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

TL;DR

The paper tackles high-confidence VaR estimation as a rare-event problem and compares two simulation-based strategies: importance sampling (IS) under a Gaussian nominal model for efficient tail probability estimation, and discrete moment matching (DMM) for robust VaR bracketing under distributional ambiguity. By separating the nominal Gaussian calibration from a heavy-tailed true data-generating process (Student-$t$ with variance matching), the study isolates the efficiency-robustness trade-off: IS is highly efficient under the nominal model but systematically underestimates the true VaR as tails thicken, while DMM yields conservative, interval-valued VaR that remain meaningful under tail misspecification albeit with reduced efficiency. DMM contracts the moment-feasible set as more moments are enforced, tightening the VaR bounds; IS relies on diagnostics like effective sample size and weight concentration to gauge stability but cannot fix misspecification-induced bias. The findings highlight that variance reduction alone is insufficient when model uncertainty about tails is significant, and practitioners should choose IS or DMM based on tail-model confidence, potentially using DMM's ambiguity-aware bounds in stressed markets.

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.

Figures (7)

• Figure 1: Moment-feasible VaR bracketing versus empirical loss CDF at $\alpha=0.99$. The vertical lines indicate the DMM lower and upper VaR bounds at the chosen moment order and the Monte Carlo VaR reference.
• Figure 2: Sensitivity of DMM VaR bounds to moment order $d$. The lower bound increases with $d$ and stabilizes, while the upper bound remains nearly constant.
• Figure 3: Calibration of IS VaR under Student-$t$ tails: IS mean VaR versus true VaR. The dashed line is the identity line; points below indicate underestimation.
• Figure 4: IS bias (IS mean minus true VaR) versus degrees of freedom $\nu$. Bias magnitude decreases as tails become lighter.
• Figure 5: IS estimator variability (standard deviation across replications) versus degrees of freedom $\nu$.