Modelling parameter uncertainty for risk capital calculation

For risk capital calculation within the framework of Solvency II the possible loss of basic own funds over the next business year of an insurance undertaking is usually interpreted as a random variable \( \varvec{X} \). If we assume that the parametric distribution family \(\left\{ \varvec{X} (\theta )|\theta \in I\subseteq \mathbb {R}^d \right\} \) is known, but the parameter \(\theta \) is unknown and has to be estimated from the available historical data, the undertaking faces parameter uncertainty. To assess methods to model parameter uncertainty for risk capital calculations we apply a criterion going back to the theory of predictive inference which has already been used in the context of Solvency II. In particular, we show that the bootstrapping approach is not appropriate to model parameter uncertainty from the undertaking’s perspective. Based on ideas closely related to the concept of fiducial inference we introduce a new approach to model parameter uncertainty. For a wide class of distributions and for common estimators including the maximum likelihood method we prove that this approach is appropriate to model parameter uncertainty according to the criterion mentioned above. Several examples demonstrate that our method can easily be applied in practice.