A generalization of the Aumann–Shapley value for risk capital allocation problems

Abstract

The paper proposes a new method to allocate risk capital to divisions or lines of business within a firm. Existing literature advocates an allocation rule that, in game-theoretic terms, is equivalent to using the Aumann–Shapley value as allocation mechanism. The Aumann–Shapley value, however, is only well-defined if a specific differentiability condition is satisfied. The rule that we propose is characterized as the limit of an average of path-based allocation rules with grid size converging to zero. The corresponding allocation rule is equal to the Aumann–Shapley value if it exists. If the Aumann–Shapley value does not exist, the allocation rule is equal to the weighted average of the Aumann–Shapley values of “nearby” capital allocation problems.