R26.P1.T4.DOWD_Topic: EXPECTED_SHORTFALL

[gargi.adhikari]

In reference to R26.P1.T4.DOWD_Topic: EXPECTED_SHORTFALL :-
For finding the ES, we find the Total area under the Tail and then divide by (1-alpha) to get the average amount of Loss under the Tail ...Shouldn't we dividing the Total area under the Tail by Alpha instead of ( 1-Alpha). Am probably missing a point here..Grateful for any insights into this...

 

[QuantMan2318]

@gargi.adhikari

We have to remember that in Dowd's version of the formula, alpha is the Confidence level and hence we have to divide by 1-alpha, which gives us the probability of exceeding the losses at a given Confidence level. I have attached an Excel based on Dowd's example showing the computation of ES under Dowd. As you can see, when we multiply the quantile (q) by the size of the difference (dp) -which is the size of the slice and divide it by 1- alpha, we get 1/10 or 0.1 which is the same as adding all the VaRs at the different Confidence levels and dividing by the number of buckets

We can also see from the above screen that David has taken the integral from alpha to 1, which means that he is taking the area or the sum of all the losses from the CI to the tail and hence the constant dp term which if combined with 1-alpha gives you the number of buckets. dp/(1-alpha) = 1/number of buckets or slices.

Please note that my example is also a discrete version (the reimann sum), when the size of the slice becomes smaller and smaller and the number of slices increases, you get the continuous version of the formula with the integrals above.

We can divide the sum of the product of the quantile and the size of the slice by alpha if alpha is the significance level and the integral is taken from zero to the significance level as is done by wikipedia.

Thanks
Mani

 

[gargi.adhikari]

@QuantMan2318 Was stuck on this ..Thank You so so much !