Expected Shortfall as an Average of Tail Quantiles

Summary

The document clarifies why expected shortfall (ES) can be described both as an average of value-at-risk (VaR) levels and as an average of losses in a tail. In its integral definition, VaR quantiles across the selected probability range receive equal weight. That does not mean loss outcomes receive equal weight: the quantile function maps those probability levels to losses of differing magnitudes.

For a continuous loss distribution, the answer expresses ES as the conditional expected loss beyond the VaR threshold, or equivalently as an integral of loss values weighted by their density in that tail. This explains why rare outcomes are included without making ES an unweighted average of tail loss values. The explanation is brief and uses a particular sign convention for losses; details at probability mass points or with alternate conventions are not discussed.

Key ideas

• Expected shortfall averages VaR quantiles over a chosen probability range.
• Equal weighting applies across quantile levels rather than across loss values.
• The loss outcomes in the tail contribute according to the distribution's density.
• For a continuous distribution, expected shortfall can be expressed as a conditional mean beyond the VaR threshold.

Tags

# Expected Shortfall and Spectral Risk Measure


# Expected Shortfall and Spectral Risk Measure












Not sure I am understanding spectral risk measures correctly.

Why is there an equal weighting scheme placed on the tail losses in expected shortfall.

Will that no bias the expected value of the loss towards the lower tail because the probability that the loss will occur is small compared to that which is closer to the p-value?

## Answer by emcor (score 1)

https://quant.stackexchange.com/a/15110

The ES definition is:

$$ES_\alpha(X)=\frac{1}{\alpha}\int_{0}^{\alpha}VaR_\beta(X)d\beta$$

This is indeed an equal weighting over each VaR, but not on the $x$-Achsis, VaR is the inverse function such that adding all possible VaR's is equally weighted, but the VaR's themselves have different magnitude over the $\alpha$-Achsis.

The formula can also be rewritten as expected value:

$$ES_\alpha(X)=E(X|X<-VaR_\alpha(X))=\frac{1}{\alpha}\int_{-\infty}^{-VaR_\alpha(X)}x \cdot f(x)\,dx$$

Therefore you can see that on the $x$-Achsis, it is a weighted average.

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