Tail Average✓ Mathematical
◆ The PatternAverage loss in the worst α % of scenarios
Expected Shortfall (ES), also called Conditional VaR, is the mean of all losses exceeding the VaR cutoff. It is sub-additive — combining portfolios never makes ES worse — so regulators prefer it to VaR.
ESα = E[ L | L > VaRα ]
For a normal distribution ES has a closed-form: ES = μ + σ · φ(zα) / (1 − α). For fat-tailed distributions Monte Carlo or historical methods are used.
// Interactive — tail threshold and expected shortfall region
Tail α %
Basel III. Banks must now report Expected Shortfall at 97.5 % under the Fundamental Review of the Trading Book (FRTB).
Pattern bridge: Expectation and conditioning connect to Probability Distributions.