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Expected Shortfall (CVaR)

Average loss in the worst tail beyond VaR.

Conditional VaR answers 'if we breach VaR, how bad is it on average?' — capturing tail severity that VaR ignores. It is a coherent risk measure (sub-additive) and is now the Basel-mandated metric for market risk.

ES_α = −E[ r | r ≤ −VaR_α ]

import numpy as np
cutoff = np.percentile(returns, 5)
cvar = -returns[returns <= cutoff].mean()