Upper Comonotonicity and Risk Aggregation under Dependence Uncertainty

TL;DR

This paper investigates how small positive dependencies between risks can dramatically influence tail risk measures, revealing that even minimal dependence can lead to near-perfect tail correlation in risk aggregation scenarios.

Contribution

It introduces the concept of regular dependence measures and demonstrates their impact on tail risk aggregation, extending understanding of dependence effects in risk management.

Findings

Small positive dependence can cause tail risks to align with perfect correlation.

Tail risk of aggregated losses can match that of fully dependent risks under minimal dependence.

Results have implications for credit risk and value at risk calculations.

Abstract

In this paper, we study dependence uncertainty and the resulting effects on tail risk measures, which play a fundamental role in modern risk management. We introduce the notion of a regular dependence measure, defined on multi-marginal couplings, as a generalization of well-known correlation statistics such as the Pearson correlation. The first main result states that even an arbitrarily small positive dependence between losses can result in perfectly correlated tails beyond a certain threshold and seemingly complete independence before this threshold. In a second step, we focus on the aggregation of individual risks with known marginal distributions by means of arbitrary nondecreasing left-continuous aggregation functions. In this context, we show that under an arbitrarily small positive dependence, the tail risk of the aggregate loss might coincide with the one of perfectly correlated losses. A similar result is derived for expectiles under mild conditions. In a last step, we discuss our results in the context of credit risk, analyzing the potential effects on the value at risk for weighted sums of Bernoulli distributed losses.