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

Abstract

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 T2 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 T2 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.