Tail Risk Measurement: Estimation, Sensitivity, Uncertainty

Regulatory frameworks such as Solvency II require non-life insurers to quantify extreme risks — most notably the 99.5% Value-at-Risk over a one-year horizon for the Solvency Capital Requirement (SCR). In practice, however, this poses a fundamental challenge: historical loss data contains little to no information about such rare events, making direct estimation inherently unreliable.

This web session addresses exactly this gap. It provides participants with a structured and practical toolkit to estimate high-confidence risk measures from limited data — and, crucially, to understand and communicate the uncertainty involved, enabling more robust risk quantification, particularly in contexts such as SCR validation and ORSA.

Starting with classical parametric approaches and kernel density estimation, the course progresses to Extreme Value Theory (EVT), with a focus on the Peaks-over-Threshold (POT) method and the Generalised Pareto Distribution (GPD). Particular attention is given to threshold selection and to Bayesian formulations in which the threshold is treated as an uncertain parameter, enabling posterior-predictive inference for high quantiles. We also consider flexible bulk-tail mixture models that combine non-parametric bulk estimation with an EVT-based tail component.

For each method, the course takes a structured perspective across four dimensions:

• the point estimator and its finite-sample properties,
• parameter uncertainty and confidence intervals,
• sensitivity to modelling assumptions, and
• overall model uncertainty.

Finally, we connect tail risk modelling to practical risk steering by linking estimated risk measures to capital allocation via the Euler (gradient) principle, enabling a decomposition into marginal risk contributions across business units or risk types.

For the core methods, participants apply estimation procedures in hands-on R exercises using real non-life insurance loss data, developing both technical proficiency and the critical judgement required to interpret results.

The course provides a critical overview of methods for estimating tail risk at high confidence levels under real-world data constraints, examining where and why they differ in their conclusions.

R exercises on real non-life insurance data illustrate the methods in practice, with particular attention to the interpretation and limitations of the resulting estimates.