How do you calculate Expected Shortfall and why is it replacing VaR in Basel regulations?

Expected Shortfall (also called CVaR or Conditional VaR) answers the question VaR ignores: 'When things go bad, how bad do they get?' VaR tells you the threshold; ES tells you the average loss beyond that threshold.

Definition:

ES at confidence level alpha = E[Loss | Loss > VaR_alpha]

At 97.5% confidence: ES = average of all losses in the worst 2.5% of scenarios.

Calculation Methods:

Method 1: Historical Simulation

Sort 1000 historical P&L scenarios. For 97.5% ES:

• Find the worst 25 scenarios (1000 x 2.5%)
• ES = average of these 25 losses

Worked Example — Pinnacle Trading Desk:

Pinnacle has 1000 days of P&L data. The 25 worst daily losses (in $M):

-8.2, -7.5, -6.9, -6.3, -5.8, -5.5, -5.2, -4.9, -4.7, -4.5,

-4.3, -4.1, -3.9, -3.8, -3.6, -3.5, -3.3, -3.2, -3.0, -2.9,

-2.8, -2.7, -2.6, -2.5, -2.4

97.5% VaR = $2.4M (the 25th worst loss — the threshold)

97.5% ES = average of all 25 = $4.24M

ES is 77% higher than VaR, showing the tail is substantially worse than the threshold.

Method 2: Parametric (Normal Distribution)

For normal returns: ES = mu + sigma x [phi(z_alpha) / (1 - alpha)]

Where phi is the standard normal PDF and z_alpha is the VaR z-score.

At 97.5%: z = 1.96, phi(1.96) = 0.0584

ES = sigma x 0.0584 / 0.025 = sigma x 2.338

Compare with VaR = 1.96 x sigma. ES is about 19% higher under normality.

Why ES Is Coherent and VaR Is Not:

A risk measure is coherent if it satisfies four axioms:

1. Monotonicity: If portfolio A always loses more than B, its risk should be higher
2. Translation invariance: Adding cash reduces risk by that amount
3. Positive homogeneity: Doubling the portfolio doubles the risk
4. Subadditivity: Risk(A+B) <= Risk(A) + Risk(B) — diversification should not increase risk

VaR fails subadditivity. It's possible to construct portfolios where the combined VaR exceeds the sum of individual VaRs. This happens with concentrated, non-normal tail risks.

Classic counterexample: Two bonds, each with 4% default probability and $100 loss on default:

• Individual 95% VaR = $0 (each defaults < 5% of the time)
• Combined portfolio: probability of at least one defaulting = 1 - 0.96^2 = 7.84%
• Combined 95% VaR = $100 > $0 + $0

VaR says combining them is riskier than holding them separately — nonsensical from a diversification standpoint. ES doesn't have this problem.

FRTB Regulatory Shift:

The Basel Committee's Fundamental Review of the Trading Book (FRTB) replaced 99% VaR with 97.5% ES as the market risk capital standard. The 97.5% ES is calibrated to be roughly equivalent to 99% VaR for normal distributions, but captures tail risk much better for non-normal distributions.

Practice ES calculations in our FRM Part II question bank.