Expected Shortfall

madcow49 · March 5th, 2009, 10:25 pm

Hello, I am trying to find the Expected Shortfall corresponding to a 95% confidence level on a lognormal distribution. Given x~LogNorm solve for Alpha. E[ x | x > LogNormInv( Alpha%) ] = LogNormInv(95%) Any idea?Many thanks

acastaldo · March 7th, 2009, 4:47 am

Are you sure your equations are correct?Should'nt it be ES = E[x | x > LNORMINV(0.95) ] This expectation, the partial expectation of a lognormal variate, is available in closed form.

teddydavis · August 28th, 2009, 8:40 pm

Hello acastaldo,Do you know of a clear textbook or other reference which walks through the derivation of this partial expectation?Thanks.

acastaldo · August 29th, 2009, 7:45 pm

teddydavis asks for a clear derivation of the partial expectation of a lognormal r.v. Here is mine:we want to findMake a change of variablesone more change of variableQEI

fizik · August 3rd, 2011, 7:57 pm

Expected Shortfall is usually defined as:Also, is typically defined through the percentile .Putting it all together:

olim275 · June 18th, 2012, 3:45 pm

Hi acastaldo,I wanted to ask if g(k) gives final partial expectation or we should multiply it with probability to get the expectation.Thanks

acastaldo · June 18th, 2012, 4:47 pm

I believe fizik's remark is correct: the integral g(k) needs to be divided by the probability of occurrence of an extreme event to give ES.