Non-Negative Risk Components

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