# Parameter uncertainty for integrated risk capital calculations based on normally distributed subrisks

## TL;DR

This paper investigates how to properly account for parameter uncertainty in risk capital calculations when summing normally distributed subrisks, proposing a new integrated model that improves confidence level approximations at both subrisk and overall risk levels.

## Contribution

It provides a theoretical proof and a new method for modeling parameter uncertainty in integrated risk capital calculations with normally distributed subrisks.

## Key findings

- Theoretical result for appropriate risk capital modeling under parameter uncertainty.
- A method to improve confidence level approximation for both subrisks and overall risk.
- Demonstrates that subrisk-level appropriateness does not guarantee overall risk appropriateness.

## Abstract

In this contribution we consider the overall risk given as the sum of random subrisks $\mathbf{X}_j$ in the context of value-at-risk (VaR) based risk calculations. If we assume that the undertaking knows the parametric distribution family subrisk $\mathbf{X}_j=\mathbf{X}_j(\theta_j)$, but does not know the true parameter vectors $\theta_j$, the undertaking faces parameter uncertainty. To assess the appropriateness of methods to model parameter uncertainty for risk capital calculation we consider a criterion introduced in the recent literature. According to this criterion, we demonstrate that, in general, appropriateness of a risk capital model for each subrisk does not imply appropriateness of the model on the aggregate level of the overall risk.\\ For the case where the overall risk is given by the sum of normally distributed subrisks we prove a theoretical result leading to an appropriate integrated risk capital model taking parameter uncertainty into account. Based on the theorem we develop a method improving the approximation of the required confidence level simultaneously for both - on the level of each subrisk as well as for the overall risk.