Risk Quantification and Bayesian Calibration: Assessing Tail Risk across Market Regimes

Abstract

Post-pandemic volatility clustering highlights the gap between static governance and quantitative risk measurement. This paper proposes a synthesized framework utilizing complementary components of classical risk measurement and Bayesian inference. First, applying Lagrangian optimization, we illustrate the sensitivity of static mean-variance allocations to regime-specific sample moments. Second, we introduce Expected Shortfall as a complementary theoretical measure and GARCH specifications to account for time-varying conditional variance. Crucially, we complement standard VaR backtesting with a Beta-Binomial conjugate prior framework to calibrate risk exposures. Utilizing S&P 500 daily returns, we demonstrate how this mechanism quantifies the uncertainty surrounding the VaR violation probability. Relative to the 1.0% expectation, the traditional 21-Day Rolling VaR yields a Bayesian breach-rate adjustment factor of 2.40x, whereas the GARCH(1,1) specification lowers this to 1.80x. Ultimately, by bridging the gap between ex-ante predictions and realized shocks, this Bayesian recalibration loop may provide a data-driven approach to ex-post risk calibration.