A tale of two allocations: Risk capital contributions versus risk contributions in the tail

RESUMO

Compares the fraction of aggregate tail risk capital allocated to a component under Conditional Tail Expectation (CTE) with the component's expected realized share of aggregate risk under Geometric Tail Expectation (GTE). The allocations typically diverge; conditions for agreement are established. Both are weighted averages of conditional risk shares, with differences explained by tail covariance. Results hold for arbitrary random vectors; among independent exponential dispersion models, exact agreement is possible only for the scaled Poisson family. Extreme-tail behaviour depends on the Expected Shortfall to Value-at-Risk ratio, which determines whether the allocations merge or separate.