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. – Ric Oct 21 '14 at 7:38
• I don't understand why you have the word spectral risk measure in the title ? – Rohit Arora Mar 22 '15 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? – Ric Oct 21 '14 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. – Ric Oct 21 '14 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 – emcor Oct 21 '14 at 9:45
• Yes, I agree about the first one ... sorry. – Ric Oct 21 '14 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 '14 at 20:27