A divergence diagnostic for monitoring the composition of default in rating-migration matrices

Reduced-form credit-portfolio models monitor risk through a single aggregate default rate, the exposure-weighted mean of origin-specific default rates across rating grades, sectors, or regions. That aggregate is only the level coordinate of the default-rate configuration: an origin-asymmetric shock that raises one origin's default rate and lowers another's by an offsetting amount leaves the aggregate unchanged while the composition of default reorganizes beneath it. This article describes a method to monitor that missing coordinate. From a decomposed migration matrix we form a normalized divergence statistic-a scalar for two origins and a vector for many-that is invariant to the common level and responds to composition rotations; we chart it with a scalar control limit and a Hotelling $T^2$ control chart on the general-K vector, bound the forecast bias a stale matrix imparts to probability of default, expected credit loss, and economic capital, and sign the rotation through a structural Merton model. We set out the two distinct central limit theorems the diagnostic rests on-one cross-sectional, governing the control limits, one temporal, governing the structural regime-and show that under a regime-switching Merton specification the reverse-hazard rate is an inverse Mills ratio, so that the direction of the rotation is itself regime-dependent. The method runs on the segment- or rating-decomposed default data institutions already report. •A normalized divergence statistic isolates the composition of default that the aggregate rate ignores. •Scalar and Hotelling $T^2$ control charts detect, time, and attribute origin-asymmetric default rotations. •A closed-form reverse-hazard rate signs the rotation and makes its direction regime-dependent.