Diversification limit of quantiles under dependence uncertainty

In this paper, we investigate the asymptotic behavior of the portfolio diversification ratio based on Value-at-Risk (quantile) under dependence uncertainty, which we refer to as “worst-case diversification limit”. We show that the worst-case diversification limit is equal to the upper limit of the worst-case diversification ratio under mild conditions on the portfolio marginal distributions. In the case of regularly varying margins, we provide explicit values for the worst-case diversification limit. Under the framework of dependence uncertainty the worst-case diversification limit is significantly higher compared to classic results obtained in the literature of multivariate regularly varying distributions. The results carried out in this paper bring together extreme value theory and dependence uncertainty, two popular topics in the recent study of risk aggregation.