# Value-at-Risk Diversification of $\alpha$-stable Risks: The Tail-Dependence Puzzle

## TL;DR

This paper investigates how tail dependence affects the limits of diversification in risk management for heavy-tailed risks, revealing that high tail dependence reduces diversification benefits and that VaR can become additive at extreme tails.

## Contribution

It provides new insights into the impact of tail dependence structures on the aggregation of VaR for $oldsymbol{ ext{α}}$-stable risks, highlighting limitations in diversification strategies.

## Key findings

- Sub-additivity violations are prevalent for risks with low tail index and positive dependence.
- Higher tail dependence levels lead to lower diversification limits across all dependence levels.
- At extreme tails, aggregated VaR becomes additive beyond a certain dependence threshold, which is lower for copulas with weaker tail dependence.

## Abstract

We consider the problem of risk diversification of $\alpha$-stable heavy tailed risks. We study the behaviour of the aggregated Value-at-Risk, with particular reference to the impact of different tail dependence structures on the limits to diversification. We confirm the large evidence of sub-additivity violations, particularly for risks with low tail index values and positive dependence. So, reinsurance strategies are not allowed to exploit diversification gains, or only a very limited amount of them. Concerning the impact of tail dependence, we find the peculiar results that for high tail dependence levels the limits to diversification are uniformly lower for all the levels of dependence, and for all levels of $\alpha<2$. The result is confirmed as we move towards extreme points in the tail: in this case, we show that at some point in the tail the aggregated VaR becomes additive above some level of dependence, but this critical dependence level is lower for copulas with lower tail dependence.