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?


1 Answer 1


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 $a$.


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