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# Expected shortfall

#### wrongsaidfred

##### Member
After reading through "Measures of Financial Risk" by Dowd, it seems like something is wrong with either the formula for expected shortfall or the description of it.

The reading states that when using expected shortfall, "all losses in the tail are given the same weight", yet the discrete formula states:

ES= (1/(1-alpha))* SIGMA[ pth highest loss * probability of pth highest loss].

Doesn't this mean that each loss is weighted by its probability? Am I reading something incorrectly or is there an inconsistency here?

I thought that the whole point of the section on generalized spectral measures is that different weights can be given to each of the possible outcomes, but it seems like that is exactly what this formula for ES is doing.

Any explanation would be greatly appreciated.

Thanks,
Mike

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi Mike,

(Sorry for delay, I was on vacation for the last week.)

This statement, "all losses in the tail are given the same weight," is true only in the special case of a simple historical simulation, where by definition, losses are simply ordered and the distribution is uniform.

Note that the correct formula for discrete ES is what you show:
ES= (1/(1-alpha))* SIGMA[pth highest loss * probability of pth highest loss]
... please note this is consistent with expected value of a discrete variable : the expected value of X = sum of [X * f(x)]

This is correct as a general form because it handles any discrete distribution (I like to keep in mind that we have two separate "steps" which are distinct: 1. specification of the distribution and 2. computing the VaR quantile or ES from the distribution). If the distribution happens to be "simple" as in a historical simulation then the above REDUCES/SIMPLIFIES to the average of the worst (1-confidence%)th losses.

But otherwise the above is general and, just as you say about a spectral measure (as Dowd points out, ES and VaR are both special cases of the spectral risk metric), the above is calculated the expected (mean) value of the discrete tail but where the distribution may not be uniform. A non-uniform distribution rules out a simple historical simulation, but that is the point of Dowd's non-parametric approaches like age-weighted historical simulation. An age-weighted HS is not uniform and it is false to say about it that the ES gives all losses equal weight.

In summary:
* The discrete distribution is the first step, so to speak. Only in the simple HS is it uniform (i.e., all outcomes equally likely or equally weighted). Other non-parametric approaches vary the weights assigned to tail losses
* The ES is the average of the tail losses in either/any case (continuous or discrete, uniform or not). The above formula handles any discrete case.
* The ES formula above is, in fact, a spectral risk measure but a special case where the non tail (e.g., 99% of the distribution in the case of a 99% ES) is given a weight of zero.

I hope that is helpful, thanks! David

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