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Expected Shortfall (CVaR)

Expected shortfall (ES), also called conditional VaR (CVaR) or expected tail loss, answers the question VaR dodges: given that the loss exceeds VaR, how large is it on average? Formally ES at confidence α\alpha is the mean loss in the worst (1−α)(1-\alpha) tail, so it sits beyond VaR and captures the shape of the tail rather than a single point. ES is a coherent risk measure: it satisfies subadditivity and so always rewards diversification. The 2019 Basel market-risk framework (FRTB) replaced 99% VaR with 97.5% expected shortfall for exactly these reasons.

Why it matters

If VaR is the height of the fence the flood clears 1 day in 100, ES is the average depth of the water on those flood days. It does not stop at saying "you will be breached"; it tells you how bad it gets on average once you are. Because it averages the whole tail, a fatter tail or a lurking catastrophe shows up in ES even when VaR looks unchanged, and because it is coherent it never penalises a genuinely diversified book.

Formulas

Expected shortfall (mean loss beyond VaR)
ESα=E ⁣[ L∣L≥VaRα ]\mathrm{ES}_{\alpha} = E\!\left[\, L \mid L \ge \mathrm{VaR}_{\alpha} \,\right]
The expected loss conditional on being in the worst (1−α)(1-\alpha) tail. ES is at least as large as VaRα\mathrm{VaR}_{\alpha} because it averages losses that all sit at or beyond the VaR threshold.
Normal (parametric) expected shortfall
ESα=μ+σ ϕ ⁣(zα)1−α\mathrm{ES}_{\alpha} = \mu + \sigma\,\frac{\phi\!\left(z_{\alpha}\right)}{1-\alpha}
For normally distributed losses with mean μ\mu and standard deviation σ\sigma, where ϕ\phi is the standard-normal density and zαz_{\alpha} the α\alpha-quantile. At 97.5% the multiplier ϕ(z)/(1−α)≈2.34\phi(z)/(1-\alpha)\approx 2.34 versus the VaR multiplier zα≈1.96z_{\alpha}\approx 1.96.

Worked examples

Scenario

A loss distribution has a 99% VaR of $10m. The four losses beyond that threshold in a stress sample are $11m, $14m, $20m and $35m. What is the (sample) expected shortfall?

Solution

ES is the average of the tail losses beyond VaR: (11+14+20+35)/4=(11+14+20+35)/4 = 20m. So while VaR reports a $10m threshold, ES tells the desk that a breach costs $20m on average, twice the VaR. This gap is the tail-severity information VaR throws away.

Scenario

A book has daily losses that are normal with mean 0 and σ=\sigma = 4m. Find the 97.5% one-day VaR and ES.

Solution

VaR0.975=1.96×4=_{0.975} = 1.96 \times 4 = 7.84m. ES0.975=4×ϕ(1.96)/0.025=4×2.338=_{0.975} = 4 \times \phi(1.96)/0.025 = 4 \times 2.338 = 9.35m. ES exceeds VaR by about 19%, the extra capital the averaged tail demands even under the thin-tailed normal assumption.

Common mistakes

  • ✗Expected shortfall is just VaR at a higher confidence level. ES averages all losses beyond the VaR threshold; it is a different functional, not VaR evaluated further out (though 97.5% ES and 99% VaR are roughly comparable under normality).
  • ✗ES is always far above VaR. The gap depends on tail shape: under a thin-tailed normal it is modest, but under fat tails ES can dwarf VaR, which is the point of using it.
  • ✗Like VaR, ES can penalise diversification. ES is coherent and subadditive, so a merged book never carries more ES than the sum of its parts.
  • ✗ES requires no distributional input. Parametric ES still assumes a loss distribution; only the historical version reads the tail average directly from data.

Revision bullets

  • •ES = mean loss in the worst (1 - alpha) tail, i.e. average loss beyond VaR
  • •Also called CVaR or expected tail loss
  • •Always at least as large as VaR; captures tail shape, not a single point
  • •Coherent: satisfies subadditivity, so it rewards diversification
  • •Basel FRTB replaced 99% VaR with 97.5% ES

Quick check

Expected shortfall at 99% confidence is best described as

Why did the Basel FRTB framework move from 99% VaR to 97.5% expected shortfall?

Connected topics

VaR DefinitionVaR LimitsCoherenceBlack Swan

Sources

  1. Acerbi & Tasche (2002)
    Acerbi, C., & Tasche, D. "Expected Shortfall: A Natural Coherent Alternative to Value at Risk." Economic Notes, 31(2), 379-388, 2002.
    Defines expected shortfall as the average tail loss and establishes its coherence.
  2. Jorion (2007), Ch. 5
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007.
    Presents conditional VaR / expected shortfall as the tail-average complement to VaR.
  3. BCBS (2019), FRTB
    Basel Committee on Banking Supervision. Minimum Capital Requirements for Market Risk (FRTB). Bank for International Settlements, 2019.
    Adopts 97.5% expected shortfall in place of 99% VaR for the market-risk capital charge.
How to cite this page
Dr. Phil's Quant Lab. (2026). Expected Shortfall (CVaR). Derivatives Atlas. https://phucnguyenvan.com/concept/frm-expected-shortfall
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