Hello David,
I think this is a really simple question but if EVT tries to “zoom in” on the LFHS loss events, and we are dealing with the loss side of the parent distribution, shouldn’t we be looking at the left tail of the distribution (since losses are negative)? The reason I ask this is because when I look at either the GEV or GPD distribution, they look more like the right tail of the parent distribution. (or is this because we are viewing the loss as a positive number here?)

and also another question from your blog on EVT:
“There are two big model “design” issues: tail size and time dependency. If we increase the tail size (index) too much, the threshold moves toward a the middle and we defeat the purpose of EVT (and the shape index, as the reciprocal, moves toward zero, and we have converge to a Gumbel distribution).”

- “if we increase the tail size(index) too much”, is this done by lowering the threshold (thus giving us more observations)?
- and just to make sure I’m correct with the concept: If tail index increase, shape parameters must decrease and thus is bad for EVT (because in EVT we look for fat-tails)?

Thanks!

Hi Jack,

1. Yes, it is because “we are viewing the loss as a positive number here.” That is typical of EVT, to define the (positive) loss in excess of some (positive) loss threshold; this would analogous, I suppose, to treating VaR as a positive loss; i.e., Jorion gives P [absolute value (loss) > VaR ] <= 1 - confidence.

In the “LDA @ Work” case (reading 48, Operational risk), Deutsche Bank illustrates this usage (losses as positive; and it’s really instructive because you can see how the *parametric* EVT distribution is “grafted onto” the *empirical* body of the distribution). They set the threshold at 50 million losses, so the EVT distribution is used only in the tail where losses (X) exceed the threshold of $50 million. This gives us a chance to study “conditional” probability:

P [ X - u<= x | X > u]

I think this EVT probability is very intstructive. First, as you say, losses are positive. Second, it’s a (Gujarati) CDF [i.e., X - u<= x ], and third, it’s a conditional distribution; i.e., what is the probability the (postive) “excess” loss (loss above threshold) will be less than value (x) conditional on X exceeding the threshold (we only care about the tail).

2. On the other 2 questions, the last 2+ years the FRM had a lame EVT reading and it forced me to write stuff like that (frankly). Their were conflicting definitions of shape, which GARP didn’t really realize far as i can tell. I recommended they replace with the Dowd chapter, which they did, and I have reflected his consistent usages in the learning XLS: http://www.bionicturtle.com/premium/spreadsheet/2.c.3._extreme_value_theory_evt/
In the XLS, I deliberately avoiding using “shape” owing to some definitional ambiguity; however, we can safely (for test purposes) consider Tail Index = Shape Index.

2b. So, when I wrote “If tail index increase, shape parameters must decrease” I was referring to the previously assigned author who defined shape as 1/tail (and in doing so, directly conflicted GARP’s test). But, I would ask you to ignore this, for the current assignment Dowd is not so confusing and GARP, if they test shape/tail will certainly assume:

Increasing the tail index (or “shape param”)—> Heavier tail

the above applies to both GEV (block maxima) and GPD (POT).

2a. In regard to “if we increase the tail size(index) too much”, is this done by lowering the threshold (thus giving us more observations)?
Um, I don’t know why I commingled tail with threshold here?

First note this must refer to peaks-over-threhsold (POT) where we have the freedom to select a threshold (u)
The point is, if we lower the (u), it would be like DB LDA case above lowering their threshold from $50 MM, to say, $30 MM. Now their extreme tail is larger and, at some point, it’s including more of the body. I would just stop there…
The tail (shape) is a parameter of the GPD distribution: it gives the GPD heavier tails.

Hope this helps, David

Hello David,
thanks so much for the detailed explanation!

Just curious, for the EVT, is the FRM weighted more towards conceptual idea or quantitative calculations? I think I’m getting a pretty good idea about what EVT is now, but as you mentioned in another forum post a few days ago (I think it was about concerns over whether to sit for the full exam or just L1), you mentioned the EVT chapter in L2 is really tough especially for candidates without good quant skills. Since I’m sitting for the full exam, that got me a little bit nervous about what’s to come in the EVT chapter..

Thanks!

Hi Jack:

Great question. Previously, the FRM has only ever tested “conceptual idea” for EVT. The L2 EVT Dowd reading includes “compute VaR and expected shortfall using the POT approach, given various parameter values.” That *clearly* implies a calculation (the use of “compute” is standard LO term), and there’s extra work to be ready for that…

I would like to include this in the clarification questions to GARP. I would be happy to share/post here the working document that I intend to send to GARP’s research director (e.g., are you really saying candidates need to know the EVT/ES calc in chapter 7; can we confirm you are defaulting to “relative” rather than absolute VaR?)?

...in the meantime, given the AIM, i think we have to plan that the L2 include this calculation, but i’d like to clarify for our customers b/c I am struck by the fact that, as a practical matter (i mentioned this on the other thread), the full exam against 150 test questions necessarily implies you’ll study for ideas that won’t be tested…in fact, there are almost as many readings as test questions, so it would seem important that candidates don’t “over-study” for questions that will not be asked. David