 VaR ad ES bewee he ad sadard copulas s much more proouced wh he heaver aled - margs comparso wh he Gaussa margs (Table 4a) The mpac of copula model creases as decreases As he value of creases o 5 he -margs behave lke a Normal dsrbuo ad he values for 99 VaR ad ES are close o hose show Table 4a as expeced Table 4b Porfolo 99 VaRQˆ ad 99 ES Ψˆ he case of - margs Copula Qˆ ( Q ˆ Q) / Q Ψˆ ( Ψ ˆ Ψ) / Ψ copula % 89-8% copula 497 % 46 % 5 5 copula % 6-3% copula 898 % 676 % copula 9-6% 493-6% copula 363 % 777 % 6 Smulao sudy of fe sample properes of MLEs I he prevous example he sample sze was very large o eglec he uceray of parameer esmaes due o fe sample sze ad check he model rsk oly I real daa he sample sze s ofe much smaller ad oe should geeral be cauous applyg he MLE procedure For small sample sze he MLE procedure could lead o bas parameer esmao ad correc esmae of parameer uceray f asympoc resuls of Theorem 4 are used Wheher he sample sze f large eough for he asympoc resuls o be accurae s model ad daa depede Here we exame he fe sample behavour of he MLEs for he copula 9

20 Assume ha he rue model s he 9 C copula e he rue parameer vecor s θ ( ) (9 ) he same as he example Seco 6 Le us smulae ( j) ( j) ( u u ) j K ad calbrae he copula usg: ) jo ML procedure for all parameers ad ) Kedall s au approxmao for whle MLEs for Repea hs smulao-calbrao procedure N mes Deoe he -h parameer esmae as θ ˆ ( ) N The he bas of he esmaor θˆ ad s mea square error (MSE) for he sample sze K ca be esmaed as N N ( ) ( ) Bas θ[ θˆ] E[ θˆ] θ ( θˆ θ) MSE[ θˆ] E[( θˆ θ) ] ( θˆ θˆ) N N wh a well kow decomposo MSE[ θˆ] Var[ θˆ] + (Basθ[ θˆ]) For he prese sudy we use N 4 samples (so ha umercal error due o fe umber of samples s o maeral) ad hree sample szes were cosdered: K 5 8 The umercal error of he averages N due o he fe umber of samples (e he sadard devao of ˆ( ) / N θ ) ca be calculaed as Var[ ˆ] θ / N Table 5a Resuls of fe sample sze sudy usg Kedall s au esmaor for ad MLEs for ad K E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] Table 5b Resuls of fe sample sze sudy usg full jo ML esmao for ad K E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] E[ ˆ ] Var[ ˆ ] MSE[ ˆ ] Table 5c Average MLE varace ave(var mle [ θ ˆ]) over 4 samples for MLEs ˆ ˆ ad ˆ K ave(var mle [ θ ˆ]) ˆ ˆ ˆ

21 Table 5a shows resuls of he sudy whe Kedall s au approxmao s used for ad MLEs for Table 5b preses he resuls for jo ML calbrao of From daa Tables 5a ad 5b he magude of he relave bases δ (E[ ˆ] ) / δ (E[ ] ) / ad δ (E[ ] ) / ca be compued Fgure 7a shows he relave bas δ ad δ as a fuco of he sample sze usg daa from Table 5a e wh Kedall s au approxmao for Fgure 7b shows he same quaes for jo ML esmao of all parameers e correspodg o Table 5b The values of he average MLE varace over 4 samples ave(var mle [ θ ˆ]) for MLEs ˆ ˆ ad ˆ are gve Table 5c mle Here for each sample Var [ ˆ( ) θ ] N s esmaed usg he observed formao marx () From he resuls Tables 5a 5b ad 5c (also see Fgure 7a ad 7b) he followg observaos ca be made: For he correlao coeffce o sgfca bas s observed for MLE (see Table 5b) eve for a small sample sze A K 5 he relave bas δ 8% whch reduces o δ % a K ad s of he same order of magude as he umercal error due o fe N ; The bas of ˆ from he Kedall s au approxmao s very small eve for small sample szes (eg δ 6% for K 5 ) umercally valdag he use of Kedall s au approxmao he calbrao procedure The bas δ does o decrease as he sample sze creases reflecg he fac ha (3) s oly a approxmao he case of he copula The varace for ˆ sll decreases wh creasg sample sze The bas for ˆ ad ˆ s sgfca a small sample sze bu decreases reasoably rapdly wh creasg sample sze A K 8 δ s less ha % for eher he case of Kedall s au approxmao or full jo calbrao whle δ s less ha % The larger bas exhbed by he hgher dof ˆ s o expeced o affec applcao as sgfcaly as he bas of ˆ sce he -dsrbuo (copula or margal) s less sesve wh respec o he dof parameer whe s large The average ave(var mle [ θ ˆ]) from Table 5c ad Var[ θ ˆ] from Table 5b are: - almos decal he case of ˆ for all sample szes cludg K 5 ; - sgfcaly dffere he case of ˆ ad ˆ for small sample szes; - of smlar magude he case of ˆ ad ˆ for sample sze 8 Ths dcaes ha asympoc Gaussa approxmao () ceraly ca o be used o esmae uceraes of ˆ ad ˆ for small sample szes bu somewha jusfed for very large samples such as of he order of cosdered ex seco The rao Var [] / MSE[] for ˆ ˆ ad ˆ s large more ha 8 % mos cases reflecg ha he bas s relavely small comparso wh he varace or uceray of he parameer esmaes I oher words he mea square error s mosly due o varace of he esmaor

22 δ δ δ Fgure 7a Magude of relave bas δ δ ad δ as a fuco of sample sze from he fe sample sudy wh Kedall s au approxmao for ad MLEs for ad δ δ δ Fgure 7b Magude of relave bas δ δ ad δ as a fuco of sample sze from he fe sample sudy wh jo ML calbrao for ad 63 Smulao sudy of model rsk for small sample szes I seco 6 a dfferece bewee copula ad sadard copula s demosraed her dffere predcos o TDC VaR ad ES usg a large sample sze For small sample szes he esmaed parameers ad cosequely he predcos o TDC VaR ad ES vares cosderably from sample o sample bu sascally sgfca behavours ca be esmaed from may samples as s doe he prevous seco Usg each of he N 4 samples (he sze of each se s fxed a K ) geeraed he sudy descrbed Seco 6 we esmae all parameers joly va ML mehod for he ad sadard copulas respecvely The Moe Carlo mehod (smlar o Seco 6) s used o calculae he 99 VaR ad ES for hese parameer esmaes ad oba N esmaes for VaR ad ES for each copula model Fally he mea E [] ad varace Var [] ca be esmaed by sample mea ad varace over 4 (he sadard error of he mea due o he fe umber of samples N ca be esmaed by Var []/ N ) ( ) Deoe he 99 VaR ad ES for he sadard copula as ˆ ( ) Q ad Ψ ˆ For he copula ( ) we use he oaos ˆ ( ) Q ad ˆ ( ) Ψ Noe ˆ ( ) Q ad Q ˆ are for he same -h sample bu ( ) ( ) ( ) dffere models ad hus ca be srogly depede Le ˆ ˆ δ Q Q Q whch s he

23 dfferece bewee he 99 quales of he sadard copula ad copula based o he same ( ) ( ) ( ) -h sample of fe sze K Smlarly for he expeced shorfall we le ˆ ˆ δ Ψ Ψ Ψ ( ) Table 6a shows he mea ad sadard devao of ˆ ( ) Q ˆ () ( ) Q δ Q ˆ ( ) Ψ ˆ () Ψ ad δ Ψ N whe N 4 ad sample szes K 5 8 he case of he sadard Normal margs Table 6b shows he same summary sascs bu for sadard -margs wh he same dof 5 (oe of he cases Table 4b) Resuls Tables 6a ad 6b show ha he mea of he 99 quale ad expeced shorfall over 4 samples for all hree fe sample szes K 5 8 are close o hose show Table 4a ad 4b for he large sample sze of K 5 However for small sample szes he sadard devaos of δ Qˆ ad δ Ψˆ are relavely large comparso wh he values of δ Qˆ ad δ Ψˆ dcag ha he model dfferece s o sascally sgfca for small samples For sample sze K 8 he sadard devaos of δ Qˆ ad δ Ψˆ become small eough so ha he model dfferece s sascally sgfca ( s more proouced he case of -margs Table 6b) Table 6a Mea ad sadard devao of he 99 VaR Qˆ ad 99 ES Ψˆ over N 4 samples for sample szes K 5 8 he case of Normal margs Qˆ Q ˆ δ Qˆ Ψˆ ˆΨ δ Ψ K 5 mea sdev K mea sdev K 8 mea sdev Table 6b Mea ad sadard devao of he 99 VaRQˆ ad 99 ES Ψˆ over N 4 samples for sample szes K 5 8 he case of -margs wh 5 Qˆ Q ˆ δ Qˆ Ψˆ ˆΨ δ Ψ K 5 mea sdev K mea sdev K 8 mea sdev Example: Foreg Exchage daa The above smulao expermes demosraed ha he mpac of usg Gaussa or sadard copulas whe he rue copula s ca be sgfca I hs seco we cosder real foreg 3

24 exchage daly daa o explore f he copula fs beer ha he Gaussa or he sadard copulas The daly foreg exchage rae daa were dowloaded from he Federal Reserve Sascal Release (hp://wwwfederalreservegov/releases) These daly daa have bee cerfed by he Federal Reserve Bak of New York as he oo buyg raes New York Cy For our example we chose USD/AUD ad USD/JPY raes Followg commo pracce see eg McNel e al (5) we use he GARCH() model o sadardze he log-reurs of he exchage raes margally The GARCH() model calculaes he curre squared volaly σ as σ ω + α ( x μ) + βσ ω α β α + β < (7) where x deoes he log-reur of a exchage rae o dae I s modelled as x () μ +σ ε (8) () where μ s he asse drf ad ε s a sequece of d radom varables referred o as he resduals The GARCH parameers ω α ad β are esmaed usg he ML mehod The he ( ) ( ) GARCH flered resduals ε ad ε of he USD/AUD ad USD/JPY raes respecvely are used o f he Gaussa copula he sadard copula ad he copula For boh he sadard copula ad he copula we esmae correlao ad dof parameers joly usg ML mehod As a eresg comparso we also use Kedall s au approxmao for correlao ad ML for dofs I he laer case he correlao coeffce s ( ) ( ) ( ) ( ) approxmaed as ˆ s( πτˆ( ε ε ) / ) where ˆ( τ ε ε ) s Kedall s au of he resduals ad he he dof parameers are fed usg he ML mehod The Gaussa copula correlao coeffce was esmaed as a lear correlao of he resduals Before ML fg he resduals were rasformed oo () doma margally usg her emprcal dsrbuos of he resduals I he frs example we use he daly exchage raes from Jauary 3 o 7 Sepember 7 a oal of 8 daa pos Table 7a shows he resuls of jo ML fg wh sadard devaos gve brackes ex o he correspodg MLEs To assess he uceray of parameer esmaors oe ca geerae may samples usg paramerc or oparamerc boosrap see eg Efro ad Tbshra (994) ad esmae he uceray of he esmaors as he smulao expermes Seco 6 Here we chose a smpler approach ad esmae he MLE varaces as [ Iˆ (ˆ)] θ / K usg he observed formao marx () e assumg asympoc Gaussa approxmao of Theorem 4 These esmaes should be adequae for sample szes of he order of ad larger as dcaed by he resuls of smulao sudy Seco 6 The obaed esmaes of he wo dof parameers ˆ 6 ad ˆ 5 of he copula are very dffere (he dfferece s approxmaely whch s very close o he smulao example Seco 6) A formal Lkelhood Rao Tes for he ad copulas (esg hypohess ha ) dcaes a very srog rejeco of he copula favour of he copula e ad are sascally dffere (correspodg ch-square es sasc p-value s 6) Table 7b shows resuls for he same case as Table 7a bu wh Kedall s au approxmao for he correlao parameer Comparso of Table 7a ad 7b shows ha he 4

25 esmaed parameers by he wo mehods mos cases are decal o wo sgfca dgs cofrmg good accuracy of he Kedall s au smplfcao The same observao was made McNel e al (5) Table 7a Copula parameers joly fed o USD/AUD ad USD/JPY daa from Ja 3 o 7 Sep 7 usg ML mehod Copula Dof Log-lkelhood Gaussa ˆ 46 N/A 464 ˆ 48() 564 () 6395 ˆ 5() 6 (5) 5 (39 ) 6993 Table 7b Copula parameers fed o USD/AUD ad USD/JPY daa from Ja 3 o 7 Sep 7 wh Kedall s au approxmao for he correlao ad MLEs for dofs Copula Dof Log-lkelhood ˆ (93) 6386 ˆ (5) (36 ) 698 Table 8a Copula parameers joly fed o USD/AUD ad USD/JPY daa from Ja o 7 Sep 7 usg ML mehod Copula Dof Log-lkelhood Gaussa ˆ 33 N/A 44 ˆ 35() ˆ 56 (85) 489 ˆ 36() 89 (6) ˆ 34 (45) 4387 ˆ Table 8b Copula parameers fed o USD/AUD ad USD/JPY daa from Ja o 7 Sep 7 wh Kedall s au approxmao for he correlao ad MLEs for dofs Copula Dof Log-lkelhood ˆ 35 ˆ 563 (45) 484 ˆ 35 9 (58) ˆ 34 (43) 4386 ˆ I he secod example we used he daly exchage raes from Jauary o 7 Sepember 7 a oal of 934 sample pos The MLEs ad her sadard devaos are show Table 8a (for jo fg) Aga he wo dof parameers of he copula are very dffere ad he dfferece s approxmaely A formal Lkelhood Rao Tes dcaes ha ad are sascally dffere (he correspodg Ch-square es sasc p-value s 4) As a comparso Table 8b shows fg resuls for he same daa bu usg Kedall s au approxmao for Oce aga he dfferece bewee he wo fg mehods s que small I s eresg o oe ha he case of -copula he error esmaed by ML mehod for ˆ s larger from he jo esmao ha from usg he Kedall s au approxmao 5

26 The above wo examples are gve for llusrao purposes oly Of course a accurae modellg of exchage rae dyamcs ca be more volved (eg me depede depedece srucure mgh be requred) bu goes beyod he scope of hs paper 7 Cocluso I hs paper we roduced ad suded he copula wh mulple dof parameers referred o as he copula Ths copula ca be regarded as a grouped copula where each group has oly oe member I has he advaages of a grouped copula flexble modellg of mulvarae depedeces ye a he same me overcomes he dffcules wh a pror choce of groups Some characerscs of hs copula bvarae case are dffere from hose of he sadard copula eg he copula s asymmerc respec o he u u axs ad he al depedece mpled by he copula depeds o boh dof parameers The al depedece s derved closed form for he copula The dfferece bewee ad sadard copulas erms of mpac o VaR ad ES of he porfolo ca be sgfca as demosraed by smulao expermes ad fg real FX daa bvarae case The porfolo VaR ad ES s show o be depede o boh copula ad margal models Sudes o he hgher dmesoal cases wll be carred ou furher work I would be eresg o see f he mpac of msrepreseg wh a sadard copula or a grouped copula ca be more proouced ha he bvarae case Sudy o fe sample properes of ML esmaor for he copula shows he Kedall s au approxmao has a small bas For dof parameers he bas due o fe sample sze ca be sgfca bu reduces farly rapdly wh creasg sample sze The sascal ad physcal ( erms of VaR ad ES) dfferece bewee copula ad sadard copula has also bee suded wh small sample szes by usg summary sascs over may depede daa samples Ths showed ha large bases dofs (for large values of dof) do o ecessarly roduce bas VaR ad ES Smulao procedure of he copula s very smple bu calbrao procedure s compuaoally more demadg ha he case of sadard copula Ths s because he copula parameers (a leas dof parameers) should be esmaed joly ad calculao of he copula desy volves d umercal egrao I he examples of fg o USD/AUD ad USD/JPY daly daa sadard copula was sascally rejeced favour of copula (e dof parameers he copula were sascally dffere) The sadard copula ad he grouped copula are subses of he copula Thus he laer ca be used for model seleco purposes (e seleco of rsk groups wh he same dof copula parameer) Effce model seleco ad parameer esmao for he copula he Bayesa ferece framework usg Markov cha Moe Carlo mehods are opcs of furher sudy Flexble modellg ca also be acheved usg skewed copulas see Demara ad McNel (5) of dsrbuos kow as mea-varace mxures Z μ + γg ( W ) + WX Here W s some radom varable μ ad γ are parameer vecors ad g ( ) s some fuco [ ) [ ) These skewed copulas ca be geeralzed by allowg W be a vecor smlar o he way we geeralzed he copula o he copula Fally we would lke o remark ha afer submsso of hs mauscrp we were made aware of he revew paper by Veer e al (7) where a possbly of he proposed copula was meoed Also a rece paper by Baachewcz ad Vaar (8) he formula for he 6

27 al depedece s correc for he case of copula due o a error equao (4) her paper (we provde a correc oe (6) wh he proof Appedx B) Ackowledgeme We would lke o hak Ross Sparks Gareh Peers ad wo aoymous referees for may cosrucve commes whch have led o mprovemes he mauscrp Appedx A Here we prove ha he explc formula (8) s deed he cdf of copula defed by s sochasc preseao (6-7) ad he desy of s gve by (9) I addo we show ha f all dofs are equal copula reduces o he sadard copula Usg Defo see Seco he copula wh mulple dofs k k s defed as he dsrbuo of radom vecor U ( ( X) ( X )) where X ( W Z W Z ) The dsrbuo of he copula ca be calculaed as follows Sce Z s from mulvarae Normal dsrbuo he codoal desy of X gve S (he radom vecor W W W W ) s kow oce S s gve) s a mulvarae Normal oo ( ϕ x S ) ϕ ( x / w x / w ) /( w w ) (A) ( Σ see also Defo 3 for ϕ Σ ad w k Gve ha S has uform () dsrbuo ad s depede from Z he ucodoal desy of X s he ad he cdf of X s H ϕ( x s) ds ϕσ( x / w x / w ) /( w ϕ ( x) w ) ds (A) x x x Σ ( x) ϕ ( x / w x / w ) ϕ ( x) dx dx dxds (A3) w w Iroducg a ew varable z x / w x / w ) (A3) ca be smplfed as ( x / w x / w Σ( z z) dz dzds Φ H ( x) ϕ ( x / w x / w ds (A4) Σ ) Usg () he copula dsrbuo (8) s readly obaed from (A4) by replacg x k wh ( u k ) Takg dervaves of (8) wh respec o u he desy fuco of he copula (9) s easly foud 7

28 Equaos (A) (8) ad (9) are also vald for he specal cases of grouped copula ad he sadard copula We show below ha f all he dofs are equal he (A) rasforms o he famlar sadard mulvarae -dsrbuo desy ad copula becomes a sadard copula Obvously he subscrp k relaed o k ca be dropped so xk / wk ( uk ) / w( s) wh w( s) / χ ( s) Chagg he egral varable (A) from s o s Ch square verse fuco χ ( s) (A) ca be re-wre as / / / e ϕ( x ) ϕσ( x / x / )( / ) d (A5) / Γ( / ) / / where ( ) /( w w ) ad e / / / /[ Γ( / )] ds / d Subsug he Normal desy ϕ Σ( z) exp( z Σ z) / (π ) de Σ s he Ch square desy o (A5) ad smplfyg oba + ( x / ) Σ ( x / ) d ϕ( x ) C exp (A6) / ( + ) / where he cosa s [ ] C ( π ) de Σ Γ( / ) Iroducg a ew varable y ( + x Σ x / ) / (A6) s rasformed o Recogsg ha he egrao ad og ha ( x / ) Σ ( x / ) ( / ) x Σ x ( + ) / y ( + ) / ( + x Σ x / ) e y dy ( ) / ϕ( x ) C (A7) + e y y ( + ) / dy s he Gamma fuco Γ(( + ) / ) subsug he cosa C ad smplfyg (A7) fally becomes he famlar expresso for he mulvarae sadard -dsrbuo desy fuco ϕ( (( + ) / ) ( + ) / Γ x ) / + x Σ x (A8) de Σ ( π ) Γ( / ) The sadard copula () ca ow be cosruced usg () Appedx B From (5) we have (see Seco ad 3 for defos ad oaos) λl ( ) lm Φ ( ) Φ ( ) + z( q s) z( q s) ds lm+ z( q s) z( q s) ds (B) q q q q 8

29 q Φ ( z ( q s) z ( q s) ) ds ds + ds I + I Φ z z q Φ z z q For bvarae ormal Φ ( z z) Φ / z (π ) exp( z / ) FN (( z z) / ) where F N () s sadard ormal dsrbuo Recall he defos z( q s) ( q) / w ( s) x ( q ) w / y ad y χ ( s) we fd z q [ w ( s) f ( x )] Thus / z Φ z y I ds e FN (( z z) / ) ds (B) z q π f ( x ) Chagg varable from s o y y he ds g ( y) dy where g () s he Ch square desy he I ( x y / x y / ) / ) x y g ( y) y e FN π f ( x) dy (B3) As q x x from lower al of -dsrbuo we have x C ( x ) / Γ[( + ) / ] Γ( / ) C ( / ) [( ) / ] Γ Γ + / ( ) /( ) ( ) /( ) From lower al of Ch square dsrbuo we have y χ So (B3) uder he lm I + + / / / / ( χ ( y)) C y C [ ( )] [ ( )] x becomes Γ / ( C C ( x y) / x y / )/ ) x y g ( y) y e FN π f ( x ) dy (B4) Γ Chagg varable x y / x ( / y) / d ( x / ) dy akg lm ad smplfyg gve lm I Ω + ( ) g + q / ( ( B ) / ) ( ) F d N (B5) where / / / Γ[( + ) / ] B C C / / Γ[( + ) / ] Smlarly we fd lm I Ω( ) e he same fuco bu wh swapped Thus + q λl ( ) Ω( ) + Ω( ) (B6) If he B ad (B5) reduces o 9

30 ( d / + ) d ( ( + )( ) / ) Ω + ( ) g + ( ) FN + (B7) gvg he well kow resul for sadard copula λl ( ) Ω( ) + ( ( + )( ) / + ) (B8) Refereces Aas K C Czado A Frgess ad H Bakke (7) Par-copula cosrucos of mulple depedece Isurace: Mahemacs ad Ecoomcs I press do:6/jsmaheco 7 Baachewcz K ad A Vaar (8) Tal depedece of skewed grouped -dsrbuos Sascs & Probably Leers Bedford T ad R M Cooke () Probably desy decomposo for codoally depede radom varables modelled by ves Aals of Mahemacs ad Arfcal Iellgece Bedford T ad R M Cooke () Ves - a ew graphcal model for depede radom varables Aals of Sascs 3 (4) 3 68 Breyma W A Das ad P Embrechs (3) Depedece srucures for mulvarae hghfrequecy daa face Quaave Face 3-4 Casella G ad RL Berger () Sascal Iferece Pacfc Grove CA: Duxbury Cherub U E Lucao ad W Vecchao (4) Copula Mehod Face Joh Wley & Sos Ld Daul S E De Gorg F Ldskog ad A McNel (3) The grouped -copula wh a applcao o cred rsk RISK Demara S ad A McNel (5) The copula ad relaed copulas Ieraoal Sascal Revew 73() -9 Efro BF ad DV Hkley (978) Assessg he accuracy of he maxmum lkelhood esmaor: observed versus expeced Fsher formao Bomerka 65: Efro BF ad RJ Tbshra (994) A Iroduco o he Boosrap Chapma & Hall New York Embrechs P A McNel ad D Srauma () Correlao ad depedece rsk maageme: properes ad pfalls I Rsk Maageme: Value a Rsk ad Beyod Eds M Dempser ad H Moffa 76-3 Cambrdge Uversy Press Fag H K Fag ad S Koz () The mea-ellpcal dsrbuos wh gve margals Joural of Mulvarae Aalyss 8-6 Joe H (997) Mulvarae Models ad Depedece Coceps Chapma&Hall Lodo Lehma EL (983) Theory of Po Esmao Wley 3

31 Luo XL PV Shevcheko ad JB Doelly (7) Addressg he mpac of daa rucao ad parameer uceray o operaoal rsk esmaes J Operaoal Rsk (4) 3-6 Mashal R M Nald ad A Zeev (3) O he depedece of equy ad asse reurs RISK McNel AJ (8) Samplg esed Archmedea copulas Joural of Sascal Compuao ad Smulao 78: McNel AJ R Frey ad P Embrechs (5) Quaave Rsk Maageme Coceps Techques ad Tools Prceo Uversy Press New Jersey Pesses R E De Docker-Kapega CW Überhuber ad DK Kahaer (983) QUADPACK a Subroue Package for Auomac Iegrao Sprger-Verlag Veer G J Bare R Kreps ad J Major (7) Mulvarae Copulas for Facal Modelg Varace : 4-9 3