Adaptive Bernstein Copulas and Risk Management
1 Adaptive Bernstein Copulas and Risk Management
Dietmar Pfeifer and Olena Ragulina Carl von Ossietzky Universität Oldenburg, Germany and Taras Shevchenko National University of Kyiv, Ukraine December 8, 2020
Abstract:
We present a constructive approach to Bernstein copulas with an admissible dis-crete skeleton in arbitrary dimensions when the underlying marginal grid sizes are smaller than the number of observations. This prevents an overfitting of the estimated dependence model and reduces the simulation effort for Bernstein copulas a lot. In a case study, we com-pare different approaches of Bernstein and Gaussian copulas w.r.t. the estimation of risk measures in risk management.
Key words: copulas, partition-of-unity copulas, Monte Carlo methods
AMS Classification:
1. Introduction
Since the pioneering paper by Serge Bernstein in 1912 [3] Bernstein polynomials have been an indispen-sable tool in calculus and approximation theory (see e.g. [14]). Bernstein copulas, which can be consid-ered as Bernstein polynomials for empirical and other copula functions, have a long tradition in non-parametric modelling of dependence structures in arbitrary dimensions, in particular with applications in risk management, and have come into a deeper focus in the recent years. There is an extensive list of re-search papers on this topic, in particular [2], [5], [6], [9], [10], [13], [16], [17], [23], [24], [25] and [26]. The monographs [8] and [11] have, in particular, devoted separate chapters to the topic of Bernstein copu-las. A very important aspect of Bernstein copulas lies in Monte Carlo simulation techniques of dependence structures, in particular in higher dimensions. The structure of such procedures ranges from very complex (see e.g. [17]) to extremely simple (see e.g. [6]) such that Monte Carlo simulations could e.g. be per-formed easily with ordinary spreadsheets, in particular in applications concerning quantitative risk man-agement. From a statistical point of view, the problem of a potential overfitting of the true underlying dependence structure with Bernstein polynomials emerges naturally when the pertaining Bernstein polynomial degree increases, i.e. with an increasing number of observations. This has been discussed extensively for instance in [9] (Chapter 3.1), [10] (Chapter 4), [22] (Chapter 3.2.1) or [26] (Remark 4). This leads to the fact that the Bernstein copula density becomes more wiggly the more empirical observations are used in the analy-sis. In comparison with classical parametric dependence models such as elliptically contoured or Archi-medean copulas, this is probably a non-desirable property. This problem has in particular been tackled seemingly first in [17] by approximating the underlying discrete skeleton for the Bernstein copula by a least-squares approach and recently in the Ph.D. Thesis [1] where cluster analytic methods were used. In the present paper, we propose a simple but yet effective approach to reduce the complexity of Bern-stein copulas in a two-step approach, namely first an augmentation step in combination with a second reduction step. The reduction step is also discussed in [24], however without a possible application to a general complexity reduction of Bernstein copulas.
2. Some important facts about multivariate Bernstein polynomials
Let f be an arbitrary bounded real-valued function on the unit cube [ ] : 0,1 dd = C with dimension . d Î Let further , , d n n be integers. The corresponding multivariate Bernstein polynomial is defined by ( ) ( ) ( )
11 110 0 1 , , : , , 1 , , , d j jjd n n d n ij idd j j d djji i d niiB f x x f x x x xin n -== = æ öæ ö ÷÷ çç ÷÷= - = Îçç ÷÷ çç ÷ ÷ç çè ø è ø å å n x C (1) with ( ) , , d n n = n (see e.g. [14], p. 51). It is known that for ( ) min , , d n n ¥ multivariate Bernstein polynomials converge to f at any point of continuity and approximate f uniformly if f is continuous on . d C Another important property of multivariate Bernstein polynomials that is perhaps less known in the mathematical community is the fact that the multivariate Bernstein polynomial density given by ( ) ( ) , , : , , , dd d dd b f x x B f x xx x ¶= ζ ¶ n n x C (2) can be written as a linear combination of statistical product beta densities. For this purpose, consider uni-variate beta densities (1 )( ; , ) : B( , ) x xf x a b a b a b - - -= for x a b < < > (3) where B( , ) a b denotes the Euler Beta-function, i.e. ( ) ( ) ( )B( , ) . a ba b a b
G ⋅ G= G +
We need a further definition to proceed.
Definition 1.
Let g be a real-valued bounded function on . d We call ( ) ( ) { } , , 0,1 : ( 1) (1 ) , , (1 ) 0 d iidd d d d d g g a b a b ee e e e e e = Î åD = - + - + - ³ å ba (4) the - D difference of g over the interval ( ] ( ] , : , d i ii a b = æ ö÷ç= ÷ç ÷çè ø ´ a b with ( ) ( ) , , , , , dd d a a b b = = Î a b and , 1 . i i a b i d < £ £ (We adopt here a notation similar as in [15], Definition 2.1, which is slightly different from the notation in [7], Definition 1.2.10.) Proposition 1.
With the above notation, the Bernstein polynomial density b f n can be represented as ( ) , , ( ; 1, ) dd n n dd j j j jji i b f x x f f x i n i - - == = = D + - å å ii bn a (5) with
11: , , and : , , . d dd d i ii in n n n æ ö æ ö++÷ ÷ç ç÷ ÷= =ç ç÷ ÷ç ç÷ ÷ç çè ø è ø i i a b
Proof.
This follows immediately from the arguments in the proof of Theorem 2.2. in [6]; compare also the line of proofs in [4]. · Example 1.
We consider the polynomial ( , ) : 2 (1 ) 3(1 ) , 0 , 1 f x y x y x y x y = - - - £ £ with n = n = In this case, the two-dimensional Bernstein polynomial
B f n differs from f due to smaller poly-nomial degrees. We have
22 3 2 3 2 2 2 3
145 14 4 97 17 7( , ) 2 9 36 9 3 18 12 9 6 y x yB f x y x xy y y xy xy x y x y = - - - - + - + - - n (6) with
145 97 17 14( , ) 6 .36 6 9 3 3 xb f x y y y xy xy = - - + + - - n (7) Note that here i i i i i i i if f f f fn n n n n n n n æ ö æ ö æ ö æ ö+ + + +÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷D = + - -ç ç ç ç÷ ÷ ÷ ÷ç ç ç ç÷ ÷ ÷ ÷ç ç ç çè ø è ø è ø è ø ii ba (8) or, in tabular form, i
0 1 0 1 0 1 i
0 0 1 1 2 2 f D ii ba - - - Tab. 1 After a little computation it is easy to see that indeed here ( ) (1 ) (1 ), .B( 1, 2 ) B( 1, 3 ) i i i ii i x x y yb f x y f i i i i - -= = - -= D + - + - å å ii bn a (9) Fig.1 Fig. 2 Fig. 3 ( , ) f x y ( , )
B f x y n ( , ) ( , ) f x y B f x y - n A direct consequence of Proposition 1 concerns the monotonicity behaviour of multivariate Bernstein polynomials.
Definition 2.
Let g be a real-valued function on . d We call g d -monotone iff g D ³ ba for all ( ) , , , d a a = a ( ) , , dd b b = Î b with , 1 . i i a b i d < £ £ It is obvious by the iterated mean value theorem that for a sufficiently smooth function g , d -monotonicity is equivalent to ( ) , , 0 d dd g x xx x ¶ ³¶ ¶ for all ( ) , , . dd x x Î (10) Note that in case that g is a d -dimensional cumulative distribution function of a probability measure P on the d -dimensional Borel - s field , d then ( ] , . d i ii g P a b = æ ö÷çD = ÷ç ÷çè ø ´ ba Proposition 2.
Let f be a real-valued d -monotone function on . d Then the corresponding multivariate Bernstein polynomial
B f n is also d -monotone. In particular, the Bernstein polynomial density b f n is a positive-linear combination of product beta densities. Proof.
By the arguments above and the notation as in Proposition 1, we have ( ) ( ) , , : , , ( ; 1, ) 0 dd n nd dd d j j j jji id b f x x B f x x f f x i n ix x - - == = ¶= = D + - ³¶ ¶ å å ii bn n a (11) which is a sufficient condition for B f n to be d -monotone. · Note that the polynomial f from Example 1 is not 2-monotone since .00126... 0 f D = - < ba with ( ) = a and ( ) = b However, the slightly modified polynomial ( , ) ( , ) 6 g x y f x y xy = + is 2-monotone since ( , ) 6 6(1 ) 36(1 ) 0 g x y y x yx y ¶ = - - + - ³¶ ¶ with the unique global minimum point ( ) ( ) , 1,1 x y = and (1,1) 0. gx y ¶ =¶ ¶ With respect to the corresponding multivariate Bernstein polynomial, we now have i
0 1 0 1 0 1 i
0 0 1 1 2 2 g D ii ba Tab. 2 which also explicitly shows that the Bernstein polynomial
B g n is 2-monotone.
3. From Bernstein polynomials to Bernstein copulas
Remark 1.
Seemingly Proposition 2 can be usefully applied to arbitrary d -dimensional cumulative distri-bution functions F concentrated on the unit cube [ ] : 0,1 dd = C (continuous or not) such that the corre-sponding multivariate Bernstein polynomial ( ) ( ) ( )
11 110 0 1 , , , , 1 , , , d j jjd n n d n ij idd j j d djji i d niiB F x x F x x x xin n -== = æ öæ ö ÷÷ çç ÷÷= - = Îçç ÷÷ çç ÷ ÷ç çè ø è ø å å n x C (12) also is a cumulative distribution function since B F n is non-negative and d -increasing with ( ) ( )
0, , 0 0, , 0
B F F = n and ( ) ( )
1, ,1 1, ,1 1.
B F F = = n In particular, the Bernstein polynomial den-sity b F n always is a (probabilistic) mixture of product beta densities as explicitly noted in [6] and [24] for Bernstein copulas. Note also that this observation was the motivation for the setup in [20]. Example 2.
We consider a two-dimensional random vector ( ) , X Y = X with a discrete distribution con-centrated on { } , x y with ( ) ( ) , 0.2, 0.7 x x = = x and ( ) ( ) , 0.3, 0.5 y y = = y given by ( ) , i j P X x Y y = = x x y y H the Heaviside step function in its original form, i.e. 0, 0( ) , .1, 0 xH x xx ì <ïï= Îíï ³ïî Then the corresponding cumulative distribution function F for the given discrete distribution is ( ) ( ) ( ) ( , ) , , , i j i jj i F x y P X x Y y H x x H y y x y = = = = = - - Î å å (13) The following graphs show the corresponding cumulative distribution function F as well as the corre-sponding Bernstein polynomials B F n and densities b F n for various choices of n according to relations (11) and (12) above. Fig. 4 Fig. 5 Fig. 6 ( , )
F x y ( , )
B F x y n ( , ) b F x y n
3, 5 n n = =
Fig. 7 Fig. 8 Fig. 9 ( , )
F x y ( , )
B F x y n ( , ) b F x y n
11, 7 n n = =
Fig. 10 Fig. 11 Fig. 12 ( , )
F x y ( , )
B F x y n ( , ) b F x y n
50, 50 n n = = The above figures clearly visualize the approximation effect of multivariate Bernstein polynomials for discrete multivariate distributions if ( ) min , , d n n gets large. In particular, the Bernstein polynomial density has spikes around the support points of the underlying discrete distribution. To simplify notation, we will use the following convention. Let d > be a natural number and ( ) , , x dd x = Î x be arbitrary. Then, for , y Î let ( ) ( ) ( )
21 1 11 1 , , , 1( ) : , , , , , , 1, , , dk k k dd y x x if ky x x y x x if k dx x y if k d - +- ì =ïïïï= < <íïï =ïïî x (14) denote the vector x where the k -th component is replaced by y . Proposition 3.
Suppose that for d > there is a cumulative distribution function [ ] [ ] : 0,1 0,1 d F with (0, , 0) 0 F = and (1, ,1) 1 F = such that for given natural numbers , , 1 d n n > we have k j j i iF n n æ öæ ö÷ç ÷ç ÷÷ç ç =÷÷ç ç ÷÷ç ÷ç ÷ç è øè ø for { } j i n j d k d Î = = where ( )
1, ,1 . d = Î Then the d -dimensional Bernstein polynomial B F n with ( ) , , d n n = n associated with F is a copula. Proof:
By Remark 1 above we know that
B F n also is a cumulative distribution function with (0, , 0) (0, , 0) 0 B F F = = n and (1, ,1) (1, ,1) 1, B F F = = n and (note that = ( ) ( ) , , 1 1(1 ) (1 ) d j jjd k kk k n n d n ij idk j jjji i dn nk kn i n ii i kki ik k k niiB F x F x xin nn n n xiF x x i x x xi in n n - == = - -= = æ öæ ö ÷÷ çç ÷÷= -çç ÷÷ çç ÷ ÷ç çè ø è øæ öæ öæ ö æ ö÷ç ÷÷ ÷çç ç÷÷= ÷ ç - = ÷ - = =çç ç÷÷÷ ÷ç çç ç÷ ÷÷ç ç÷ç ÷çè ø è øè øè ø å å å å n (15) for
1, , k d = and 0 1 x £ £ ( k n x is the expectation of the Binomial distribution with k n trials and suc-cess probability x ). The marginal distributions induced by B hence are continuous uniform, which means that B is indeed a copula. · Note that Proposition 3 was already implicitly formulated in [6] and [17], see also [8], Chapter 4.1.2. We reformulate the corresponding statements there in an appropriate way.
Corollary 1.
Let ( ) , , d U U = U be a discrete random vector whose marginal component i U follows a discrete uniform distribution over { } : 0,1, , 1 i i T n = - with integers 1, 1, , . i n i d > = Then the multi-variate Bernstein polynomial
B F n derived from the cumulative distribution function F for the scaled ran-dom vector
11 , , d d
UU n n æ ö++ ÷ç ÷= ç ÷ç ÷çè ø V given by ( ) ( ) ( ) , , , , , , , d d d d d F x x P V x V x x x = £ £ = Î x C is a copula. The corresponding Bernstein copula density b F n is given by ( ) ( ) ( ) ( ) , , , , ( ; 1, ), , , . dd n n dd d j j j j d dji i b F x x P i i f x i n i x x - - == = = = + - Î å å n U C (16) Proof.
For , 1, , j j i T j d
Î = we have ( ) , , , , , , d dd d dd d i ii iF P V V P U i U in n n n æ ö æ ö÷ ÷ç ç÷ ÷= £ £ = < <ç ç÷ ÷ç ç÷ ÷ç çè ø è ø and hence ( ) ( )
11 , , , , . d dd dd d i ii iF P V V P i in n n n æ ö++ ÷ç ÷D = < £ < £ = =ç ÷ç ÷çè ø ii ba U · Remark 2.
We call
B F n the Bernstein copula induced by U . In coincidence with [17] we also call U the discrete skeleton of the Bernstein copula B F n and the number d n n ´ ´ its grid size. If V is an arbitrary discrete random vector over : , d ii T = = ´ T we call V an admissible discrete skeleton if the marginal distribu-tions are discrete uniform. So every admissible skeleton over T induces a corresponding Bernstein copula via the multivariate Bernstein polynomial of its rescaled cumulative distribution function. The corre-sponding Bernstein copula density is a mixture of product beta kernels with weights given by the individ-ual probabilities representing the admissible skeleton.
4. Empirical Bernstein copulas
Bernstein copulas can be easily constructed from independent samples , , , n n Î X X of d -dimensional random vectors with the same intrinsic dependence structure and the same marginal distribu-tions. For simplicity, we assume here that the marginal distributions are continuous in order to avoid ties in the observations. The simplest way to construct an empirical Bernstein copula is on the basis of De-heuvel’s empirical copula [7] in the form of a cumulative distribution function which can be represented by an admissible discrete skeleton derived from the individual ranks , 1 , , 1, , ij r i d j n = = of the observation vectors ( ) , , , 1, , , j j dj x x j n = = x given by the order statistics , , , , i i in i r i r i r x x x < < < i.e. ij r k = iff ij x is the k -largest value of the i -th observed component . The discrete skeleton U is here given by a random vector over { } : 0,1, , 1 d n = - T with a discrete uniform distribution over the n support points , , n s s where ( )
1, , 1 , 1, , . j j dj r r j n = - - = s Since the empirical copula converges in distri-bution to the true underlying copula when n ¥ it follows that the corresponding empirical Bernstein copula does so likewise, cf. [8], Chapter 4.1.2. This provides – in the light of relation (16) – in particular a simple way of generating samples from an empirical Bernstein copula by Monte Carlo methods in two steps: Step1: Select an index N randomly and uniformly among 1, , . n Step2: Generate d independent beta distributed random variables , , d V V (also independent of N ) where i V follows a beta distribution with parameters iN r and 1 , 1, , . iN n r i d + - = Then ( ) : , , d V V = V is a sample point from the empirical Bernstein copula. This has also been observed in [24], but was known earlier, see e.g. [6]. In what follows we will discuss the data set presented in [17], Section 3 in more detail. Example 3.
The following table contains the ranks for observed insurance data from windstorm ( 1) i = and flooding ( 2) i = losses in central Europe for 34 consecutive years discussed in [17]. ij r j
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 171 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 i
2 12 5 31 7 24 18 17 3 2 19 10 9 21 15 14 4 6 ij r j
18 19 20 21 22 23 24 25 26 27 28 28 30 31 32 33 341 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 i
2 34 1 23 11 29 33 13 8 20 32 28 22 16 26 25 30 27 Tab. 4 The following graphs show some plots for the empirical Bernstein copula.
Fig. 13 Fig. 14 Fig. 15 support points of scaled skeleton copula density contour plot simulation of 5.000 copula pairs It is clearly to be seen from the shape of the contour lines in Fig. 14 that the empirical Bernstein copula density is quite bumpy here, e.g. in comparison with the Gaussian copula density fitted to the data set above.
Fig. 16 Fig. 17 Fig. 18 support points of scaled skeleton Gaussian copula density simulation of 5.000 Gaussian contour plot copula pairs From a practical point of view, it might therefore be desirable to adapt the empirical Bernstein copula to a smaller support set : : d di ii i
T T * *= = = Ì = ´ ´
T T for the underlying discrete skeleton. This was the central idea in [17]. The disadvantage of the method proposed in that paper is, however, that the number of support points of * U gets dramatically larger and is typically of exponential order with increasing grid sizes. This is due to the fact that the number of support points is usually in the range of ( ) d ii n * = = T because by the specific method of least squares used there, most support points of * T will get a positive weight. We therefore propose a simpler way how to find a smaller discrete approximating skeleton * U with an arbi-trary given grid size in the subsequent chapter.
5. Adaptive Bernstein copula estimation
We start with the individual ranks ij r of the observation vectors ( ) , , , 1, , . j j dj x x j n = = x Let U de-note the canonical admissible discrete skeleton as described in the preceding chapter, derived from the empirical copula. Our aim is to find a good approximating admissible discrete skeleton * U with a given grid size d n n ´ ´ where the i n are typically smaller than n . We proceed in two steps:
1. Step: Augmentation
Select an integer M such that all , 1, , i n i d = are divisors of M , for instance their least common multi-ple. We construct pseudo-ranks ij r + in the following way: , : 1 1, , , 1, . ij ji M jr r M j i d j M nM + é ùê úê úê ú æ öé ù ÷çê ú= ⋅ ⋅ + - = = ⋅÷ç ÷ç ÷ê úè øê ú (17) Here { } : min | , x m x m x é ù = Î £ Îê ú stands for “rounding up”. Let ( ) , , d U U + + + = U be the uni-formly discretly distributed random vector over { } d M n ⋅ - with support points , , M n ⋅ s s where ( )
1, , 1 , 1, , . j j dj r r j n M + + = - - = ⋅ s Note that the probability mass is M n ⋅ for each support point, and that + U is an admissible discrete skeleton.
2. Step: Reduction
Construct the final ranks ij r * in the following way: : , 1, , , 1, , . ij iij r nr i d j M nn M +* é ù⋅ê ú= = = ⋅ê ú⋅ê úê ú (18) It follows from the above definition, that there will be replicates in the final ranks and that ij r * - takes values in the set { } i i T n * = - A point ( ) , , d s s = s will be a support point of the final admissi-ble skeleton * U if there exist final ranks such that ( )
1, , , , d j d j r r = s for some { } , , 1, , . d j j M n Î ⋅
The probability mass attached to s is given by the number KM n ⋅ where K is the number of rank combina-tions ( )
1, , , , d j d j r r that lead to the same s . This also enables very simple Monte Carlo realisations of the corresponding Bernstein copula as described in chapter 4 by first selecting an index N randomly and uni-formly among 1, , M n ⋅ and then by proceeding as in step 2 there with all of the ij r * . Note that the above augmentation step creates permutations of the set { }
1, ,
M n ⋅ in each component so that the pseudo-ranks ij r + actually lead to an admissible discrete skeleton, cf. [6], chapter 4. The mathematical correctness of the reduction step follows from the proof of Proposition 2.5 in [24]. In the augmentation step, M -wise partial permutations would not change the result but would create a more “random” augmentation of the original ranks. Example 4.
Consider the following rank table with n = ij r i
1 2 probabilities ( ) , p r r
1 1 3 0.2 2 2 4 0.2 3 3 1 0.2 4 4 2 0.2 j
5 5 5 0.2
Tab. 5 We want to create approximate final ranks with n = and n = Both numbers are not a divisor of n , so we choose M = ⋅ = We show a part of the resulting pseudo-ranks: ij r + i
1 2 probabilities ( ) , p r r + +
1 12 36
2 11 35
3 10 34
13 24 48
14 23 47
15 22 46
25 36 12
26 35 11
27 34 10
58 51 51
59 50 50 j
60 49 49
Tab. 6 For the final ranks we obtain the following table: ij r * i
1 2 probabilities ( ) , p r r * *
1 1 2 0.1 2 1 3
3 2 1 0.25 4 2 2
5 2 3
6 2 4 0.05 7 3 2 j
8 3 4 0.2
Tab. 7 From ( , ) 0.3, 1, 2, 3 j p i j i = = = å and ( , ) 0.25, 1, 2, 3, 4 i p i j j = = = å we see that the induced skeleton is indeed admissible. The following graphs show the corresponding copula densities ( ) , c x x U and ( ) , c x x * U induced by U and . * U Fig. 19 Fig. 20 ( ) , c x x U ( ) , c x x * U Seemingly the shape of both densities is similar, reflecting the structure of the original ranks quite well. However, the density c * U is less wiggly than the density , c U as was intended. Note also that a reduction of complexity for copulas in the sense discussed here is also an essential topic in [12], chapter 3; see in particular Fig. 2 there. However, the underlying problem of a consistent reduc-tion of complexity is not really discussed there.
6. Applications to risk management
In this chapter we first want to investigate the data set of Example 3 in more detail. It was the basis data set in [17]. In particular, we want to discuss the effect of different adaptive Bernstein copula estimations on the estimation of risk measures like Value at Risk which is the basis for Solvency II, for instance.
In [17], the number n = of the original observations was first reduced to n n = = by a least squares technique. The resulting optimal discrete skeleton with probabilities , , ij p i T j T * * Î Î is pre-sented here with { }
T T * * = = ij p i
0 1 2 3 4 5 6 7 8 99 j Tab. 8 An application of the adaptive strategy described in the preceding chapter gives alternatively the follow-ing less complex table. Here we have chosen M = Tab. 9 Seemingly the number of support points for the adaptive probabilities ij p * are much less than before. The following graphs show contour plots for the corresponding Bernstein copula densities. Here c de-notes the Bernstein copula density derived from Tab.8, c denotes the Bernstein copula density derived from Tab. 9. In the first case we have chosen 5, M = in the second case M = ij p * i
0 1 2 3 4 5 6 7 8 99 j Fig. 21 Fig. 22 ( ) , c x x ( ) , c x x Seemingly the differences are only marginal. However, in comparison with Fig. 14, the smoothing effect of the adaptive procedure is clearly visible. The next graphs show contour plots for adaptive Bernstein copula densities c and c with the choices n n = = and n n = = respectively. Fig. 23 Fig. 24 ( ) , c x x ( ) , c x x In the next step, we compare estimates for the risk measure Value at Risk VaR a with the risk level a = – corresponding to a return period of 200 years – on the basis of a Monte Carlo study with 1,000,000 repetitions each for the aggregated risk of windstorm and flooding losses. We consider the full Bernstein copula of Example 3 with n n = = as well as the adaptive Bernstein copulas with n n = = n n = = and n n = = For the sake of completeness, we also add estimates from the Gaussian copula, the independence as well as the co- and countermonotonicity copulas (see [8], p.11 for definitions). The following graphs show the support points of the underlying adaptive scaled discrete skeletons as well as 5,000 simulated pairs of the adapted Bernstein copulas.
Fig. 25 Fig. 26 n n = = Fig. 27 Fig. 28 n n = = Fig. 29 Fig. 30 n n = = The following table provides estimated values of the risk measures from the Monte Carlo simulations which are given in Mio. monetary units. For the marginal distributions, the assumptions in [17] were used. grid type
34 34 ´
10 10 ´ ´ ´ Gaussian independence comonotonic countermonotonic
VaR 1,348 1,334 1,356 1,369 1,386 1,349 1,500 1,327
Tab. 10 Seemingly the comonotonicity copula provides the largest
VaR -estimate due to an extreme tail de-pendence while the countermonotonicity copula provides the smallest
VaR -estimate. Surprisingly, the
VaR -estimates for the adaptive Bernstein copulas do not differ very much from each other (at most 2.5%) and are almost identical to the estimate from the independence copula here. Note that the
VaR -estimate for the Gaussian copula is slightly larger. Significant differences are, however, visible if we look at the densities for the aggregated risk. The fol-lowing graphs show empirical histograms for these densities under the models considered above, from 100,000 simulations each.
Fig. 31 Fig. 32
Bernstein, grid type
34 34 ´ Bernstein, grid type
10 10 ´ Fig. 33 Fig. 34
Bernstein, grid type ´ Bernstein, grid type ´ Fig. 35 Fig. 36
Gaussian copula independence copula
Fig. 37 Fig. 38 comonotonicity copula countermonotonicity copula
Note that the histogram for the full Bernstein copula has two peaks, whereas the other histograms show a more smooth behaviour. Finally, we discuss a high-dimensional data set that was also analyzed in [21]. It represents economically adjusted windstorm losses in 19 adjacent areas in Central Europe over a time period of 20 years. The losses are given in Mio. monetary units. Year Area 1 Area 2 Area 3 Area 4 Area 5 Area 6 Area 7 Area 8 Area 9 Area 10 1 23.664 154.664 40.569 14.504 10.468 7.464 22.202 17.682 12.395 18.551 2 1.080 59.545 3.297 1.344 1.859 0.477 6.107 7.196 1.436 3.720 3 21.731 31.049 55.973 5.816 14.869 20.771 3.580 14.509 17.175 87.307 4 28.99 31.052 30.328 4.709 0.717 3.530 6.032 6.512 0.682 3.115 5 53.616 62.027 57.639 1.804 2.073 4.361 46.018 22.612 1.581 11.179 6 29.95 41.722 12.964 1.127 1.063 4.873 6.571 11.966 15.676 24.263 7 3.474 14.429 10.869 0.945 2.198 1.484 4.547 2.556 0.456 1.137 8 10.02 31.283 21.116 1.663 2.153 0.932 25.163 3.222 1.581 5.477 9 5.816 14.804 128.072 0.523 0.324 0.477 3.049 7.791 4.079 7.002 10 170.725 576.767 108.361 41.599 20.253 35.412 126.698 71.079 21.762 64.582 11 21.423 50.595 4.360 0.327 1.566 64.621 5.650 1.258 0.626 3.556 12 6.38 28.316 3.740 0.442 0.736 0.470 3.406 7.859 0.894 3.591 13 124.665 33.359 14.712 0.321 0.975 2.005 3.981 4.769 2.006 1.973 14 20.165 49.948 17.658 0.595 0.548 29.35 6.782 4.873 2.921 6.394 15 78.106 41.681 13.753 0.585 0.259 0.765 7.013 9.426 2.18 3.769 16 11.067 444.712 365.351 99.366 8.856 28.654 10.589 13.621 9.589 19.485 17 6.704 81.895 14.266 0.972 0.519 0.644 8.057 18.071 5.515 13.163 18 15.55 277.643 26.564 0.788 0.225 1.230 26.800 64.538 2.637 80.711 19 10.099 18.815 9.352 2.051 1.089 6.102 2.678 4.064 2.373 2.057 20 8.492 138.708 46.708 3.68 1.132 1.698 165.6 7.926 2.972 5.237
Tab. 11
Year Area 11 Area 12 Area 13 Area 14 Area 15 Area 16 Area 17 Area 18 Area 19 1 1.842 4.100 46.135 14.698 44.441 7.981 35.833 10.689 7.299 2 0.429 1.026 7.469 7.058 4.512 0.762 14.474 9.337 0.740 3 0.209 2.344 22.651 4.117 26.586 3.920 13.804 2.683 3.026 4 0.521 0.696 31.126 1.878 29.423 6.394 18.064 1.201 0.894 5 2.715 1.327 40.156 4.655 104.691 28.579 17.832 1.618 3.402 6 4.832 0.701 16.712 11.852 29.234 7.098 17.866 5.206 5.664 7 0.268 0.580 11.851 2.057 11.605 0.282 16.925 2.082 1.008 8 0.741 0.369 3.814 1.869 8.126 1.032 14.985 1.390 1.703 9 0.524 6.554 5.459 3.007 8.528 1.920 5.638 2.149 2.908 10 9.882 6.401 106.197 44.912 191.809 90.559 154.492 36.626 36.276 11 1.052 8.277 22.564 8.961 19.817 16.437 25.990 2.364 6.434 12 0.136 0.364 28.000 7.574 3.213 1.749 12.735 1.744 0.558 13 1.990 15.176 57.235 23.686 110.035 17.373 7.276 2.494 0.525 14 0.630 0.762 25.897 3.439 8.161 3.327 24.733 2.807 1.618 15 0.770 15.024 36.068 1.613 6.127 8.103 12.596 4.894 0.822 16 0.287 0.464 24.211 38.616 51.889 1.316 173.080 3.557 11.627 17 0.590 2.745 16.124 2.398 20.997 2.515 5.161 2.840 3.002 18 0.245 0.217 12.416 4.972 59.417 3.762 24.603 7.404 19.107 19 0.415 0.351 10.707 2.468 10.673 1.743 27.266 1.368 0.644 20 0.566 0.708 22.646 6.652 14.437 63.692 113.231 7.218 2.548
Tab. 12 A statistical analysis of the data shows a good fit to lognormal ( , ) m s -distributions for the losses per Area , 1, ,19. k k = The parameters m k and s k for Area k can thus be estimated from the log data by calculating means and standard deviations. Parameter Area 1 Area 2 Area 3 Area 4 Area 5 Area 6 Area 7 Area 8 Area 9 Area 10 m k s k Tab. 13
Parameter Area 11 Area 12 Area 13 Area 14 Area 15 Area 16 Area 17 Area 18 Area 19 m k –0.3231 0.3815 3.0198 1.7488 3.0409 1.5501 3.0700 1.2444 0.9378 s k Tab. 14
As expected, insurance losses in locally adjacent areas show a high degree of stochastic dependence, which can also be seen from the following correlation tables. Correlations above 0.9 are highlighted.
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A1 1 0.46 0.03 0.16 0.47 0.20 0.35 0.49 0.41 0.24 0.78 0.64 0.91 0.63 0.85 0.66 0.30 0.67 0.56 A2 0.46 1 0.64 0.78 0.67 0.36 0.51 0.76 0.57 0.51 0.58 -0.04 0.59 0.84 0.68 0.58 0.87 0.77 0.90 A3 0.03 0.64 1 0.93 0.41 0.26 0.11 0.16 0.33 0.16 0.08 -0.09 0.13 0.64 0.25 0.10 0.74 0.14 0.35 A4 0.16 0.78 0.93 1 0.54 0.36 0.16 0.25 0.43 0.19 0.22 -0.10 0.30 0.79 0.36 0.19 0.84 0.32 0.49 A5 0.47 0.67 0.41 0.54 1 0.41 0.35 0.51 0.84 0.63 0.59 0.02 0.64 0.67 0.59 0.50 0.58 0.71 0.67 A6 0.20 0.36 0.26 0.36 0.41 1 0.07 0.11 0.28 0.19 0.28 0.14 0.31 0.42 0.24 0.27 0.39 0.27 0.40 A7 0.35 0.51 0.11 0.16 0.35 0.07 1 0.44 0.27 0.19 0.48 -0.07 0.46 0.35 0.45 0.91 0.64 0.61 0.49 A8 0.49 0.76 0.16 0.25 0.51 0.11 0.44 1 0.50 0.75 0.61 -0.03 0.54 0.47 0.71 0.53 0.40 0.75 0.90 A9 0.41 0.57 0.33 0.43 0.84 0.28 0.27 0.50 1 0.66 0.68 -0.01 0.52 0.60 0.50 0.41 0.46 0.65 0.63 A10 0.24 0.51 0.16 0.19 0.63 0.19 0.19 0.75 0.66 1 0.33 -0.12 0.27 0.28 0.43 0.24 0.23 0.45 0.65 A11 0.78 0.58 0.08 0.22 0.59 0.28 0.48 0.61 0.68 0.33 1 0.19 0.79 0.65 0.80 0.73 0.43 0.84 0.74 A12 0.64 -0.04 -0.09 -0.10 0.02 0.14 -0.07 -0.03 -0.01 -0.12 0.19 1 0.44 0.21 0.28 0.17 -0.12 0.13 0.03 A13 0.91 0.59 0.13 0.30 0.64 0.31 0.46 0.54 0.52 0.27 0.79 0.44 1 0.71 0.86 0.74 0.47 0.76 0.65 A14 0.63 0.84 0.64 0.79 0.67 0.42 0.35 0.47 0.60 0.28 0.65 0.21 0.71 1 0.74 0.54 0.79 0.68 0.72 A15 0.85 0.68 0.25 0.36 0.59 0.24 0.45 0.71 0.50 0.43 0.80 0.28 0.86 0.74 1 0.69 0.47 0.71 0.75 A16 0.66 0.58 0.10 0.19 0.50 0.27 0.91 0.53 0.41 0.24 0.73 0.17 0.74 0.54 0.69 1 0.63 0.77 0.64 A17 0.30 0.87 0.74 0.84 0.58 0.39 0.64 0.40 0.46 0.23 0.43 -0.12 0.47 0.79 0.47 0.63 1 0.59 0.64 A18 0.67 0.77 0.14 0.32 0.71 0.27 0.61 0.75 0.65 0.45 0.84 0.13 0.76 0.68 0.71 0.77 0.59 1 0.86 A19 0.56 0.90 0.35 0.49 0.67 0.40 0.49 0.90 0.63 0.65 0.74 0.03 0.65 0.72 0.75 0.64 0.64 0.86 1 correlations between original losses in adjacent areas
Tab. 15
A1 A2 A3 A4 A5 A6 A7 A8 A9 A10 A11 A12 A13 A14 A15 A16 A17 A18 A19 A1 1 0.27 0.30 0.16 0.17 0.45 0.28 0.32 0.32 0.29 0.67 0.51 0.76 0.34 0.67 0.74 0.18 0.21 0.29 A2 0.27 1 0.48 0.66 0.39 0.37 0.71 0.69 0.52 0.64 0.30 -0.02 0.45 0.66 0.58 0.45 0.73 0.74 0.78 A3 0.30 0.48 1 0.70 0.40 0.31 0.42 0.51 0.58 0.53 0.18 0.07 0.21 0.32 0.54 0.26 0.47 0.21 0.57 A4 0.16 0.66 0.70 1 0.77 0.47 0.46 0.47 0.59 0.49 0.18 -0.13 0.33 0.50 0.47 0.18 0.76 0.43 0.54 A5 0.17 0.39 0.40 0.77 1 0.59 0.30 0.20 0.49 0.39 0.28 0.08 0.35 0.56 0.44 0.16 0.55 0.36 0.41 A6 0.45 0.37 0.31 0.47 0.59 1 0.14 0.01 0.36 0.34 0.33 0.12 0.48 0.46 0.48 0.37 0.59 0.17 0.50 A7 0.28 0.71 0.42 0.46 0.30 0.14 1 0.52 0.27 0.40 0.45 -0.07 0.31 0.31 0.46 0.62 0.63 0.58 0.57 A8 0.32 0.69 0.51 0.47 0.20 0.01 0.52 1 0.64 0.81 0.27 -0.02 0.38 0.35 0.56 0.35 0.28 0.62 0.63 A9 0.32 0.52 0.58 0.59 0.49 0.36 0.27 0.64 1 0.78 0.40 0.19 0.27 0.50 0.44 0.30 0.33 0.57 0.61 A10 0.29 0.64 0.53 0.49 0.39 0.34 0.40 0.81 0.78 1 0.21 -0.02 0.21 0.37 0.52 0.30 0.31 0.53 0.81 A11 0.67 0.30 0.18 0.18 0.28 0.33 0.45 0.27 0.40 0.21 1 0.47 0.49 0.45 0.60 0.67 0.20 0.45 0.39 A12 0.51 -0.02 0.07 -0.13 0.08 0.12 -0.07 -0.02 0.19 -0.02 0.47 1 0.44 0.21 0.24 0.46 -0.23 0.25 0.05 A13 0.76 0.45 0.21 0.33 0.35 0.48 0.31 0.38 0.27 0.21 0.49 0.44 1 0.55 0.60 0.71 0.37 0.39 0.24 A14 0.34 0.66 0.32 0.50 0.56 0.46 0.31 0.35 0.50 0.37 0.45 0.21 0.55 1 0.59 0.43 0.57 0.58 0.53 A15 0.67 0.58 0.54 0.47 0.44 0.48 0.46 0.56 0.44 0.52 0.60 0.24 0.60 0.59 1 0.59 0.36 0.35 0.63 A16 0.74 0.45 0.26 0.18 0.16 0.37 0.62 0.35 0.30 0.30 0.67 0.46 0.71 0.43 0.59 1 0.38 0.43 0.39 A17 0.18 0.73 0.47 0.76 0.55 0.59 0.63 0.28 0.33 0.31 0.20 -0.23 0.37 0.57 0.36 0.38 1 0.52 0.56 A18 0.21 0.74 0.21 0.43 0.36 0.17 0.58 0.62 0.57 0.53 0.45 0.25 0.39 0.58 0.35 0.43 0.52 1 0.60 A19 0.29 0.78 0.57 0.54 0.41 0.50 0.57 0.63 0.61 0.81 0.39 0.05 0.24 0.53 0.63 0.39 0.56 0.60 1 correlations between log losses in adjacent areas
Tab. 16 The following results have been achieved by Monte Carlo studies of 1,000,000 simulations each, based on various choices of the grid constants i n m º for fixed numbers m . First we show scatterplots of each 5,000 simulated adaptive Bernstein copula points for selected area combinations with high correlations (Area 1 vs. Area 13, Area 3 vs. Area 4, Area 7 vs. Area 16) and a low correlation (Area 3 vs. Area 18). For comparison purposes, we start with m = which corresponds to an extreme overfitting of the given data. Fig. 39 Fig. 40
Fig. 41 Fig. 42 The following scatterplots correspond to the choice m = which represents the ordinary Bernstein cop-ula approach. Fig. 43 Fig. 44
Fig. 45 Fig. 46 The following scatterplots correspond to the choice m = which represents a slightly adapted Bernstein copula approach. Fig. 47 Fig. 48
Fig. 49 Fig. 50 The subsequent scatterplots correspond to the choice m = which represents a moderately adapted Bernstein copula approach. Fig. 51 Fig. 52
Fig. 53 Fig. 54 The final scatterplots correspond to the choice m = which represents a strongly adapted Bernstein cop-ula approach. Fig. 55 Fig. 56
Fig. 57 Fig. 58 As can clearly be seen, the choice of m influences significantly the shape of the adapted Bernstein copula. With decreasing magnitude of m , we see a more uniform distribution of adapted Bernstein copula points, as was expected. The following table shows estimated VaR -estimates depending on the choice of m . m
100 20 17 13 7
VaR 2,842 2,247 2,204 2,105 1,878
Tab. 17 In contrast to the analysis of Example 3 (cf. Tab. 10) we see here that the choice of grid constants has a major influence on the estimated risk measure. The overfitted
VaR for m = is more than 50% higher than the VaR for m =
7. Conclusion
Adaptive Bernstein copulas are an interesting tool for smoothing or, if desired, also sharpening the em-pirical dependence structure in particular in risk management applications when the number of observa-tions and dimensions is moderate to large. The possibility of a smoothing of the dependence structure prevents in particular a kind of overfitting to copula models. In particular, the choice of the grid constants in the reduction procedure is arbitrary, the selected grid constants need not to be divisors of the number of observations. The method presented here also enables Monte Carlo studies for the comparison of different estimates of risk measures or the shape of the aggregate risk distribution. If the various estimates for the risk measure do not differ much for several adaptive strategies, this could make a sensitivity analysis for instance under Solvency II more profound. In other cases, when significant differences in the estimation of risk measures become apparent under various adaptive strategies, one should be cautious with a mere statistical risk assessment. Anyway, a kind of a worst case analysis derived from different approaches could be helpful here. The method of reducing (or, if desired, sharpening) the complexity in the rank structures of the data might also be applied to more general partition-of-unity copulas, see [18], [19] and [21]. With such copulas, tail dependence can be introduced to the dependence models which cannot be obtained by Bernstein copulas alone due to the boundedness of the corresponding densities.
Acknowledgements:
The critical comments of the referees which lead to a more clear presentation of the content are gratefully acknowledged.
Funding:
The publication of this paper was funded by the University of Oldenburg.
References [1]
J-L. Avomo Ngomo (2017): Entwicklung und Implementierung eines Verfahrens zur Optimierung des Speicheraufwands bei Bernstein- und verwandten Copulas. Ph.D. Thesis, Universität Oldenburg. http://oops.uni-oldenburg.de/3457/ [2]
G.J. Babu, A.J. Canty and Y.P. Chaubey (2002): Application of Bernstein polynomials for smooth estimation of a distribution and density function. J. Stat. Plann. Inf. 105, 377 – 392. [3]
S. Bernstein (1912): Démonstration du théorème de Weierstrass fondée sur le calcul des probabilités. Commun. Soc. Math. Kharkow (2), 13 (1912-13), 1-2. [4]
P.L. Butzer (1953): On two-dimensional Bernstein polynomials. Can. J. Math. 5,107 – 113. [5]
C. Cheng (1995): The Bernstein polynomial estimator of a smooth quantile function. Stat. Probab. Letters 24, 321 – 330. [6]
C. Cottin and D. Pfeifer (2014): From Bernstein polynomials to Bernstein copulas. J. Appl. Funct. Anal. 9(3-4), 277 – 288. [7]
P. Deheuvels (1979): La fonction de dépendance empirique et ses propriétés. Un test nonparamétrique d’indépendance. Acad. Roy. Belg. Bull. Cl. Sci. 65(5), 274 – 292. [8] F. Durante and C. Sempi (2016): Principles of Copula Theory. CRC Press, Taylor & Francis Group, Boca Raton. [9]
Z. Guan (2016): Efficient and robust density estimation using Bernstein type polynomials. J. Non-param. Stat. 28, 250 – 271. [10]
Z. Guan (2017): Bernstein polynomial model for grouped continuous data. J. Nonparam. Stat. 29, 831 – 848. [11]
R. Ibragimov and A. Prokhorov (2017): Heavy Tails and Copulas. Topics in Dependence Modelling in Economics and Finance. Worls Scientific, Singapore. [12]
R.R. Junker, F. Griessenberger and W. Trutschnig (2021): Comp. Stat. and Data Analysis 153 (2021), 107058. [13]
A. Leblanc (2012): On estimating distribution functions using Bernstein polynomials. Ann. Inst. Stat. Math. 64, 919 – 943. [14]
G.G. Lorentz (1986): Bernstein Polynomials. 2 nd Ed., Chelsea Publishing Company, N.Y. [15]
A. Masuhr and M. Trede (2020): Bayesian estimation of generalized partition of unity copulas. De-pend. Model. 2020 (8),119–131 [16]
D. Pfeifer and J. Nešlehová (2003): Modeling dependence in finance and insurance: the copula ap-proach. Blätter der DGVFM Band XXVI, Heft 2 , 177 - 191. [17]
D. Pfeifer, D. Straßburger and J. Philipps (2009): Modelling and simulation of dependence structures in nonlife insurance with Bernstein copulas. Paper presented on the occasion of the International ASTIN Colloquium June 1 – 4, 2009, Helsinki. http://arxiv.org/abs/2010.15709 [18]
D. Pfeifer, H.A. Tsatedem, A. Mändle and C. Girschig (2016): New copulas based on general parti-tions-of-unity and their applications to risk management. Depend. Model. 4, 123 – 140. [19]
D. Pfeifer, A. Mändle and O. Ragulina (2017): New copulas based on general partitions-of-unity and their applications to risk management (part II). Depend. Model. 5, 246 – 255. [20]
D. Pfeifer and O. Ragulina (2018): Generating VaR Scenarios under Solvency II with Product Beta Distributions. Risks 6(4), 122. [21]
D. Pfeifer, A. Mändle O. Ragulina and C. Girschig (2019): New copulas based on general partitions-of-unity (part III) – the continuous case. Depend. Model. 7, 181 – 201. [22]
D. Rose (2015): Modeling and estimating multivariate dependence structures with the Bernstein copula. Ph.D. Thesis, Ludwig-Maximilians-Universitaet, Munich. https://edoc.ub.uni-muenchen.de/18757/ [23]
A. Sancetta, S.E. Satchell (2004): The Bernstein copula and its applications to modelling and ap-proximations of multivariate distributions. Econometric Theory 20(3), 535 – 562. [24]
J. Segers, M. Sibuya and H. Tsukahara (2017): The empirical beta copula. J. Multivar. Anal. 155 (C), 35 – 51. [25]
N.O.Vil'chevskii and G.L.Shevlyakov (2001): On the Bernstein polynomial estimators of distributio-and quantile functions. J. of Math. Sciences 105 (6), 2626 – 2629. [26][26]