Hi there
@emilioalzamora1 !
I am honored to talk to you, I have seen the amazing clarity of the responses given by you in the forums.
If I am not wrong, you are asking why the difference arises between the two methods, right?
I would like to hazard a guess here based on my understanding of the situation (it may be wrong also)
In Option 1, you are finding out the position of the 5th worst loss by computing the
position of the percentile, the location of the percentile is based on
(n+1)*percentile to find out the number of data lying under it, here, in your case, it is 5.05. There are different variations of the same rule, we can either use the formula above which I think incorporates the effect of an additional degree of freedom. Or, as I have posted in the attached Excel sheet, we can also take the 95% value of the number of the 6 worst losses, which we get as 5.7
The difference between both the above approaches is that the former method weighs the location in favor of the simple average HS location of 5th worst loss itself while the latter weighs it in favor of the 6th worst loss, i.e., asks us to find or shift the value closer to the 6th position. This approach can be seen to be applied in some companies.
A twist to the above story is that, a more acceptable approach would be to take the 6th worst loss of -3.20% as the 5 observations are assumed to be included in the tail.
Linda Allen's approach is to assign a curve based weight to the observations as opposed to the linear or simple weights that is implicitly assumed in the formulae above. We can see that the location of the values for the 5% percentile varies if we use the approach given by Linda Allen. The weight used by her, which is [(1-Lambda)*Lambda^(days-1)]/(1-Lambda^K) weights the most recent observations (the recent memory ) with a higher weight and those in the distant past to be having a smaller weight (or memory) and hence the day structure of the losses causes the 5th percentile to shift towards the -3.6% and -3.4%
As far as your final question goes, we can see that considering the minor difference in the percentage losses among the different methods, it doesn't really matter in practice which method is adopted. Although, David's approach incorporating the central mass position of the observation is the most theoretically sound and Linda's approach is the second best, companies would always like to keep it short and simple in the interests of practical difficulties encountered in implementation and operation.
Therefore, Many companies use Option 1 or the variation of it that I pointed out or even the simple Average HS, it just varies based on the Risk policy of the company to weight it to the closer (higher) of the two worst losses or the farther observation. In many cases, just take the 5th or the 6th observation as we cannot be too nitty-picky or precise as to an uncertain number- a best estimate works in 90% of the cases!
Hope this helped and I think
@David Harper CFA FRM can give greater clarity here
Thanks
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