Expected shortfall BIS convention

The definition of ES used by the BIS Basel Committe as part of the Trading book review that replaced VAR with ES seems to be referring to what in this syllabus is TVAR, can anyone confirm and why are the two used interchangeably because Sweeting calculates ES by diving by the whole observations in the distribution and TVAR by dividing by the number of observations in the tail?

 

I'm pretty sure you're right. What the BIS is calling ES is what Sweeting calls TVaR. Furthermore, FINMA (the Swiss regulator) use the same terminology as the BIS when specifying the Swiss Solvency Test. In my experience, this is terminology used most often in real world applications. Why Sweeting and the profession chose to use different definitions is not clear to me.

 

It is in some cases, but we can’t always define every term we use and it’s unhelpful when everyone is using one definition of a term and someone else comes along and defines it differently because it creates confusion. I challenge anyone to cite an authoritative reference other than Sweeting, Profession and ActEd materials that define Expected Shortfall in the way Sweeting does. Is this just a case of everyone else doing one thing and the Profession doing another? And if so, why?!

 

Hi td290, it's not just Sweeting even Wikipedia explains ES differently from the Basel accord;-

http://en.wikipedia.org/wiki/Expected_shortfall

So there are three different definitions of ES one by Wikipedia, Basel accord and Sweeting

Solution is stick to Sweeting to pass, but the Sweeting approach doesn't make sense. Atleast Wikipedia can explain what they are trying to say, as for Sweeting what are you doing by looking at the mass of a probability distribution above the VaR and then dividing by all observations?

 

I can't find a precise mathematical definition of expected shortfall in any of the Basel documents. All they say is that it is "the expected value of those losses beyond a given confidence level," which seems to me to be in agreement with the Wikipedia definition. Furthermore it is clear from some of the responses to the Basel consultation papers (see e.g. Embrechts et al "An Academic Response to Basel 3.5" https://people.math.ethz.ch/~embrecht/ftp/Basel_3_5.pdf) that others also believe that the intended definition of ES is that given by Wikipedia.

So I stick with my original assertion that it does seem to be one definition for Sweeting/ActEd and one for Basel/Wikipedia/everyone else.

 

hI td290, I agree it's a taxonomy thing. Wikipedia doesn't explicitly agree with the Basel accord it defines TVaR as "It quantifies the expected value of the loss given that an event outside a given probability level has occurred." which is the same as your definition of ES from Basel and different to the Wikipedia definition of ES I posted in my prior post.

http://en.wikipedia.org/wiki/Tail_value_at_risk

Infact I now see where Sweeting is coming from with his def of ES (Although it is different from the Basel accord);-

from Wikipedia link to TVaR;-

So I think, Sweeting + Wikipedia agree on what TVaR and ES are, while the Basel accord seems to have TVaR = ES (for some reason).

 

No, I believe the Wikipedia entry on ES and Basel are saying the same thing, specifically:\[\textrm{ES}={\mathbb E}\left[X|X>\operatorname{VaR}_p (X)\right]\] where \(X\) signifies the loss distribution and \(p\) the probability level. This is the definition I find in every context except ST9. In fact, this is the definition in the Embrechts textbook that was on the original ST9 reading list. It is Sweeting who is at variance with all of these for some reason.

 

Wikipedia says that's the case only for continuous distributions.

 

Yes, I was considering continuous distributions because things can get quite complicated if you start including discrete distributions as well. Even so, it leads to the same conclusion. The key definition to consider is:\[\operatorname{ES}_\alpha=\frac{1}{1-\alpha}\int_\alpha^1 {\operatorname{VaR}_\beta (X)d\beta}\]
On this, Wiki and Basel agree, but it doesn't match Sweeting's headline definition of ES. That applies to any type of distribution.