Measuring adequately the benefit of diversification in the extreme quantiles: An inquiry into covariation on the brink of catastrophe

The aim of this work is to better understand the nature of covariation in the vicinity of extremes on financial data and assess whether the usual assumptions and covariation measures fits the actual data. For simplicity, we consider pairs of random variables. In order to identify the shape of the covariation all along the distribution, and particularly as the extreme quantiles are approached, we describe the contribution of each of the variables from a random couple to the quantiles of the weighted sum of these variables. This approach makes sense since it can be interpreted in terms of Value-at-Risk in a financial institution: the VaR of the sum of variables may represent the capital requiremet for a diversified conglomerate, while the sum of VaR of the variables would correspond to the capital requirements for the components of the conglomerate, without taking diversification into account. The ratio of these two quantities appears as a good measure of both the benefit of diversification and the decorrelation of variables. We thus compare the values of quantiles and ratio taken from a representative dataset to the values obtained from various simulations relying on the usual assumptions. The result of this comparison is that the usual assumptions do not correctly model the covariation of the real-word data. In particular, the usual assumptions tend to exaggerate the correlation in the vicinity of extreme loss while the benefit of diversification is uniform across distribution. Additional simulations and modelling assumptions may be required to assess the generality of this result.