Quantitative Methods & RiskRisk Measures: Volatility, Drawdown, VaR & CVaR

Expected Shortfall (CVaR) & Where VaR Fails

12 min read

What happens once you cross the line

VaR tells you the threshold; Expected Shortfall tells you the average loss when you breach it. Where VaR is silent about the tail, Expected Shortfall — also called Conditional VaR, CVaR, or expected tail loss — is precisely the average of the losses beyond the VaR point. It captures tail magnitude, it is sub-additive (so combining positions never makes it larger than the sum of their separate values, as a coherent risk measure requires), and it is the reason regulators moved the trading-book capital framework from VaR to Expected Shortfall.

Formula

Expected Shortfall (CVaR)

ES_α = E[ L | L ≥ VaR_α ]

VaR

A single quantile — the threshold loss
Silent about how bad the tail gets
Not sub-additive in general (can penalize diversification)
Easy to game by hiding risk beyond the threshold
Simple to explain to a committee

Expected Shortfall (CVaR)

The average loss across the whole tail
Captures tail magnitude directly
Sub-additive — a coherent risk measure
Harder to game; sensitive to the far tail
The Basel trading-book standard

expected_shortfall.py

python

1import numpy as np
2 
3def expected_shortfall(pnl, confidence=0.95):
4    """Average loss in the worst (1-confidence) tail. Positive number."""
5    pnl = np.asarray(pnl, dtype=float)
6    cutoff = np.percentile(pnl, (1 - confidence) * 100)
7    tail = pnl[pnl <= cutoff]          # the losing tail
8    return -tail.mean() if len(tail) else 0.0
9 
10es95 = expected_shortfall(daily_pnl, 0.95)
11print(f"1-day 95% Expected Shortfall = ${es95:,.0f}")
12# ES is ALWAYS >= VaR: it averages the tail VaR only touches.

ES averages the tail that VaR merely points at. Reporting both is standard practice: VaR for the threshold, ES for the severity beyond it.

Tip

Always quote the trio

A complete one-line risk summary is VaR + Expected Shortfall + a discrete stress scenario. VaR gives the threshold, ES gives the average tail severity, and the stress scenario gives a concrete 'if 2008 / March 2020 / the SNB unpeg repeated' loss that no probabilistic measure fully captures. Any one of the three alone is incomplete.

Coherent risk measure

Term

A risk measure satisfying four axioms — monotonicity, translation invariance, positive homogeneity, and sub-additivity. Sub-additivity (risk of the whole ≤ sum of risks of the parts) is the one VaR can violate and Expected Shortfall always satisfies. It is the formal reason ES is preferred for capital.

Knowledge check

Why do regulators and desks prefer Expected Shortfall (CVaR) over VaR for tail risk?