Adaptive Bernstein Copulas and Risk Management

We present a constructive approach to Bernstein copulas with an admissible discrete 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 compare different approaches of Bernstein and Gaussian copulas w.r.t. the estimation of risk measures in risk management.


Introduction
Since the pioneering paper by Serge Bernstein in 1912 [3] Bernstein polynomials have been an indispensable tool in calculus and approximation theory (see e.g. [14]). Bernstein copulas, which can be considered as Bernstein polynomials for empirical and other copula functions, have a long tradition in nonparametric 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 research papers on this topic, in particular [2], [5], [6], [9], [10], [13], [16], [17], [22], [23], [24] and [25]. The monographs [8] and [11] have, in particular, devoted separate chapters to the topic of Bernstein copulas.
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 performed easily with ordinary spreadsheets, in particular in applications concerning quantitative risk management.
From a statistical point of view, the problem of a potential overfitting of the true underlying dependence structure with Bernstein polynomials emerges naturally. Clearly the Bernstein copula density becomes more wiggly the more empirical observations are used in the analysis. In comparison with classical parametric dependence models such as elliptically contoured or Archimedean copulas, this is probably a nondesirable 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 Bernstein 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 [23], however without a possible application to a general complexity reduction of Bernstein copulas.

Some important facts about multivariate Bernstein polynomials
Let f be an arbitrary bounded real-valued function on the unit cube can be written as a linear combination of statistical product beta densities. For this purpose, consider univariate beta densities where B( , ) a b denotes the Euler Beta-function, i.e.

( )
We need a further definition to proceed.
£ £ (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 ( )  3 3 Note that here (8) or, in tabular form,   ( , ) 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 .
It is obvious by the iterated mean value theorem that for a sufficiently smooth function g, d-monotonicity is equivalent to Note that in case that g is a d-dimensional cumulative distribution function of a probability measure P on Proposition 2. Let f be a real-valued d-monotone function on . Proof. By the arguments above and the notation as in Proposition 1, we have which also explicitly shows that the Bernstein polynomial B g n is 2-monotone.

From Bernstein polynomials to
also is a cumulative distribution function since B F n is non-negative and d-increasing with In particular, the Bernstein polynomial density b F n always is a (probabilistic) mixture of product beta densities as explicitly noted in [6] and [23] for Bernstein copulas. Note also that this observation was the motivation for the setup in [20].
with a discrete distribution con- The following graphs show the corresponding cumulative distribution function F as well as the corresponding Bernstein polynomials B F n and densities b F n for various choices of 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 1 d > be a natural number and denote the vector x where the k-th component is replaced by y.
such that for given natural numbers 1 Proof: By Remark 1 above we know that B F n also is a cumulative distribution function with for 1, , k d =  and 0 1 x £ £ ( k n x is the expectation of the Binomial distribution with k n trials and success 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] is a copula. The corresponding Bernstein copula density b F n is given by Proof. For , 1, , we call V an admissible discrete skeleton if the marginal distributions 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 corresponding Bernstein copula density is a mixture of product beta kernels with weights given by the individual probabilities representing the admissible skeleton.

Empirical Bernstein copulas
Bernstein copulas can be easily constructed from independent samples 1 , , , n n Î X X   of ddimensional random vectors with the same intrinsic dependence structure and the same marginal distributions. 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 Deheuvel'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 This has also been observed in [23], 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]. The following graphs show some plots for the empirical Bernstein copula. It is clearly to be seen 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. in [17]. The disadvantage of the method proposed there is, however, that the number of support points of * U gets dramatically larger and is typically of exponential order with increasing sample size. We therefore propose a simpler way how to find a smaller discrete approximating skeleton * U with an arbitrary given grid size in the subsequent chapter.

Adaptive Bernstein copula estimation
We start with the individual ranks ij r of the observation vectors ( ) 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 1 d n ń where the i n are typically smaller than n. We proceed in two steps:

Step: Augmentation
Select an integer M such that all , 1, , i n i d =  are divisors of M, for instance their least common multiple. We construct pseudo-ranks ij r + in the following way: Note that the probability mass is 1 M n ⋅ for each support point, and that + U is an admissible discrete skeleton.

Step: Reduction
Construct the final ranks ij r * in the following way: It follows from the above definition, that there will be replicates in the final ranks and that 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 [23].
In the augmentation step, M-wise partial permutations would not change the result but would create a more "random" augmentation of the original ranks.   , 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 reduction of complexity is not really discussed there.

Applications to risk management
In this chapter we 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 34 n = of the original observations was first reduced to 1 2 10 n n = = by a least squares technique. The resulting optimal discrete skeleton with probabilities 1 2 , , 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 1 c denotes the Bernstein copula density derived from Tab. 8,2 c denotes the Bernstein copula density derived from Tab. 9    In a final step, we compare estimates for the risk measure Value at Risk VaR a with the risk level 0.5% 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 1 2 34 n n = = as well as the adaptive Bernstein copulas with 1 2 10, n n = = 1 2 5 n n = = and 1 2 4. 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. The following

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 following graphs show empirical histograms for these densities under the models considered above, from 100,000 simulations each. Bernstein, grid type 34 34 Bernstein, grid type 10 10  Note that the histogram for the full Bernstein copula has two peaks, whereas the other histograms show a more smooth behaviour.

Conclusion
Adaptive Bernstein copulas are an interesting tool for smoothing the empirical dependence structure in particular in risk management applications when the number of observations is moderate to large. This prevents in particular a kind of overfitting to dependence models. They also enable Monte Carlo studies for the comparison of different estimates of risk measures or the shape of the aggregate risk distribution. If the different estimates for the risk measure do not differ much for various adaptive strategies, this could be helpful for a profound sensitivity analysis under Solvency II.
The method of reducing the complexity in the rank structures of the data could also be applied to 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 due to the boundedness of the corresponding densities.