مدل سازی وابستگی در بیمه های غیر عمر با استفاده از وسیله اتصال برنشتاین

This paper illustrates the modeling of dependence structures of non-life insurance risks using the Bernstein copula. We conduct a goodness-of-fit analysis and compare the Bernstein copula with other widely used copulas. Then, we illustrate the use of the Bernstein copula in a value-at-risk and tail-value-at-risk simulation study. For both analyses we utilize German claims data on storm, flood, and water damage insurance for calibration. Our results highlight the advantages of the Bernstein copula, including its flexibility in mapping inhomogeneous dependence structures and its easy use in a simulation context due to its representation as mixture of independent Beta densities. Practitioners and regulators working toward appropriate modeling of dependences in a risk management and solvency context can benefit from our results.

Using copulas in risk management has become popular in both academia and practice recently. Copula applications are presented for modeling dependence between stock returns (Jondeau and Rockinger, 2006), CDO pricing (Hofert and Scherer, 2011), currency option pricing (Salmon and Schleicher, 2006), or internal risk models (Eling and Toplek, 2009); further areas of application can, e.g., be found in Genest et al. (2009a). However, several popular copulas, such as elliptical and Archimedean copulas, exhibit a certain degree of symmetry or are restricted to certain correlation structures, which is not always suitable for risk modeling in practice. The Bernstein copula, which has only recently received attention in an insurance context (Pfeifer et al., 2009), has the potential to overcome these drawbacks while still being applicable in higher dimensions. It is a flexible, non-parametric copula capable of approximating any copula arbitrarily well and thus may serve as a model for an unknown underlying dependence structure (Sancetta and Satchell, 2004). We explore this flexibility in a realistic environment by fitting the Bernstein copula to empirical claims data from six lines of business and simulating the aggregate value-at-risk and tail-value-at-risk. Our data are from lines of business driven by exposure to natural perils. The resulting portfolio is difficult to model since different lines are combined, resulting in an inhomogeneous dependence structure. Goodness-of-fit is an aspect usually not considered in the literature on copula modeling and calibration (Embrechts, 2009). Thus, this paper contributes to the literature by implementing the Bernstein copula in a higher dimension, assessing its goodness-of-fit in the modeling of dependence structures of non-life insurance risks, and illustrating its use in a simulation context. For this purpose we use the representation of the Bernstein copula as a mixture of Beta densities.1 This representation facilitates an efficient random sampling algorithm, which has, to our knowledge, so far not been applied in the context of the Bernstein copula. To preview our results, we show that the Bernstein copula performs especially well when multiple risk classes with inhomogeneous dependence structure are combined. We conclude that the Bernstein copula is a promising alternative for modeling dependence structures in internal risk models. Practitioners working on calibration and implementation of such models, as well as regulators responsible for the validation of internal risk models, can benefit from our results. The remainder of this paper is organized as follows. In Section 2, we present the analyzed copulas, parameter estimation, and sampling algorithms. In Section 3, we introduce the data. Section 4 reports the results of the goodness-of-fit analysis. In Section 5, we show the value-at-risk and tail-value-at-risk simulation results. We conclude in Section 6.

This paper illustrates the modeling of dependence structures of non-life insurance risks using the Bernstein copula. We conduct a goodness-of-fit analysis to assess the Bernstein copula’s fit compared to that of other widely used copulas (parametric elliptical and Archimedean, as well as other non-parametric copulas) and illustrate its usage in a simulation context. Real-world claims data are used to calibrate the dependence structure. Goodness-of-fit analyses are not often made in copula modeling, but are important for interpreting the results accurately. We use a blanket test based on the Cramér–von Mises test statistic and bootstrapping for approximation of pp-values. Our results show that the Bernstein copula is a flexible approach, but not the solution to all modeling problems. For example, in our six-dimensional dataset we found other approaches to have a better fit. However, the fit of the Bernstein copula can be improved by increasing the grid size. We also illustrate that the Bernstein copula performs well when used with inhomogeneous or less symmetric datasets. Additionally, simulating different risk measures according to the fitted dependence structure illustrates that inadequate modeling may yield an over- or underestimation of the risk situation compared to the alternative with the best fit. A general benefit of the Bernstein copula is its applicability in higher dimensions and the usage of all available information, which is especially not the case for NACs, in which an aggregate parameter is employed. Thus even though the Bernstein copula’s performance depends on data and calibration, our results are important for insurance companies and regulators who rely on stochastic models for determining solvency capital requirements under Solvency II. The results may also be useful in conducting the “Own Risk and Solvency Assessment” that is required by Solvency II. This analysis also highlights several practical issues regarding the implementation and application of the Bernstein copula that need further research and that are of high importance for practitioners working on the implementation of such models. The representation of the Bernstein copula as a mixture of Beta distributions greatly increases sampling efficiency, but the choice of grid size is an important aspect influencing fit and variance of the estimator. Some limitations apply to this analysis. The number of data points used to calibrate the dependence structure is relatively small, but as these are realistic datasets and calibration of internal models is usually conducted on a monthly or yearly basis, insurance companies face the same limitations. We choose the grid size equal for each dimension and equidistant; relaxing this condition could further improve estimation results. Additionally, further tests of statistical fit that do not rely on the empirical copula as benchmark could be insightful. Overall, the Bernstein copula is a promising modeling alternative and can enrich the set of copulas used in internal risk models, especially in those cases where the dependence structure is inhomogeneous, not extremely highly correlated, and data are sparse. The empirical part of this paper demonstrates that these are frequently found conditions and thus provides a sound motivation for application of the Bernstein copula.