Geometric turbulence: a geodesic-regression crisis indicator for equity covariance dynamics, with evidence from African markets

Abstract

The covariance matrix of a basket of assets is a symmetric positive-definite object that evolves on a curved manifold, not on a flat vector space. Standard linear regression on its vectorised entries ignores that geometry and, in periods of market stress, can return fits that fail to be positive definite. We study geodesic regression on the symmetric positive-definite cone $\SPD(n)$ under the log-Euclidean and affine-invariant metrics, with the rolling sample covariance of a basket of log-returns as data. The fitted geodesic's velocity norm is proposed as a geometry-aware turbulence indicator. On daily data for four equity baskets- a ten-stock historical S\&P sub-basket (2006-2024), a current S\&P~100 mega-cap basket, the top ten listings of the Johannesburg Stock Exchange (JSE), and the ten most-traded Egyptian Exchange (EGX) tickers the indicator peaks cleanly on known stress episodes (the 2008 global financial crisis, the 2020 COVID crash, the 2022 Fed rate-hike cycle, and the October-November 2019 Egyptian political-economic tension episode) without any calibration. On hold-out evaluation in crash windows, log-Euclidean regression achieves 150 to 300 times smaller geodesic mean squared error than Euclidean ordinary least squares, because the latter produces non-SPD forecasts that blow up under the intrinsic metric. A naive velocity-aware de-risking strategy reduces maximum drawdown by 2.5 percentage points at the cost of 1.4 percentage points of annual return, consistent with a slow-moving indicator that signals stress reliably but translates imperfectly into tactical trading. All data, code, and experiments are reproducible from the companion repository.