Ordering and Inequalities for Mixtures on Risk Aggregation

TL;DR

This paper explores how different ways of mixing marginal distributions affect the size of risk aggregation sets and the bounds of risk measures, with implications for regulatory capital and portfolio diversification.

Contribution

It introduces ordering relations between aggregation sets under distribution and quantile mixtures, revealing how homogenization impacts model uncertainty and risk bounds.

Findings

More homogeneous marginals increase aggregation set size.

Order relation on VaR under quantile mixture with monotone densities.

Numerical visualizations support theoretical results and conjectures.

Abstract

Aggregation sets, which represent model uncertainty due to unknown dependence, are an important object in the study of robust risk aggregation. In this paper, we investigate ordering relations between two aggregation sets for which the sets of marginals are related by two simple operations: distribution mixtures and quantile mixtures. Intuitively, these operations ``homogenize" marginal distributions by making them similar. As a general conclusion from our results, more ``homogeneous" marginals lead to a larger aggregation set, and thus more severe model uncertainty, although the situation for quantile mixtures is much more complicated than that for distribution mixtures. We proceed to study inequalities on the worst-case values of risk measures in risk aggregation, which represent conservative calculation of regulatory capital. Among other results, we obtain an order relation on VaR under quantile mixture for marginal distributions with monotone densities. Numerical results are presented to visualize the theoretical results and further inspire some conjectures. Finally, we provide applications on portfolio diversification under dependence uncertainty and merging p-values in multiple hypothesis testing, and discuss the connection of our results to joint mixability.