Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulas

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1 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Prof. Dr. Detmar Pfefer Dept. of Mathematcs, Unversty of Olenburg an AON Benfel, Hamburg Dr. Doreen Straßburger Dept. of Mathematcs, Unversty of Olenburg an mgm consultng partners, Hamburg Jörg Phlpps Dept. of Mathematcs, Unversty of Olenburg e-mals: etmar.pfefer@un-olenburg.e oreen.strassburger@un-olenburg.e joerg.phlpps@un-olenburg.e Abstract: In ths paper we revew Bernsten an gr-type copulas for arbtrary mensons an general gr resolutons n connecton wth screte ranom vectors possessng unform margns. We further suggest a pragmatc way to ft the epenence structure of multvarate ata to Bernsten copulas va gr-type copulas an emprcal contngency tables. Fnally, we scuss a Monte Carlo stuy for the smulaton an PML estmaton for aggregate epenent losses form observe wnstorm an floong ata.

2 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas. Introucton The use of copulas for moellng an smulaton purposes especally n nonlfe nsurance (nternal moels) has attane ncreasng nterest n the recent years, see e.g. [3], Chapter 5 an the references gven there. However, the scusson of potental copula moels has so far mostly focusse on ether the ellptcal case (e.g. Gaussan an t-copula) or the Archmean case (e.g. Gumbel-, Clayton-, Frank-copula an others). Although the use of Bernsten polynomals n one an more varables or n one an more mensons (especally Bézer curves an surfaces) has a long traton n numercal analyss an computer ae esgn, t seems that the true mpact of Bernsten polynomals on copula moels has been scovere only more recently, frst n the framework of approxmaton theory (see e.g. [], [5], [7], [8]) an later n partcular n connecton wth applcatons n fnance (see e.g. [2], [3], [0], []). Bernsten copulas possess several benefts compare to the tratonal approaches: Bernsten copulas allow for a very flexble, non-parametrc an essentally non-symmetrc escrpton of epenence structures also n hgher mensons Bernsten copulas approxmate any gven copula arbtrarly well Bernsten copula enstes are gven n an explct form an can hence be easly use for Monte Carlo smulaton stues. In ths paper, we take a specal smple look on the constructon of Bernsten copulas through screte ranom vectors wth unform margns, an pont out ther connecton to gr-type copulas scusse n [4] (also calle checkerboar copulas n [2] an [5]). Ths vew whch can also be foun n [2] an [5], however restrcte to the bvarate case, opens a pragmatc approach to ft the epenence structure of observe ata to Bernsten copulas va gr-type copulas an (multvarate) contngency tables. As an example, we present a Monte Carlo stuy on the aggregate rsk strbuton for epenent wnstorm an floong losses. 2. Some smple mathematcal facts on Bernsten polynomals an copulas Lemma. Let Further, m k m k Bmkz (,, ) = z( z), 0 z, k = 0,, m. Then we have k mb( m, k, z) z= for k = 0,, m. 0 B( mkz,, ) = mbm [ (, k, z) Bm (, kz), ] for k = 0,, m z wth the conventon Bm (,, z) = Bm (, mz, ) = 0.

3 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Proof: 0 m m ( k ) ( m mb( m, k, z) z m Beta( k, m k) m Γ + Γ k) = + = k k Γ ( m + ) mm ( )! k!( m k )! = =. k!( m k )! m! Further, for 0 < k< m, m m Bmkz (,, ) = k z ( z ) ( m k ) z ( z ) z k k k m k k m k m m = m z ( z) m z ( z) k k k ( m ) ( k ) k m k [ (,, ) (,, )] = m B m k z B m k z whch, by the above conventon, also hols for k { m} 0,. Theorem. For let U = U,, U be a ranom vector whose margnal component U follows a screte unform strbuton over T : = 0,,, m wth, =,,. Let further enote Then ( ) p k k P U k (,, ): = { = } for all ( m m = { } ) m k,, k T. = cu (,, u) : = pk (,, k) mbm (, k, u), ( u,, u) [ 0, k= 0 k = 0 = efnes the ensty of a -mensonal copula, calle Bernsten copula. We call c the Bernsten copula ensty nuce by U. ] Proof. For fxe j we obtan, accorng to the Lemma above, m m (,, ) j = (,, ) (,, ) j ( j, j, j) cu u u pk k mbm k u mbm k u u 0 k= 0 k = 0 = 0 j m m (,, ) (,, ) = pk k mbm k u k= 0 k = 0 = j j 2

4 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas for ( u u u u ) [ ],,,,, 0, j j+ m mj mj+ m mj = p( k,, k ) mb( m, k, u k= 0 kj = 0 kj+ = 0 k= 0 kj= 0 = j m mj mj+ m = P { U k } = mb( m, k, u k= 0 kj = 0 kj+ = 0 k= 0 = = j j ( j j+ ) [,, j, j+,, ] = : c u,, u, u,, u [,,,.e. j, j +,, ] c s also a Bernsten copula, but of menson nstea of. (Note that for j =, the symbol [,, j, j+,, ] reas [2,, ], lkewse for j =, corresponngly for the vectors of varables.) Contnung ntegraton accorng to the remanng varables except for the varable for fxe r, we en up wth mr [ r] c ur c u u u ur ur+ u P Ur kr mrb mr k u 0 0 kr = 0 for all ( ) = (,, ) = ( = ) (,, ) r r m =,, =,, = ( ) = mr mr mr r k m k mrb( mr kr ur) B( mr kr ur) ur ur kr 0m = r kr= 0 k= 0 k unform strbuton over ] u [0, r whch proves that the r-th margnal ensty of c s that of a contnuous [ ] 0,, for every r. u r ) ) Note that the lne of proof above shows that f = ( V V ) V,, s a ranom vector wth jont Bernsten copula ensty c as above, then also any partal ranom vector ( V V ) n< an < < n possesses a Bernsten copula ensty [,, c ] n gven by,, wth n m m ( ) n n n [,, ] n n c u, u { } (,, ), (,, ) [ 0, ]. = n P U = k m B m k u u u n k = 0 k = 0 = = n Ths means that the Bernsten copula ensty ranom vector ( U,, ). U n [,, n ] c s just the ensty nuce by the partal By ntegraton, we obtan the Bernsten copula C nuce by U as x x m m C( x,, x ): = c( u,, u ) u u = P { U < k } B( m, k, x ), 0 0 k= 0 k = 0 = = 3

5 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas for ( x x ) [ ],, 0,. Ths can be verfe by partal fferentaton of C, usng the above Lemma, an some rearrangements n the summaton: m m C( x,, x ) = P { U k } < B( m, k, x ) m B( m, k, x ) B( m, k, x ) u r r r r r r r r k= 0 k = 0 = = r m m = P U k U k < = B m k x m B m k x whch, by teraton, fnally leas to { } { } (,, ) (,, ) r r r r r r k= 0 k = 0 = = r r u u C( x,, x ) = c( x,, x ), ( x,, x ) [ 0,]. There s also a natural relatonshp between Bernsten an gr-type copulas as scusse n [2], [5] an [4]. We refer to a slghtly more general setup here. Defnton. Uner the assumptons of the above theorem efne the ntervals kj kj+ Ik :,,, k = j= mj m for all possble choces ( k,, k) T. Then the functon = j m m * c : m p( k,, k ) = = k= 0 k = 0 Ik,, k s the ensty of a -mensonal copula, calle gr-type copula nuce by U. Here enotes the ncator ranom varable of the set A, as usual. A A natural nterpretaton of ths copula s as follows: a ranom vector W = ( W W ) follows a gr-type copula ff the contonal strbuton fulflls the contons ( U = (,, ) ) = U ( k, ) W P k k I,k for all ( k,, k ) T, =,, where U ( B) enotes the contnuous unform strbuton over a -mensonal Borel set B wth postve Lebesgue measure an ( k,, k) Ik U= W,,k (.e., U enotes n some sense the coornates of W w.r.t. the gr nuce by the ). Ik,, k Hence the Bernsten copula nuce by U can be regare as a naturally smoothe verson of the gr-type copula nuce by U, replacng the ncator functons 4

6 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas ( u,, u ) ( u ) by the polynomals Bm ( k u) ( u u) [ ] = Ik,, k k k, + = m m =,,,,, 0,. Example. The followng graphs show the smoothng effect n case =. m = 5 m = 0 Natural generalzatons of Bernsten an gr-type copulas are obtane f we look at sutable parttons of unty,.e. famles of non-negatve functons { φ( mk,, ) 0 k m, m } efne on the unt nterval [ 0, ] wth the followng propertes (see e.g. [5] or [7]): φ ( mku,, ) u= for k = 0,, m m 0 m φ( mk,, ) = for m. k= 0 In ths case, a -mensonal copula ensty c φ nuce by U s gven by m m φ c ( u,, u) : = P { U = k} mφ ( m, k, u), ( u,, u ) [ 0,. ] k= 0 k = 0 = = The copula tself s accorngly gven by m m φ C ( u,, u) : = P { U < k} φ ( m, k, u), ( u,, u) [ 0,]. k= 0 k = 0 = = (cf. [2] an [5] for the bvarate case wth m = m ). 2 5

Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Note that m k φ( mku,, ) = Bm (, ku, ) = u( u) k m k n case

Suppose that for menson = 2 the jont strbuton of U = ( U, U 2 ) s gven by the followng table.

7 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Note that m k φ( mku,, ) = Bm (, ku, ) = u( u) k m k n case of Bernsten copulas an = ( mku,, ) k k φ +, m m ( u) for 0 k m, m n case of gr-type copulas. Example 2. Suppose that for menson = 2 the jont strbuton of U = ( U, U 2 ) s gven by the followng table. ( U = ( j) ) P, ,02 0 0,08 0,5 0 0,03 0,2 0,0 j 2 0,3 0,07 0, ,0 0,5 0 0 The graphs below show jontly the gr-type an the Bernsten copula ensty nuce by U. 3. Fttng emprcal ata to gr-type an Bernsten copulas In ths secton, we shall restrct ourselves to the case = 2, metho propose here works accorngly n any menson. for smplcty. However, the Suppose that a bvarate ata set of observatons s gven, for nstance a tme seres of (economcally ajuste) wnstorm an floong losses. One possble way to extract the 6

Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas epenence structure from the ata s the emprcal copula scatterplot, whch s a plot of the jont relatve ranks of the ata.

8 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas epenence structure from the ata s the emprcal copula scatterplot, whch s a plot of the jont relatve ranks of the ata. The followng fgure shows such a plot for a seres of n = 34 observaton years. ranks of wnstorm losses In a frst step, we want to ft these ata to a gr-type copula wth a gven gr resoluton, say m = m2 = m= 0. Countng the relatve frequency of the ata ponts n each of the m m2 = 00 target cells, we obtan the followng contngency table a j (matrx notaton: = row nex, j = column nex; roune to 3 ecmal places). upper cell bounary 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9,0 sum,0 0,000 0,000 0,000 0,000 0,000 0,029 0,029 0,029 0,000 0,000 0,009 0,9 0,029 0,000 0,000 0,000 0,000 0,000 0,029 0,000 0,000 0,029 0,009 0,8 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,029 0,029 0,059 0,02 0,7 0,000 0,029 0,000 0,000 0,000 0,029 0,000 0,000 0,029 0,000 0,009 0,6 0,000 0,029 0,029 0,029 0,000 0,000 0,000 0,029 0,000 0,000 0,02 0,5 0,000 0,029 0,000 0,029 0,000 0,000 0,000 0,000 0,029 0,000 0,009 0,4 0,029 0,000 0,000 0,000 0,029 0,029 0,029 0,000 0,000 0,000 0,02 0,3 0,000 0,000 0,000 0,059 0,000 0,000 0,000 0,029 0,000 0,000 0,009 0,2 0,029 0,029 0,000 0,000 0,059 0,000 0,000 0,000 0,000 0,000 0,02 0, 0,000 0,000 0,059 0,000 0,000 0,029 0,000 0,000 0,000 0,000 0,009 sum 0,009 0,02 0,009 0,02 0,009 0,02 0,009 0,02 0,009 0,009 Obvously, the observe margnal sums are not equal to = We therefore conser the m 0. followng optmzaton problem, n orer to approxmate the contngency table a j by a unform contngency table x j : 7

9 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas k = j= mn! m m 2 ( xj aj) subject to = j= m m x = x j = = an x, 0 for k, =,, m m 0 k The explct soluton of such a problem s n general not straghtforwar to fn, although there exsts a soluton ue to the Karush-Kuhn-Tucker theorem from optmzaton theory. Usng a sutable software package lke octave (a publc oman computer algebra system), we obtan the followng soluton (roune to 3 ecmal places); see the coe lstng n the Appenx. upper cell bounary 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9,0 sum,0 0,005 0,00 0,003 0,000 0,005 0,026 0,03 0,026 0,00 0,00 0, 0,9 0,034 0,000 0,002 0,000 0,004 0,000 0,030 0,000 0,00 0,030 0, 0,8 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,020 0,025 0,054 0, 0,7 0,005 0,030 0,002 0,000 0,005 0,025 0,00 0,000 0,03 0,00 0, 0,6 0,00 0,027 0,028 0,02 0,00 0,000 0,000 0,022 0,000 0,000 0, 0,5 0,005 0,030 0,002 0,025 0,005 0,000 0,00 0,000 0,03 0,00 0, 0,4 0,028 0,000 0,000 0,000 0,028 0,09 0,025 0,000 0,000 0,000 0, 0,3 0,005 0,00 0,003 0,054 0,005 0,000 0,002 0,026 0,002 0,002 0, 0,2 0,03 0,009 0,000 0,000 0,042 0,004 0,009 0,005 0,009 0,009 0, 0, 0,005 0,00 0,06 0,000 0,005 0,025 0,00 0,000 0,00 0,00 0, sum 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, A more pragmatc way to fn at least a goo suboptmal soluton that can be easly mplemente e.g. n spreasheets s as follows. Conser the above optmzaton problem wthout the non-negatvty contons frst. The equvalent Lagrange problem (whch leas to a system of lnear equatons) s easy to solve an gves the (general) soluton x j a a a = a + + for, j=,, m, j j m m 2 m m where the nex means summaton, as usual. For the ata set above, we thus obtan 8

10 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas upper cell bounary 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9,0 sum,0 0,002-0,00 0,002-0,00 0,002 0,029 0,032 0,029 0,002 0,002 0, 0,9 0,032-0,00 0,002-0,00 0,002-0,00 0,032-0,00 0,002 0,032 0, 0,8-0,00-0,004-0,00-0,004-0,00-0,004-0,00 0,026 0,029 0,058 0, 0,7 0,002 0,029 0,002-0,00 0,002 0,029 0,002-0,00 0,032 0,002 0, 0,6-0,00 0,026 0,029 0,026-0,00-0,004-0,00 0,026-0,00-0,00 0, 0,5 0,002 0,029 0,002 0,029 0,002-0,00 0,002-0,00 0,032 0,002 0, 0,4 0,029-0,004-0,00-0,004 0,029 0,026 0,029-0,004-0,00-0,00 0, 0,3 0,002-0,00 0,002 0,058 0,002-0,00 0,002 0,029 0,002 0,002 0, 0,2 0,029 0,026-0,00-0,004 0,058-0,004-0,00-0,004-0,00-0,00 0, 0, 0,002-0,00 0,06-0,00 0,002 0,029 0,002-0,00 0,002 0,002 0, sum 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Seemngly, ths soluton s not feasble snce t contans negatve entres. A smple way to a: = mn x, j m an overcome ths problem s a cell-wse atve correcton wth { j } xj + a consecutve normng; the fnal resultng contngency table y j = s gven by + a m upper cell bounary 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9,0 sum,0 0,004 0,002 0,004 0,002 0,004 0,024 0,026 0,024 0,004 0,004 0, 0,9 0,026 0,002 0,004 0,002 0,004 0,002 0,026 0,002 0,004 0,026 0, 0,8 0,002 0,000 0,002 0,000 0,002 0,000 0,002 0,022 0,024 0,046 0, 0,7 0,004 0,024 0,004 0,002 0,004 0,024 0,004 0,002 0,026 0,004 0, 0,6 0,002 0,022 0,024 0,022 0,002 0,000 0,002 0,022 0,002 0,002 0, 0,5 0,004 0,024 0,004 0,024 0,004 0,002 0,004 0,002 0,026 0,004 0, 0,4 0,024 0,000 0,002 0,000 0,024 0,022 0,024 0,000 0,002 0,002 0, 0,3 0,004 0,002 0,004 0,046 0,004 0,002 0,004 0,024 0,004 0,004 0, 0,2 0,024 0,022 0,002 0,000 0,046 0,000 0,002 0,000 0,002 0,002 0, 0, 0,004 0,002 0,048 0,002 0,004 0,024 0,004 0,002 0,004 0,004 0, sum 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, Note that ths matrx was use to fee the octave workng sheet as an ntal soluton. The quaratc error between the contngency table y j an the orgnal a j s 0,00279 whle the quaratc error for the optmal soluton s 0, an hence only very slghtly smaller. For the remaner of the paper, we shall therefore use the contngency table y j, for smplcty; the optmal contngency table wll prouce manly the same results here. 9

ranom vector U = ( U, U nucng the gr-type an Bernsten copulas, smlar as n

emprcal copula scatterplot, we have to efne ( ) PU, U2 j ym, j + = = = for,

jont plot of the gr-type an Bernsten copula ensty nuce by U plot of the

11 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas The table y j can be use to efne the jont strbuton of the screte ranom vector U = ( U, U nucng the gr-type an Bernsten copulas, smlar as n [2]. Note that n 2) orer to obtan a physcally correct corresponence to the emprcal copula scatterplot, we have to efne ( ) PU, U2 j ym, j + = = = for, j= 0,, m. The followng graphs show the resultng copula enstes. jont plot of the gr-type an Bernsten copula ensty nuce by U plot of the Bernsten copula ensty contour plot of the Bernsten copula ensty, wth orgnal scatterplot supermpose 0

12 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas 4. Smulatng from Bernsten copulas Snce Bernsten copula enstes are polynomals n varables, they are boune over the unt cube [ 0, by a constant M > 0, say whch makes a stochastc smulaton qute easy. ] The most convenent way s an applcaton of the multvarate acceptance-rejecton metho (see e.g. [4], secton 2.5.): Step : generate + nepenent unformly strbute ranom numbers u,, u +. Step 2: check whether cu (,, u) > Mu. + If so, go to Step 3, otherwse go to Step. Step 3: use ( u,, u ) as a sample from the Bernsten copula. The average rate of samples obtane by ths proceure s / M, as usual. Note that n our example, M = 2,35 s suffcent. From the 34 year tme seres of the logarthms of wnstorm an floong losses above the followng margnal strbutons were estmate, on the bass of a Q-Q-plot ( μ = locaton parameter, σ = scale parameter): Q-Q-plots for log losses; left: wnstorm, rght: floong Log wnstorm losses Log floong losses Dstrbuton Gumbel Normal Parameters μ = 6,367 μ = 6,625 σ = 0,8872 σ = 0,9777 I.e., the wnstorm losses are consere to be Fréchet strbute wth extremal nex α= / σ=,27 an the floong losses are consere to follow a lognormal strbuton. The followng graphs show the results of a fourfol Monte Carlo smulaton for the aggregate rsk (wnstorm an floong) on the bass of 000 pars of ponts smulate from Bernsten copulas accorng to secton 3 an the margnal strbutons specfe above. The four cases consere are:

13 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas re lne: Bernsten copula on the bass of a 4 x 4 gr (smlar to secton 3) green lne: Bernsten copula on the bass of a 0 x 0 gr (exact ata from secton 3) blue lne: nepenence case orange lne: Gaussan copula estmate from orgnal ata Dscusson: All four copula moels are qute close n the range of a return pero of 50 years. Sgnfcant fferences occur for hgher return peros. It s nterestng to observe that the PML estmates on the bass of Bernsten copulas le between the nepenence case (lower boun) an the Gaussan copula (upper boun) n the range of 60 to 95 years return pero. The two Bernsten copula approaches are surprsngly close n the range up to a return pero of 00 years, although the copula enstes are clearly fferent. 2

Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas 4x4 gr 0x0 gr contour plot of Bernsten copula enstes Note that the coarser gr

Ths effect results n a substantally hgher PML estmate for return peros above 70 years for the Bernsten copula on the 4x4 gr, even hgher than the Gaussan copula

Usng Bernsten or gr-type copulas for moellng epenence structures gves n general a better ft to local unsymmetres than other (classcal) copulas can acheve, but a

14 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas 4x4 gr 0x0 gr contour plot of Bernsten copula enstes Note that the coarser gr prouces a sngle peak of the ensty n the rght top corner whle the fner gr prouces two stnct peaks there. Ths effect results n a substantally hgher PML estmate for return peros above 70 years for the Bernsten copula on the 4x4 gr, even hgher than the Gaussan copula above 200 years return pero. Concluson. Usng Bernsten or gr-type copulas for moellng epenence structures gves n general a better ft to local unsymmetres than other (classcal) copulas can acheve, but a goo compromse has to be foun between the number of ata ponts an the unerlyng gr resoluton. Also, as s ponte out n [2], both types of copulas show a zero upper tal epenence snce the enstes are boune. However, snce Monte Carlo stues as performe here are always fnte, ths problem can be reuce for practcal purposes by choosng a hgher gr resoluton. 3

15 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Acknowlegement. We woul lke to thank Lena Reh for some stmulatng scussons on the topc of copulas an pontng out some of the references to us. Appenx octave source coe for the KKT-optmzaton problem from secton 3 functon x=bernstenopt(a,x) %A s the contngency table [a_j] obtane by the ata an x s the ntal value. %A possble ntal value s the approxmatve soluton presente n the paper m=length(a); %reshapng matrces to vectors a=-vec(a); X0=vec(x); % postvty lb=[:m^2]'*0; % equalty constrant b=[2:2*m]'*0+/m; % equalty constrant column sum= B=[ ]; for =:m Bnew=[ ]; for j=2: Bnew=[Bnew,[:m]*0]; en Bnew=[Bnew,[:m]*0+]; for j=+:m Bnew=[Bnew,[:m]*0]; en B=[B;Bnew]; en % equalty constrant row sum= for =2:m Bnew=[ ]; C=[:m]*0; C()=; for j=:m Bnew=[Bnew,C]; en B=[B;Bnew]; en % octave quaratc optmzaton tool [X, OBJ, INFO, LAMBDA] = qp (X0, eye(m^2), a, B, b, lb, [ ], [ ], eye(m^2), [ ]); %reshapng vectors to matrces x=reshape(x,m,m); 4

16 Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas References [] T. BOUEZMARNI, J. V.K. ROMBOUTS, A. TAAMOUTI (2008): Asymptotc propertes of the Bernsten ensty copula for epenent ata. CORE scusson paper 2008/45, Leuven Unversty, Belgum. [2] V. DURRLEMAN, A. NIKEGHBALI, T. RONCALLI (2000): Copulas approxmaton an new famles. Groupe e Recherche Opératonelle, Crét Lyonnas, France, Workng Paper. [3] V. DURRLEMAN, A. NIKEGHBALI, T. RONCALLI (2000): Whch copula s the rght one? Groupe e Recherche Opératonelle, Crét Lyonnas, France, Workng Paper. [4] D. PFEIFER, D. STRAßBURGER (2005): Depenence matters! Paper presente at the 36th Internatonal ASTIN Colloquum, September 4 7, 2005, ETH Zürch. [5] T. KULPA (999): On approxmaton of copulas. Internat. J. Math. & Math. Sc. 22, [6] A. MCNEIL, R. FEY, P. EMBRECHTS (2005) : Quanttatve Rsk Management. Concepts, Technques, Tools. Prnceton Unversty Press, Prnceton, N.J. [7] X. LI, P. MIKUSIŃSKI, H. SHERWOOD, M.D. TAYLOR (997): On approxmaton of copulas. In: V. Beneš an J. Štěpán (Es.), Dstrbutons wth Gven Margnals an Moment Problems, Kluwer Acaemc Publshers, Dorrecht. [8] X. LI, P. MIKUSIŃSKI, H. SHERWOOD, M.D. TAYLOR (998): Strong approxmaton of copulas. J. Math. Anal. Appl. 225, [9] R.Y. RUBINSTEIN, D.P. KROESE (2008) : Smulaton an the Monte Carlo Metho. Wley, N.Y. [0] SANCETTA, S.E. SATCHELL (2004): The Bernsten copula an ts applcatons to moellng an approxmatons of multvarate strbutons. Econometrc Theory 20(3), [] M. SALMON, C. SCHLEICHER (2007): Prcng multvarate currency optons wth copulas. In: Copulas. From Theory to Applcaton n Fnance, J. Rank (e.), Rsk Books, Lonon,

Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulas

Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulas Moellng an smulaton of epenence structures n nonlfe nsurance wth Bernsten copulas Prof. Dr. Detmar Pfefer Dept. of Mathematcs, Unversty of Olenburg an AON Benfel, Hamburg Dr. Doreen Straßburger Dept. of