What is Conditional VaR (CVaR / Expected Shortfall), and why did Basel III replace VaR with ES for market risk capital?

Conditional VaR (CVaR), also called Expected Shortfall (ES), measures the average loss in the worst alpha-percent of scenarios. While VaR tells you the threshold loss at a given confidence level, ES tells you how bad it gets beyond that threshold.

Formula (Continuous)

ES_alpha = E[Loss | Loss > VaR_alpha]

For a discrete sample of n sorted losses (L_1 >= L_2 >= ... >= L_n) at alpha = 1%:

ES_{1%} = (1 / floor(n x 0.01)) x SUM of the worst floor(n x 0.01) losses

Worked Example

Dunbar Fixed Income Fund has 1,000 daily P&L observations. At the 99% confidence level:

• VaR_{99%} = $4.2 million (the 10th worst loss out of 1,000)
• The 10 worst daily losses (in $M): 9.1, 7.8, 6.5, 5.9, 5.3, 5.0, 4.8, 4.6, 4.4, 4.2

ES_{99%} = (9.1 + 7.8 + 6.5 + 5.9 + 5.3 + 5.0 + 4.8 + 4.6 + 4.4 + 4.2) / 10 = $5.76M

VaR says: "On 99% of days, losses won't exceed $4.2M."

ES says: "On the 1% of worst days, losses average $5.76M."

Why ES Replaced VaR in Basel III (FRTB)

Subadditivity Example

Consider two portfolios X and Y:

• VaR(X) = $5M, VaR(Y) = $5M
• VaR(X + Y) could be $11M (superadditive!) — VaR says combining them increases risk
• ES(X) = $7M, ES(Y) = $7M
• ES(X + Y) will always be <= $14M — ES correctly reflects diversification benefit

FRTB Specifics:

• Basel moved from 99% VaR (10-day) to 97.5% ES for internal models
• The confidence level was lowered because ES already captures tail severity
• Stressed ES is calibrated to the worst 12-month period in the bank's history

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