CVaR / Expected Shortfall Formula
Conditional Value-at-Risk, also called Expected Shortfall, is the average loss in the worst tail beyond the VaR threshold. Where VaR asks how bad a loss could be at a confidence level, CVaR asks how bad it is on average when that level is breached. CVaR is a coherent risk measure (it is subadditive), unlike VaR, and Basel rules have moved capital requirements onto it.
Formula
Copy the exact expression or work through it step by step below.
CVaR_alpha = E[ -R | R <= -VaR_alpha ]
Empirical: CVaR_alpha = average of the worst (1 - alpha) fraction of losses Variables
alpha
Confidence level
The same confidence level used for VaR, typically 0.95 or 0.975. CVaR averages over the worst (1 - alpha) fraction of outcomes, so a 95% CVaR averages the worst 5% of losses.
VaR_alpha
Value-at-Risk threshold
The quantile loss at the confidence level. CVaR is the expected loss conditional on the loss equaling or exceeding this threshold, so CVaR is always at least as large as VaR.
R
Portfolio return
The random periodic return. Only its realizations in the loss tail (R at or below minus VaR) enter the CVaR average.
E[ . | . ]
Conditional expectation
The mean taken over only the tail scenarios beyond VaR. Empirically it is the simple average of the worst (1 - alpha) fraction of the sorted losses.
Step By Step
- 1
Sort the historical returns from worst to best and identify the (1 - alpha) tail.
At 95% over 100 observations, the tail is the worst 5 returns.
- 2
Read the VaR as the boundary of that tail.
The 5th-worst return is -2.6%, so VaR is 2.6%.
- 3
Average all losses in the tail, including the ones deeper than VaR.
The worst 5 returns are -5.0%, -4.2%, -3.5%, -3.0%, -2.6%.
- 4
Report the absolute value of that tail average as CVaR, scaling by portfolio value if needed.
Mean of the worst 5 is -3.66%, so CVaR is 3.66%.
Worked Example
One-day 95% CVaR on a 1,000,000 book, worst 5 of 100 daily returns
Worst five daily returns
-5.0%, -4.2%, -3.5%, -3.0%, -2.6%
Confidence level
95%
Portfolio value
1,000,000
The 95% tail over 100 observations is the worst 5 returns. VaR = 2.6% (the 5th-worst, the tail boundary). CVaR = average of the worst 5 = (5.0 + 4.2 + 3.5 + 3.0 + 2.6)/5 = 18.3/5 = 3.66%. CVaR currency = 0.0366 x 1,000,000 = 36,600.
One-day 95% CVaR of 36,600 versus a VaR of 26,000. CVaR is 41% larger because it accounts for how deep the tail actually runs, not just where it begins. This is why risk committees and Basel III favor expected shortfall: two books with the same VaR can have very different CVaR if one has a longer left tail.
Common Variations
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Sources & References
- Coherent Measures of Risk — Artzner, Delbaen, Eber, Heath, Mathematical Finance (1999)
- Optimization of Conditional Value-at-Risk — R. Tyrrell Rockafellar and Stanislav Uryasev, Journal of Risk (2000)
Related Content
Keep the topic connected
Historical VaR Formula
The historical VaR formula: the empirical quantile of past returns at a confidence level. A distribution-free Value-at-Risk method, with an example.
Parametric VaR Formula
The parametric VaR formula: Value-at-Risk from the mean, volatility, and a normal z-score. The variance-covariance method, with a worked example.
Expected Shortfall (CVaR)
Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.
Value at Risk (VaR)
Value at Risk: the loss threshold you'll exceed with probability α. Why historical VaR is brittle and what it doesn't tell you about the tail.