# Expected Shortfall (CVaR)

> Expected shortfall, or CVaR, is the average loss on the worst days beyond the VaR threshold. Learn the formula, a worked example and why regulators adopted it.

Source: https://learn.tradelabsai.com/portfolio/expected-shortfall/  
Track: Portfolio and Performance · Level: Advanced · Updated: 2026-10-03  
Publisher: TradeLabs AI (https://tradelabsai.com). Education, not financial advice.  
Cite as: TradeLabs Learn, "Expected Shortfall (CVaR)", https://learn.tradelabsai.com/portfolio/expected-shortfall/

Value at risk tells you the threshold a loss should not cross on most days, but not how bad things get when it does. Expected shortfall (ES), also called conditional value at risk (CVaR), fills that gap. It is the average loss on the days when losses exceed the VaR threshold. If 99% VaR is the line, expected shortfall measures how deep the water is beyond it. Because it captures tail severity, expected shortfall has become the preferred tail risk measure for many risk managers and, since the Basel Committee's Fundamental Review of the Trading Book, for bank market risk capital.

## Definition

```
Expected shortfall at 97.5% = Average of losses worse than the 97.5% VaR
```

Historically, take the worst 2.5% of daily outcomes and average them.

## A historical example

**Example: Calculating expected shortfall from history**
A portfolio has 1,000 days of historical returns applied to today's positions. For 99% confidence, look at the worst 1%, which is the 10 worst days. Suppose their losses are $42,000, $38,000, $35,000, $33,000, $31,000, $30,000, $29,000, $28,500, $28,000 and $27,500. The 99% VaR is around $27,500, near the boundary of the worst 1%. The 99% expected shortfall is the average of the ten, which is $322,000 divided by 10, or $32,200. ES is higher than VaR and reflects the size of the bad days, not just where they start.

## Expected shortfall under a normal distribution

For normally distributed returns, ES can be calculated directly. In multiples of daily volatility:

| Confidence | VaR | Expected shortfall |
|---|---|---|
| 95% | 1.645 | 2.063 |
| 97.5% | 1.960 | 2.338 |
| 99% | 2.326 | 2.665 |

Under normality, 97.5% ES is about the same as 99% VaR. That is one reason regulators chose 97.5% ES to replace 99% VaR: similar under calm assumptions, but more sensitive to fat tails. With real fat tailed returns, ES rises much more than VaR. See [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/) and [Normal Distribution](https://learn.tradelabsai.com/math/normal-distribution/).

## Why ES is preferred

| Property | VaR | Expected shortfall |
|---|---|---|
| Measures tail severity | No | Yes |
| Sub additive (diversification never increases risk) | Not always | Yes |
| Sensitive to fat tails | Weakly | Strongly |
| Harder to game | Less | More |
| Easy to backtest | Yes | Harder |

The sub additivity property means ES is a coherent risk measure in the sense defined by Artzner and coauthors in 1999: combining portfolios never shows more risk than the sum of their separate risks.

## Weaknesses

- **Needs tail data:** estimates rest on few observations and can be noisy.
- **Backtesting is harder** than for VaR, since it averages tail outcomes.
- **Still backward looking** when based on historical windows.
- **Model dependent** in parametric and Monte Carlo forms.

## Using expected shortfall

1. **Set limits** on ES alongside VaR. See [Risk, Position, Loss and Drawdown Limits](https://learn.tradelabsai.com/portfolio/risk-limits/).
2. **Optimise portfolios** to minimise CVaR, which handles non normal returns and options well. See [Portfolio Optimization](https://learn.tradelabsai.com/portfolio/portfolio-optimization/).
3. **Compare strategies** with similar volatility but different tails, such as option selling versus trend following. See [Skewness and Kurtosis](https://learn.tradelabsai.com/math/skewness-and-kurtosis/).
4. **Combine with stress tests** for scenarios beyond history. See [Stress Testing and Scenario Analysis](https://learn.tradelabsai.com/portfolio/stress-testing/).

## Traders and tail risk

For individual traders, a practical version is to average your worst 5% of daily or weekly results. If that figure would threaten your account or your discipline, position sizes are too large. See [Position Sizing](https://learn.tradelabsai.com/risk/position-sizing/) and [Risk of Ruin](https://learn.tradelabsai.com/risk/risk-of-ruin/).

## Frequently asked questions

### What is expected shortfall?

The average loss on the days when losses exceed the value at risk threshold, measuring how severe tail losses are.

### What is the difference between VaR and CVaR?

VaR is the loss threshold at a confidence level; CVaR, or expected shortfall, is the average loss beyond that threshold.

### Why did regulators switch to expected shortfall?

It captures tail severity, reacts more to fat tails and is sub additive, addressing key weaknesses of VaR revealed in the 2008 crisis.

Next, learn to test portfolios against extreme scenarios in [Stress Testing and Scenario Analysis](https://learn.tradelabsai.com/portfolio/stress-testing/).

## Continue learning

- Next lesson: [Stress Testing and Scenario Analysis](https://learn.tradelabsai.com/portfolio/stress-testing/)
- Previous lesson: [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/)
- Related: [Value at Risk (VaR)](https://learn.tradelabsai.com/portfolio/value-at-risk/): Value at risk estimates the loss a portfolio should not exceed with a given confidence over a set period. Learn the three methods, an example and the limits.
- Related: [Fat Tails](https://learn.tradelabsai.com/math/fat-tails/): Fat tails mean extreme market moves happen far more often than the normal curve predicts. Learn the evidence, the causes, how to measure them and how to manage them.
- Related: [Stress Testing and Scenario Analysis](https://learn.tradelabsai.com/portfolio/stress-testing/): Stress testing asks how a portfolio would fare in extreme but plausible events. Learn historical and hypothetical scenarios and reverse stress tests.
- Related: [Portfolio Volatility and VaR Calculator](https://learn.tradelabsai.com/tools/var-calculator/): Free value at risk calculator. Enter portfolio value, daily volatility, confidence level and horizon to estimate VaR and expected shortfall in dollars.
- Related: [Risk, Position, Loss and Drawdown Limits](https://learn.tradelabsai.com/portfolio/risk-limits/): Risk limits turn a risk policy into hard rules on position size, exposure, daily loss and drawdown. Learn how to set them, enforce them and avoid mistakes.