Expected Shortfall Conventions for Loss Signs and Tail Quantiles

Summary

The document addresses two integral expressions for expected shortfall that appear to use different probability ranges. It explains that they can describe the same tail-risk measure when authors use different sign conventions for returns and losses. Under a positive-loss convention, the expression averages upper-tail Value at Risk quantiles from the confidence level toward one. Under a negative-loss convention, it uses the corresponding lower-tail quantiles and integrates from zero to the tail probability.

The key is to check both how the random variable is signed and whether the probability level represents a confidence level or its complementary tail probability. Value at Risk may likewise be defined as a quantile or as its negative, depending on convention. The answer gives a conceptual reconciliation, not a worked numerical example, and assumes the quantile and probability notation are interpreted consistently. Readers should therefore verify definitions before comparing formulas across sources.

Key ideas

• The two expected shortfall integrals can express the same measure under different sign conventions.
• Positive losses use upper-tail quantiles, while negative losses use lower-tail quantiles.
• Confidence levels and complementary tail probabilities must be matched when comparing formulas.
• Value at Risk may be reported as a quantile or the negative of a quantile, depending on convention.

Tags

# What is the difference between these two Expected Shortfall definitions?


# What is the difference between these two Expected Shortfall definitions?












I have come across different ways expected shortfall is defined. e.g. $$ES_a(X)=\frac{1}{1-a}\int_a^1VaR_b(X)db$$

and $$ES_a(X)=\frac{1}{a}\int_0^aVaR_b(X)db$$ e.g. on Wikipedia's article.

Are these different?

$VaR$ itself is somewhere defined as simply the quantile and somewhere as negative of quantile.

Could you shed some light on whether it is simply inconsistency of some authors, or is there some deeper reasons?

## Answer by David Harper (score 3, accepted)

https://quant.stackexchange.com/a/16938

These are identical definitions of ES.

It's just a matter of expressing losses as negatives or positives.

First definition

Notice the integral bounds are $a$ and $1$: losses are positive; this is so-called Loss(+)/Profit(-).

Here alpha might be 95%, as in 95% confidence VaR or ES.

Second definition

Losses are negative, and the corresponding quantile is 5%; the integral bounds are $0$ and $a$.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.