VaR vs Expected Shortfall in one line: VaR is the loss you expect to be exceeded on 1 day in 100. Expected Shortfall (ES) is the average loss on those bad days. FRTB moved internal-model capital from 99% VaR to 97.5% ES because VaR says nothing about how bad the tail is, and can even punish diversification.
1. What each number measures
Take a vector of P&L scenarios from historical simulation or Monte Carlo. Sort them from worst to best. The 99% VaR is the loss at the 1% quantile. With 1,000 scenarios that is roughly the 10th worst. ES at the same level is the average of the 10 worst. VaR tells you where the tail starts. ES tells you what is inside it.
$$\mathrm{VaR}_\alpha = \inf\{x : P(L \le x) \ge \alpha\}, \qquad ES_\alpha = \frac{1}{1-\alpha}\int_\alpha^1 \mathrm{VaR}_u\,du$$
Directly proportional relationships to remember: a wider P&L distribution raises both numbers; a fatter tail raises ES much faster than VaR; a higher confidence level raises both; ES is always at least as large as VaR at the same level.
2. Same VaR, very different books
Illustrative example: two desks both report a 99% one-day VaR of 10 (in millions) on 1,000 scenarios. Desk A's 10 worst scenarios average a loss of 12. Desk B's 10 worst average 40, because it sells deep out-of-the-money options whose losses explode beyond the quantile. VaR rates them identically. ES says Desk B is more than three times riskier in the tail. This is the tail blindness that regulators wanted to remove.
3. VaR can punish diversification
A risk measure should never say that combining two positions is riskier than the sum of the parts. VaR sometimes does. Take two independent bonds, each losing 100 with probability 4% and nothing otherwise, and look at 95% confidence.
| Position | 95% VaR | 95% ES |
|---|---|---|
| Bond 1 alone | 0 | 80 |
| Bond 2 alone | 0 | 80 |
| Both together | 100 | 103.2 |
| Sum of the two alone | 0 | 160 |
Each bond alone has VaR 0, because a 4% default chance is below the 5% tail. Together the chance of at least one default is 7.84%, so portfolio VaR jumps to 100: larger than the sum of the parts, which is a violation of subadditivity. ES gives 103.2 for the pair, below the stand-alone sum of 160, so it rewards diversification as it should. ES is a coherent risk measure; VaR is not in general.
4. How much bigger is ES than VaR?
It depends on the tail. In units of standard deviation:
| Distribution | 97.5% VaR | 97.5% ES | 99% VaR | 99% ES |
|---|---|---|---|---|
| Normal | 1.96 | 2.34 | 2.33 | 2.67 |
| Student-t, 4 d.o.f., same variance | 1.96 | 2.82 | 2.65 | 3.69 |
For a normal distribution, 97.5% ES (2.34) is almost the same as 99% VaR (2.33). That is exactly why FRTB chose 97.5%: capital stays comparable for light tails, but heavy-tailed books are charged more, as the second row shows.
5. What FRTB actually does with ES
- Confidence level: 97.5% one-tailed ES replaces 99% VaR in the internal models approach.
- Liquidity horizons: risk factors are shocked over 10, 20, 40, 60 or 120 days depending on how fast you could really exit. The cascade is built from nested ES numbers:
$$ES = \sqrt{ES_T(P)^2 + \sum_{j\ge 2}\Big(ES_T(P,j)\sqrt{\tfrac{LH_j-LH_{j-1}}{T}}\Big)^2}, \quad T = 10$$
- Stress calibration: ES is computed on the worst 12-month window of a reduced set of risk factors and scaled to the full set, so capital cannot fall just because markets have been calm.
- Backtesting stays on VaR: ES is not elicitable, meaning there is no simple scoring rule that identifies the correct ES. So FRTB tests desks on 99% and 97.5% VaR exceptions. Use ES for capital and VaR exceptions for model credibility.
6. Practical estimation issues
- Tail data: 97.5% on 250 scenarios means a tail of 6.25 observations, so the fractional observation must be weighted correctly. Naive averaging of the worst 6 or 7 is a classic bug.
- Noise: ES from a few tail points is noisier than VaR from the quantile. Use more scenarios, stressed windows or filtered historical simulation.
- Full revaluation: tail scenarios are where delta approximations fail, so reprice options in the tail rather than using Greeks.
7. Common mistakes
- Describing VaR as the maximum loss. It is a quantile, not a worst case.
- Comparing 99% VaR and 97.5% ES as if they were the same number. They only match for a normal distribution.
- Scaling ES by the square root of time. FRTB uses liquidity-horizon cascades instead.
- Forgetting that ES needs the same data quality as VaR plus more tail observations.
8. Interview answers you can say out loud
Why did Basel move from VaR to ES? "VaR ignores tail severity and is not subadditive. ES is coherent and averages the tail, so it penalises tail risk and rewards diversification. 97.5% was chosen so that for normal returns capital stays close to 99% VaR."
Why is ES not backtested directly? "It is not elicitable, so there is no simple test for it. Regulators backtest VaR exceptions at 99% and 97.5%, and rely on ES for capital."
Can ES be lower than VaR? "No. ES is the average loss beyond VaR, so it is at least as large."
FAQ
Is 97.5% ES riskier than 99% VaR? For light-tailed books they are about equal. For heavy-tailed books ES is higher, which is the point.
Does ES replace VaR on the desk? Not entirely. Many banks still use VaR for daily limits and backtesting while ES drives regulatory capital.