Risk measures that are not subadditive may penalize the aggregation of risk, by inducing portfolio requirements that are larger than those of undiversified positions. This happens for instance for Value-at-Risk (VaR), as well as convex shortfall risk measures. In this paper we characterize the potential for superadditivity that any risk measure may exhibit, considering both dependence uncertainty as well as the effect of portfolio size.
It is shown that for the wide majority of risk measures of use or interest this corresponds to calculating the smallest dominating coherent risk measure (SDCRM). We show that this risk measure often exists and is identified with the notion of extreme-aggregation risk measure introduced in this paper. Explicit results are provided for the class of distortion risk measures, where the SDCRM is again a distortion risk measure and for  the class of shortfall risk measures, where the SDCRM is given by an expectile.

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Cordiali saluti,

Gabriele Stabile