An Impossibility Theorem on Capital Allocation

Abstract

Two natural and potentially desirable 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 satised by, respectively, the Euler allocation
rule and the stress allocation rule. We prove an impossibility theorem which 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.

Keywords: Euler allocation, stress scenarios, top-down consistency, shrinking independence

JEL Classification: C70