Conditional Tail Expectation Decomposition and Conditional Mean Risk Sharing for Dependent and Conditionally Independent Losses

Conditional tail expectations are often used in risk measurement and capital allocation. Conditional mean risk sharing appears to be effective in collaborative insurance, to distribute total losses among participants. This paper develops analytical results for risk allocation among different, correlated units based on conditional tail expectations and conditional mean risk sharing. Results available in the literature for independent risks are extended to correlated ones, in a unified way. The approach is applied to mixture models with correlated latent factors that are often used in practice. Conditional Monte Carlo simulation procedures are proposed in that setting.