Reducing the Contamination: Variance Reduction Strategies for Monte Carlo ES Backtests

Monte Carlo estimation of Expected Shortfall contaminates the Acerbi-Székely (2014) backtest statistic with computational noise. Nagy (2026, Contaminated by Construction) proved that the test statistic's variance decomposes as \(\text{Var}_{\text{returns}} + \text{Var}_{\text{MC}}\), and showed that exact computation eliminates \(\text{Var}_{\text{MC}}\) for lognormal portfolios. This paper addresses the complementary problem: reducing \(\text{Var}_{\text{MC}}\) when exact computation is unavailable.

We derive an estimator for \(\text{Var}_{\text{MC}}\) that requires no additional simulation beyond the standard ES computation, and analyze how each major class of variance reduction techniques affects the decomposition. Antithetic variates reduce \(\text{Var}_{\text{MC}}\) by a factor of \(1.5\)–\(2\times\). Control variates with an analytical proxy achieve \(3\)–\(10\times\) reduction when a good control is available. Importance sampling with exponential tilting targets the tail directly, achieving the largest reductions (\(5\)–\(50\times\)) but requiring careful calibration. Quasi-Monte Carlo methods using Sobol sequences improve convergence from \(O(1/\sqrt{M})\) to \(O((\log M)^d / M)\) in dimension \(d\), and as a deterministic method, also resolve the reproducibility problem identified in the companion paper. A simulation study using Student-\(t(5)\) and correlated lognormal portfolios validates all analytical predictions.

Keywords: Expected Shortfall, backtesting, Monte Carlo, variance reduction, quasi-Monte Carlo, Sobol sequences, importance sampling, regulatory capital

JEL Classification: G32, C15, C63