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

Transcription

1 Moelling an simulation of epenence structures in nonlife insurance with Bernstein copulas Prof. Dr. Dietmar Pfeifer Dept. of Mathematics, University of Olenburg an AON Benfiel, Hamburg Dr. Doreen Straßburger Dept. of Mathematics, University of Olenburg an mgm consulting partners, Hamburg Jörg Philipps Dept. of Mathematics, University of Olenburg

2 . Introuction o Classical copula concepts: elliptical (Gauß, t, ) Archimeian (Gumbel, Clayton, Frank, ) o Approximation theory: Bernstein polynomials in one an more variables or in one an more imensions (Bézier curves an surfaces) Bernstein copulas: Bernstein copulas allow for a very flexible, non-parametric an essentially non-symmetric escription of epenence structures also in higher imensions Bernstein copulas approximate any given copula arbitrarily well Bernstein copula ensities are given in an explicit form an can hence be easily use for Monte Carlo simulation stuies.

3 2. Some simple mathematical facts on Bernstein polynomials an copulas m k m k Lemma. Let Bmkz (,, ) = z( z), 0 z, k = 0,, m. k Then we have Further, mb( m, k, z) z= for k = 0,, m. 0 B ( mkz,, ) = mbm [ (, k, z ) Bm (, kz, )] for k = 0,, m z with the convention Bm (,, z) = Bm (, mz, ) = 0. 2

4 Theorem. For let U = ( U,, U ) be a ranom vector whose marginal component U follows a iscrete uniform istribution over T : = { 0,,, m } i with m, i=,,. Let further enote i i i Then p k k P U k (,, ): = { = } for all ( ) i i i= k,, k T. i i= m m (,, ): = (,, ) (,, ), (,, ) [ 0,] cu u pk k mbm k u u u i i i i k= 0 k = 0 i= efines the ensity of a -imensional copula, calle Bernstein copula. We call c the Bernstein copula ensity inuce by U. 3

5 By integration, we obtain the Bernstein copula C inuce by U as x x m m C( x,, x ): = c( u,, u ) u u = P { U < k } B( m, k, x ), i i i i i 0 0 k= 0 k = 0 i= i= for ( x x ) [ ],, 0,. Remark: if = ( V V ) V,, is a ranom vector with joint Bernstein copula ensity c then also any partial ranom vector ( Vi Vi ) possesses a Bernstein copula ensity,, with n< an i n < < in [ i,, i n ] c given by, n,,,,, 0,. m m n n [ i,, in ] c u u P U k m B m k u u u i ( ) i n i { } ( ) ( ) [ ] i = n i = i i i i i i n ki = 0 k 0 i n = = = 4

6 Definition. Uner the assumptions of the above theorem efine the intervals kj kj+ Ik :,,, k = j= mj m for all possible choices ( k,, k) Ti. Then the function i= j m ( ) m * c : = mi p k,, k Ik,, k i= k= 0 k = 0 is the ensity of a -imensional copula, calle gri-type or checkerboar copula inuce by U. Here A enotes the inicator ranom variable of the set A, as usual. Interpretation: = ( W W ) W,, follows a gri-type copula iff ( W k,, k ) = U ( k,, k ) i for all ( ) W P I I k,, k T, i= i where U ( ) enotes the continuous uniform istribution over a Borel set with positive Lebesgue measure. 5

7 Hence the Bernstein copula inuce by U can be regare as a naturally smoothe version of the gri-type copula inuce by U, replacing the inicator functions ( u,, u ) ( u ) by the polynomials = Ik,, k k i k, i + i i= mi m i Bm ( i, ki, ui), ( u,, u) [ 0, ]. i= 6

8 Example. The following graphs show the smoothing effect in case =. m = 5 m = 0 7

9 Natural generalizations of Bernstein an gri-type copulas are obtaine if we look at suitable partitions of unity, i.e. families of non-negative functions { φ( mk,, i) 0 k m, m } efine on the unit interval [ ] properties: 0, with the following φ ( mku,, ) u= for k = 0,, m m 0 m φ( mk,, i ) = for m. k= 0 In this case, a -imensional copula ensity c φ inuce by U is given by m m φ c ( u,, u ): = P { U = k } mφ ( m, k, u ), ( u,, u ) [ 0, ]. i i i i i i k= 0 k = 0 i= i= 8

10 The copula itself is accoringly given by m m φ C ( u,, u ): = P { U < k } φ ( m, k, u ), ( u,, u ) [ 0, ]. i i i i i k= 0 k = 0 i= i= Note that m φ( mku,, ) = Bm (, ku, ) = u( u) k k m k in case of Bernstein copulas an = φ( mku,, ) k k+, m m ( u) for 0 k m, m in case of gri-type copulas. 9

11 Note further that any such family of functions { φ( mk,, i) 0 k m, m } inuces immeiately a new family { φ (,, i ) 0, } but fixe K with similar properties via K mk k m m for arbitrary, since obviously K K φ ( mk,, i): = φ( K mk, k+ j, i ) for k = 0,, m j= 0 K K φk ( mkuu,, ) = φ( KmKk, + juu, ) = =, k= 0,, m 0 j= 0 0 j= 0 K m m m K m K m φ ( mk,, i) = φ( K mk, k+ j, i) = φ( K mi,, i ) =, m. K k= 0 j= 0 k= 0 i= 0 0

12 For Bernstein copulas, this generalization has a irect impact on the smoothing effect pointe out in the above example. The following two graphs show this effect for K = 3 an K = 0. The case K = is shown as a thin black line, for comparison. m = 5 K = 3 K = 0

13 3. Fitting empirical ata to gri-type an Bernstein copulas o In this section: case = 2, for simplicity. However, the metho propose here works accoringly in any imension. o Example ata set: a 34-year time series of (economically ajuste) winstorm an flooing losses One possible way to extract the epenence structure from the ata is the empirical copula scatterplot, which is a plot of the joint relative ranks of the ata. The following figure shows such a plot for a series of n = 34 observation years. 2

14 ranks of winstorm losses empirical copula scatter plot 3

15 o Fit these ata to a gri-type copula with a given gri resolution, say m= m2 = m= 0, by counting the relative frequency of the ata points in each of the m m2 = 00 target cells contingency table a ij (matrix notation: i = row inex, j = column inex; roune to 3 ecimal 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 4

16 o observe marginal sums are not equal to m = optimization problem: 0 ik i= j= min! m m i= j= ( x ) 2 ij aij subject to m m x = x j = = an x, k 0 for k, =,, m m 0 The explicit solution of such a problem is in general not straightforwar to fin, although there exists a solution ue to the Karush-Kuhn-Tucker theorem from optimization theory. Using a suitable software package like octave (a public omain computer algebra system), we obtain the following solution (roune to 3 ecimal places); see the coe listing in the Appenix of the paper. 5

17 upper cell bounary 0, 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9,0 sum,0 0,003 0,000 0,002 0,000 0,003 0,027 0,032 0,027 0,003 0,003 0, 0,9 0,032 0,000 0,00 0,000 0,002 0,000 0,03 0,000 0,002 0,03 0, 0,8 0,000 0,000 0,000 0,000 0,000 0,000 0,000 0,020 0,025 0,055 0, 0,7 0,003 0,027 0,002 0,000 0,003 0,027 0,003 0,000 0,032 0,003 0, 0,6 0,000 0,025 0,029 0,02 0,000 0,000 0,000 0,025 0,000 0,000 0, 0,5 0,003 0,028 0,002 0,025 0,003 0,000 0,003 0,000 0,032 0,003 0, 0,4 0,027 0,000 0,000 0,000 0,027 0,02 0,026 0,000 0,000 0,000 0, 0,3 0,003 0,000 0,002 0,054 0,003 0,000 0,003 0,028 0,003 0,003 0, 0,2 0,025 0,020 0,000 0,000 0,055 0,000 0,000 0,000 0,000 0,000 0, 0, 0,003 0,000 0,06 0,000 0,003 0,026 0,002 0,000 0,002 0,002 0, sum 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, optimal resulting contingency table 6

18 A more pragmatic way to fin at least a goo suboptimal solution that can be easily implemente e.g. in spreasheets is as follows. Consier the above optimization problem without the non-negativity conitions first. The equivalent Lagrange problem (which leas to a system of linear equations) is easy to solve an gives the (general) solution x ij a j ai 2 = aij + for i, j=,, m, 2 m m m where the inex i means summation, as usual. For the ata set above, we thus obtain 7

19 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, o solution is not feasible since it contains negative entries cell-wise aitive correction with a: min { xij i, j m} norming = an consecutive 8

20 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, xij + a final suboptimal contingency table y ij = 2 + m a 9

21 For imension 2, > with the inex sets I : = {,, m} an, for i {,, m} an k k k =,,, I ( i) : {,, m} = { i} {,, m}, the corresponing Lagrange optimization problem has the solution k ( xi i ai i ) i i I min! ( ) x [ k] ( ik) : = xi i = ( i i ) I ( i ) m k k x a a i 2 subject to i,, m, k =,, (*) for { } = ( ) + for ( i i ) { m} i i i i [ k] k m k= k m,,,,. 20

22 Proof: putting the graient of the Lagrange function 2 m L= ( xi ) 2 ( ) i a i i λ k, i x k [ k] i + k ( i i ) I k= ik = m to zero results in the m aitional equations (besies the sie conitions (*)) L x i i ( x a ) = λ = 0 i i i i k, ik k= for all ( ) i i I (**). These two sets of equations are solve by λ ( i ) a = an m m [ k] k ki, k x a λ i i i i k, ik k= = i,, m, k =,,. for { } k 2

23 Note also that in the special case = 2, we have ( ) = an a ( j) = a for i j { m} a i a [] i [2] j,,,. The above solution can be use as an initial solution for either the multiimensional Karush-Kuhn-Tucker approach or the simplifie version escribe above, giving y x + a = + ma with a = { xi i i i m} i i i i : min,,. 22

24 Any of the contingency tables above can be use to efine the joint istribution of the U = U, U inucing the gri-type an Bernstein copulas. iscrete ranom vector ( ) 2 contour plot of the (suboptimal) Bernstein copula ensity, with empirical copula scatterplot superimpose 23

25 4. Simulating from Bernstein copulas o Bernstein copula ensities are polynomials, hence boune over the unit cube [ 0,] by a constant M > 0 multivariate acceptance-rejection metho o average rate of samples obtaine by this proceure is /M Step : generate + inepenent uniformly istribute ranom numbers u,, u +. Step 2: check whether ( ) cu,, u > Mu +. If so, go to Step 3, otherwise go to Step. Step 3: use ( u,, u ) as a sample from the Bernstein copula. 24

26 o in the winstorm / flooing example: M = 2,35 o Q-Q-plots from the 34 year time series of the logarithms of winstorm an flooing losses (μ = location parameter, σ = scale parameter): winstorm 25

27 flooing 26

28 Log winstorm losses Log flooing losses Distribution Gumbel Normal Parameters μ = 6,367 μ = 6,625 σ = 0,8872 σ = 0,9777 o winstorm losses are consiere to follow a Fréchet istribution with extremal inex α= / σ=,27 o flooing losses are consiere to follow a lognormal istribution 27

29 The following graphs show the results of a fourfol Monte Carlo simulation for the aggregate risk (winstorm an flooing) on the basis of 000 pairs of points simulate from Bernstein copulas an the marginal istributions specifie above. The four cases consiere are: re line: Bernstein copula on the basis of a 4 x 4 gri green line: Bernstein copula on the basis of a 0 x 0 gri blue line: inepenence case orange line: Gaussian copula estimate from original ata 28

30 4x4 gri, approx. sol. 0x0 gri, approx. sol. 0x0 gri, opt. sol. contour plot of Bernstein copula ensities 29

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36 References [] T. BOUEZMARNI, J. V.K. ROMBOUTS, A. TAAMOUTI (2008): Asymptotic properties of the Bernstein ensity copula for epenent ata. CORE iscussion paper 2008/45, Leuven University, Belgium. [2] V. DURRLEMAN, A. NIKEGHBALI, T. RONCALLI (2000): Copulas approximation an new families. Groupe e Recherche Opérationelle, Créit Lyonnais, France, Working Paper. [3] T. KULPA (999): On approximation of copulas. Internat. J. Math. & Math. Sci. 22, [4] X. LI, P. MIKUSIŃSKI, H. SHERWOOD, M.D. TAYLOR (997): On approximation of copulas. In: V. Beneš an J. Štěpán (Es.), Distributions with Given Marginals an Moment Problems, Kluwer Acaemic Publishers, Dorrecht. [5] A. SANCETTA, S.E. SATCHELL (2004): The Bernstein copula an its applications to moelling an approximations of multivariate istributions. Econometric Theory 20(3), [6] M. SALMON, C. SCHLEICHER (2007): Pricing multivariate currency options with copulas. In: Copulas. From Theory to Application in Finance, J. Rank (e.), Risk Books, Lonon,