Who this entry is for — Anyone who wants to know not only where the estimated loss tail begins, but also how severe its outcomes are on average.
Expected Shortfall at level α aggregates loss-distribution quantiles above α:
ESα(L) = 1 / (1 − α) × ∫ from α to 1 of VaRu(L) du
For a continuous distribution with no probability mass at the threshold, this can be read as average loss conditional on outcomes reaching or exceeding VaR. A 97.5% ES therefore describes the average of the worst 2.5% of the estimated distribution. It is not the worst possible outcome.
In plain terms — VaR marks the edge of the tail; ES summarises average severity inside the tail under the same assumptions.
VaR and ES compared
| Aspect | VaR | Expected Shortfall |
|---|---|---|
| Question | Where is the selected quantile? | How severe is the tail beyond that level on average? |
| Tail information | Does not measure the size of outcomes beyond the threshold | Aggregates them into a tail average |
| Worst case | No | No |
| Model dependence | Yes | Yes, with particular sensitivity to sparse extreme data |
| Coherence | Not generally subadditive | Properly defined ES is a coherent risk measure |
ES being greater than VaR at the same level does not show that one is “correct”: they answer different questions about the same estimated distribution.
Discontinuous distributions and the name CVaR
When a distribution has probability mass exactly at the quantile, the
intuitive expression E[L | L ≥ VaRα] can place too much weight on outcomes at
the threshold. The general definition must include only the portion of that
mass needed to make up tail probability 1 − α; the quantile-integral
representation avoids the ambiguity.
CVaR, Conditional Value at Risk, tail mean and Expected Shortfall are often used as synonyms, particularly for continuous distributions. The literature contains variants that may diverge at discontinuities. A page, model or report should therefore state its definition instead of inferring equivalence from the label alone.
Estimation, control and limitations
ES needs the same coordinates as VaR: scope, unit, horizon, level, date, data, revaluation method and dependency assumptions. Its tail estimate can be especially unstable because it uses a small fraction of the observations.
- a historical window may omit relevant stress;
- a parametric distribution may misdescribe asymmetry and tails;
- a simulation inherits the model's dynamics, correlations and pricing;
- illiquidity, jumps, defaults and costs may be absent or understated;
- validation and alternative benchmarks reduce uncertainty but do not remove it.
ES should therefore be read with stress testing, concentration analysis and model-risk controls.
Basel's parameter is not a universal default
In its internal-models market-risk framework, Basel uses daily one-tailed 97.5% ES, calibrated in part to a stress period and scaled across liquidity horizons. Those specifications serve a particular prudential application. They do not automatically determine the level, horizon or method for a fund, broker, trader or other portfolio.
Typical mistake — Calling ES an “average maximum loss”. It is an average over a region of an estimated distribution, not an upper bound.
Sources
- Basel Framework — MAR33, calculation of expected shortfall
- Basel Committee — Explanatory note on the minimum capital requirements for market risk
- Acerbi and Tasche — On the coherence of Expected Shortfall
- Artzner, Delbaen, Eber and Heath — Coherent Measures of Risk