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<p>Risk aggregation under Solvency II requires appropriate modelling of dependencies between risk factors. The variance–covariance method used in the Standard Formula, based on fixed correlations, often fails to capture nonlinear and tail dependencies, leading to biased Solvency Capital Requirement (SCR) estimates. However, existing frameworks clearly lack data-driven methods that can flexibly model such complex dependencies. This study proposes a nonparametric deep neural network (DNN) approach that identifies marginal distributions and the copula function governing aggregated risk. Using data from German and Austrian non-life insurers’ Solvency and Financial Condition Reports (SFCRs) from 2016 to 2023, we compare three frameworks: the variance–covariance method used in the Standard Formula, a vine copula model, and the proposed DNN model. Model fit is evaluated using the energy distance metric, which quantifies multivariate distributional differences. Results show that the DNN approach improves dependence modeling accuracy and yields more realistic diversification effects (DE) for a selected set of four non-life segments, thereby supporting a more reliable SCR assessment. The framework offers a flexible, data-driven enhancement to internal model methodologies under Solvency II.</p>

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Nonlinear dependencies in Solvency II: risk aggregation with deep neural networks

  • 期刊论文

  • 发表日期：

    2026年2月10日

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  • Anna Denkowska,
  • Krystian Szczȩsny,
  • Stanisław Wanat

摘要

Risk aggregation under Solvency II requires appropriate modelling of dependencies between risk factors. The variance–covariance method used in the Standard Formula, based on fixed correlations, often fails to capture nonlinear and tail dependencies, leading to biased Solvency Capital Requirement (SCR) estimates. However, existing frameworks clearly lack data-driven methods that can flexibly model such complex dependencies. This study proposes a nonparametric deep neural network (DNN) approach that identifies marginal distributions and the copula function governing aggregated risk. Using data from German and Austrian non-life insurers’ Solvency and Financial Condition Reports (SFCRs) from 2016 to 2023, we compare three frameworks: the variance–covariance method used in the Standard Formula, a vine copula model, and the proposed DNN model. Model fit is evaluated using the energy distance metric, which quantifies multivariate distributional differences. Results show that the DNN approach improves dependence modeling accuracy and yields more realistic diversification effects (DE) for a selected set of four non-life segments, thereby supporting a more reliable SCR assessment. The framework offers a flexible, data-driven enhancement to internal model methodologies under Solvency II.