# Modelling Extremal Dependence for Operational Risk by a Bipartite Graph

## TL;DR

This paper develops a bipartite graph-based statistical model for operational risk that captures heavy-tailed loss distributions and dependence structures, enabling accurate tail risk estimation and capital allocation.

## Contribution

It introduces a novel bipartite graph model for operational risk that explicitly accounts for heavy tails and dependence, with new estimation methods tested on real and simulated data.

## Key findings

- Reliable tail risk estimates with limited data
- Dependence quantification significantly impacts capital allocation
- Model effectively captures heavy-tailed operational losses

## Abstract

We introduce a statistical model for operational losses based on heavy-tailed distributions and bipartite graphs, which captures the event type and business line structure of operational risk data. The model explicitly takes into account the Pareto tails of losses and the heterogeneous dependence structures between them. We then derive estimators for individual as well as aggregated tail risk, measured in terms of Value-at-Risk and Conditional-Tail-Expectation for very high confidence levels, and provide also an asymptotically full capital allocation method. Estimation methods for such tail risk measures and capital allocations are also proposed and tested on simulated data. Finally, by having access to real-world operational risk losses from the Italian banking system, we show that even with a small number of observations, the proposed estimation methods produce reliable estimates, and that quantifying dependence by means of the empirical network has a big impact on estimates at both individual and aggregate level, as well as for capital allocations.