A new skewness adjustment for Solvency II SCR standard formula

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Under the Solvency Capital Requirement (SCR) formula of European Solvency II, the required capital for an insurer is calculated as the Value-at-Risk of the aggregate loss, utilizing a modular approach with a set of fixed correlations among business lines. This square-root method, while well-aligned with the multivariate normal model, does not easily extend to general loss distributions which are often right-skewed or heavy-tailed. To address this limitation, Sandström (Solvency II: Calibration for skewness, SAJ, 2007(2)) introduced a skewness adjustment term in the tail measure to improve SCR estimation. However, estimating the portfolio skewness remains challenging as it requires third-order cross-moments among the risks, which are generally impossible to derive from the prescribed correlations. In this paper, we propose a method to estimate portfolio skewness using the Normal Power approximation. By leveraging high-order cross-moments of the standard normal variable, our proposed method effectively calibrates portfolio skewness, while maintaining the mandated correlations. Additionally, we apply extreme value theory to estimate the Expected Shortfall within our framework. Our method is consistent with the current SCR framework and well-suited for insurance supervision that requires a balance between precision and simplicity. Our simulation study shows that the new approach performs significantly better than existing alternatives.