Expected shortfall (ES) is a risk measure that quantifies the average loss beyond a certain threshold. Unlike value-at-risk (VaR), which only considers a single quantile, ES incorporates the entire tail of the loss distribution. This makes ES a more comprehensive and accurate measure of risk, particularly for portfolios with fat tails and a high potential for extreme losses.
Key Facts
- Expected shortfall, also known as conditional value-at-risk (CVaR), is a risk measure that quantifies the average loss beyond a certain threshold.
- Expected shortfall is sub-additive, meaning that the expected shortfall of a portfolio is generally lower than the sum of the expected shortfall of its individual components.
- The sub-additivity of expected shortfall holds even when the underlying profit/loss distributions are discontinuous.
- The sub-additivity property of expected shortfall is utilized in optimizing expected shortfall by finding efficient algorithms.
- The sub-additivity of expected shortfall is in contrast to value-at-risk (VaR), which is not sub-additive.
- When adding two distributions, the overall worst cases in the output distribution are not as bad as the overall sum of the worst cases of the individual distributions.
- VaR is a one-off case, so it can be worse than the individual distributions, while expected shortfall measures the overall cases and is not as bad as the sum of individual distributions.
- The expected shortfall can be approximated by taking the average of the worst losses and multiplying it by -1.
- When a loss function can be decomposed into losses from two sub-portfolios, the expected shortfall of the combined portfolio is generally lower than the sum of the expected shortfall of the individual sub-portfolios.
Sub-Additivity of Expected Shortfall
A crucial property of ES is its sub-additivity. This means that the ES of a portfolio is generally lower than the sum of the ES of its individual components. This property holds even when the underlying profit/loss distributions are discontinuous.
The sub-additivity of ES can be explained intuitively. When adding two distributions, the overall worst cases in the output distribution are not as bad as the overall sum of the worst cases of the individual distributions. This is because the worst cases in each distribution are unlikely to occur simultaneously.
In contrast, VaR is not sub-additive. This is because VaR only considers a single quantile, and the worst cases in the individual distributions can occur simultaneously in the combined distribution.
Implications of Sub-Additivity
The sub-additivity of ES has several important implications. First, it means that ES is a more conservative risk measure than VaR. This is because ES considers the entire tail of the loss distribution, while VaR only considers a single quantile.
Second, the sub-additivity of ES makes it more suitable for optimizing risk. By minimizing the ES of a portfolio, one can also minimize the ES of its individual components. This is not the case for VaR, which is not sub-additive.
Conclusion
Expected shortfall is a sub-additive risk measure that is more comprehensive and accurate than VaR. This makes ES a more suitable measure of risk for portfolios with fat tails and a high potential for extreme losses. The sub-additivity of ES also makes it more suitable for optimizing risk.
References
- Comparative analyses of expected shortfall and value-at-risk under market stress
- Why is Expected Shortfall, not VaR, Sub-additive — a simple & intuitive explanation
- Subadditivity of Expected Shortfall
FAQs
What is expected shortfall?
Expected shortfall (ES) is a risk measure that quantifies the average loss beyond a certain threshold. It is also known as conditional value-at-risk (CVaR).
Is expected shortfall additive?
No, expected shortfall is not additive. This means that the ES of a portfolio is generally lower than the sum of the ES of its individual components.
Why is expected shortfall sub-additive?
Expected shortfall is sub-additive because the worst cases in the individual distributions are unlikely to occur simultaneously in the combined distribution.
What are the implications of expected shortfall sub-additivity?
The sub-additivity of expected shortfall makes it a more conservative risk measure than VaR and more suitable for optimizing risk.
How can expected shortfall be used to optimize risk?
By minimizing the ES of a portfolio, one can also minimize the ES of its individual components.
What is the difference between expected shortfall and value-at-risk (VaR)?
Expected shortfall considers the entire tail of the loss distribution, while VaR only considers a single quantile. This makes ES a more comprehensive and accurate measure of risk, particularly for portfolios with fat tails and a high potential for extreme losses.
Is expected shortfall always less than VaR?
No, expected shortfall is not always less than VaR. However, ES is generally lower than VaR for portfolios with fat tails and a high potential for extreme losses.
When should expected shortfall be used instead of VaR?
Expected shortfall should be used instead of VaR when one is concerned about the risk of extreme losses. This is particularly important for portfolios with fat tails and a high potential for extreme losses.