Notes on Correlation Stress Tests

This note outlines an approach to stress testing of covariance of financial time series, in the context of financial risk management. It discusses how the geodesic distance between covariance matrices implies a notion of plausibility of covariance stress tests. In this approach, correlation stress tests span a submanifold of constant determinant of the Fisher–Rao manifold of covariance matrices. A parsimonious geometrically invariant definition of arbitrarily large correlation stress tests is proposed, and a few examples are discussed.

💡 Research Summary

The paper proposes a rigorous geometric framework for correlation stress testing in financial risk management, based on the Fisher‑Rao information geometry of multivariate normal distributions. It begins by distinguishing two notions of a “correlation break”: (i) an unexpected joint movement of market factors relative to their historical covariance, and (ii) a realised correlation matrix that deviates substantially from the forecasted one. The authors focus on the second notion and model it as a change in the forecast covariance matrix.

A stress test is defined as a one‑parameter family Σ(t) of symmetric positive‑definite matrices, with Σ(0) the unstressed covariance. To isolate pure correlation effects, the authors impose the constraint that the determinant of Σ(t) remains constant for all t. This determinant‑preserving condition is equivalent to keeping the generalized variance (Wilks’ statistic) or the entropy of the associated Gaussian distribution unchanged, and it ensures that the stress is independent of the choice of basis for the risk factors.

Within the Fisher‑Rao manifold, the distance between two covariance matrices Σ₁ and Σ₂ is given by
 d²(Σ₁,Σ₂)=½∑ₖ