Efficiency versus Robustness under Tail Misspecification: Importance Sampling and Moment-Based VaR Bracketing

Abstract

Value-at-Risk (VaR) estimation at high confidence levels is inherently a rare-event problem and is particularly sensitive to tail behavior and model misspecification. This paper studies the performance of two simulation-based VaR estimation approaches, importance sampling and discrete moment matching, under controlled tail misspecification. The analysis separates the nominal model used for estimator construction from the true data-generating process used for evaluation, allowing the effects of heavy-tailed returns to be examined in a transparent and reproducible setting. Daily returns of a broad equity market proxy are used to calibrate a nominal Gaussian model, while true returns are generated from Student-t distributions with varying degrees of freedom to represent increasingly heavy tails. Importance sampling is implemented via exponential tilting of the Gaussian model, and VaR is estimated through likelihood-weighted root-finding. Discrete moment matching constructs deterministic lower and upper VaR bounds by enforcing a finite number of moment constraints on a discretized loss distribution. The results demonstrate a clear trade-off between efficiency and robustness. Importance sampling produces low-variance VaR estimates under the nominal model but systematically underestimates the true VaR under heavy-tailed returns, with bias increasing at higher confidence levels and for thicker tails. In contrast, discrete moment matching yields conservative VaR bracketing that remains robust under tail misspecification. These findings highlight that variance reduction alone is insufficient for reliable tail risk estimation when model uncertainty is significant.