Spectral FX: Eigenvalue Classification of Currency Dynamics

We show that vector autoregression (VAR), cointegration, geometric Brownian motion (GBM), and the Ornstein--Uhlenbeck (OU) process are four eigenvalue regimes of a single linear stochastic differential equation \(dX = AX\,dt + \Sigma\,dW\). The eigenvalue spectrum of the drift matrix \(A\) classifies each mode as mean-reverting (\(\lambda < 0\)), random walk (\(\lambda = 0\)), or controlled (\(\mu(t)\) piecewise constant). We apply PCA to the return covariance matrix of a 15-currency FX universe (vs EUR) over 1,300 trading days (2021--2026) to estimate the empirical factor structure. The covariance eigenvalues---distinct from the drift eigenvalues but sharing the same eigenvector basis under stationarity---yield an effective dimension \(K_{\text{eff}} = \bigl(\sum \lambda_k\bigr)^2 / \sum \lambda_k^2 = 10.5\), indicating 10--11 independent risk factors. A pure carry strategy achieves a Sharpe ratio of 0.51; a spectral carry-tilt strategy achieves 0.39 SR with comparable maximum drawdown (−21.3% vs. −22.0%), trading off return for improved risk characteristics. The dead-currency problem (EUR adoption) maps to eigenvalue collapse: as \(\sigma_k \to 0\), the mode vanishes from the decomposition.