Journal of Risk

Nonnegative risk components

Jeremy Staum

  • We propose new methods to decompose risk into a sum of nonnegative components.
  • The proposed methods give significantly different answers from the usual method.
  • The proposed methods have good theoretical properties that the usual method lacks.


We propose two new methods for attributing the risk of a portfolio or system to its components, in particular when it is required to produce nonnegative risk components that sum to the risk of the portfolio or system as a whole. The first method attributes risk entirely to losses, taking profits for granted. The second method allows profits in some scenarios to offset losses in other scenarios to a certain extent, but not in a way that could yield a negative risk component.We illustrate these methods by applying them to an example in which we attribute a firm's expected shortfall to business units within the firm and an example in which we attribute systemic risk to banks.We prove that, under appropriate conditions, the methods proposed have some game-theoretic properties that are desirable for risk attribution.

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