An impossibility theorem on capital allocation

Two natural and potentially useful properties for capital allocation rules are top-down consistency and shrinking independence. Top-down consistency means that the total capital is determined by the aggregate portfolio risk. Shrinking independence means that the risk capital allocated to a given business line should not be affected by a proportional reduction of exposure in another business line. These two properties are satisfied by, respectively, the Euler allocation rule and the stress allocation rule. We prove an impossibility theorem that states that these two properties jointly lead to the trivial capital allocation based on the mean. When a subadditive risk measure is used, the same result holds for weaker versions of shrinking independence, which prevents the increase in risk capital in one line, when exposure to another is reduced. The impossibility theorem remains valid even if one assumes strong positive dependence among the risk vectors.