# 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?

• Please add a reference to a paper or a web page with formulas. The question is unclear to me. Oct 21, 2014 at 7:38
• I don't understand why you have the word spectral risk measure in the title ? Mar 22, 2015 at 18:12

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.

• Good answer but are you sure abou the denominator in the second integral? Oct 21, 2014 at 9:32
• And I think in your setting the upper bound of the integral should be $-VaR$ and in the first integral also $-VaR$ or the quantile directly. Oct 21, 2014 at 9:34
• @Richard I changed the second integral. The first integral is the definition of CVaR: en.wikipedia.org/wiki/Expected_shortfall#Formal_definition Oct 21, 2014 at 9:45
• Yes, I agree about the first one ... sorry. Oct 21, 2014 at 10:40
• Thank you for the help, gents. I was forgetting that the returns are already factored into the probability density function.
– user13327
Oct 21, 2014 at 20:27