Expected Shortfall (CVaR)

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#

Expected shortfall under a normal distribution#

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

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 and Normal Distribution.

Why ES is preferred#

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.
2. Optimise portfolios to minimise CVaR, which handles non normal returns and options well. See Portfolio Optimization.
3. Compare strategies with similar volatility but different tails, such as option selling versus trend following. See Skewness and Kurtosis.
4. Combine with stress tests for scenarios beyond history. See Stress Testing and Scenario Analysis.

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 and 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.