finance_risk

We study the bias-variance trade-off in Expected Shortfall estimation, comparing spectral (order-statistic) methods with parametric and bootstrap Monte Carlo. We prove that spectral ES is the minimum-variance unbiased estimator (MVUE) in the class of model-free risk estimators and characterize the Pareto frontier between bias and variance.

What are the fundamental limits of estimating coherent risk measures from finite data? We establish four structural results.