Risk contributions: duality and sensitivity

Abstract

Given a decomposition of a portfolio P as a sum of K components, practitioners commonly decompose the risk of P as a corresponding sum of risk contributions. In this paper, we prove two theorems about risk contributions. The first theorem concerns a form of duality identified in Grinold [J. Portfolio Manage. 2011, 37(2), 15–30], which may be described as follows. When we view a portfolio decomposition as a coordinate representation of the portfolio with respect to a given vector-space basis, then there is a natural dual basis with respect to which there is an alternative decomposition, here referred to as the dual decomposition. The dual decomposition gives the same contributions to risk as the original decomposition. The first theorem gives necessary and sufficient conditions for a change of basis to preserve risk contributions, and shows that all such changes of basis can be explained in terms of dual decompositions. The second theorem explores sensitivity of portfolio risk to a risk regime change and indicates that large risk contributions or large risks of the components of a decomposition may be harbingers of high sensitivity. This provides a motivation for the practice of reporting both the risk contributions and the risks of the components in a decomposition.

Keywords:

• Risk contribution
• Contribution to risk
• Risk decomposition
• Risk attribution
• Constraint attribution
• Alternating bilinear forms

JEL Classification:

• 60J10
• 15A18
• 15A51

Acknowledgements

The author thanks Ketan Prajapati and other participants at the Society of Quantitative Analysts Half Day Conference on November 10, 2016, in New York, for many helpful comments. He also thanks the referees for their careful reading and generous guidance.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

This paper represents the views of the author and not necessarily those of QS Investors.

1 A similar symmetry applies when only rows of F, G, H, and L are permuted, but this will not serve our purpose because, when we apply lemma 5.6 in the proofs of proposition 5.7 and theorem 5.9, the matrix L will be symmetric and it will be convenient to preserve this symmetry.