Finite-Sample Conformal Risk Bounds for Joint Value-at-Risk and Expected-Shortfall Forecasting Under Non-Exchangeable Financial Time Series

Abstract

Financial tail-risk observations are non-exchangeable: serial dependence and regime shifts make their joint law depend on the time ordering, invalidating the exchangeability that standard conformal guarantees assume, and expected shortfall is not elicitable on its own, so a forecaster cannot be calibrated to it as a quantile is to its coverage. We ask whether a black-box value-at-risk and expected-shortfall forecaster can be calibrated under such dependence while retaining finite-sample guarantees. We tune a single inflation parameter by conformal risk control on a bounded monotone loss that couples value-at-risk breach frequency with breach magnitude normalised by the model’s predicted value-at-risk–expected-shortfall gap; the guarantee is thus for a tail-gap-normalised exceedance-severity surrogate, and its expected-shortfall reading depends on the predicted gap being a sound tail-gap estimate. Under exchangeability, the method gives finite-sample expected-risk control; for dependent data we invoke a non-exchangeable swap-distance bound and add, for separated calibration points, a regime-drift bound with an explicit cumulative β-mixing cost, plus a high-probability realised-path statement and a heavy-tail rate of order D(p−1)/p. Building regimes causally from previous-month FRED-MD vintages across eight exchange rates, a Bitcoin series, and the GIFT-Eval finance domain, the weighted controller attains a 2.51% violation rate and a Fissler–Ziegel score of 0.431 against 0.441 and 0.439 for the strongest conformal baselines—an incremental gain, not significant at the 5% level, that concentrates in turbulent regimes and at matched capital, supporting calibration of a joint frequency-and-normalised-severity budget rather than distribution-free control of the expected-shortfall forecast itself.