Risk using Student T

Discussion on Value-at-Risk Models
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FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Risk using Student T

Postby FatTail » Sun Nov 16, 2008 10:31 am

A few questions/thoughts about using Student T for risk

(a) Estimating the correct degrees of freedom (tail index) from data
- what are best methods - Hill estimator other etc
- it seems a practical way to do it would be to use the ratio of ES/VAR from the empirical data for a relevant pecentile and look up which degree of freedom (tail index) that would imply from a t distribution. Is that a method that is used?

(b) If you add up n independent t distributions are there any formulas or rules of thumb about the degrees of freedom (tail index) of the combined distribution

(c) For Degrees of freedom of 1 this is a special case = Cauchy where mean and variance are infinite. But also it is self similar/stable. For degrees of freedom =2 this is case where variance is infinite - which is also true for Levy alpha stable distributions. Any ideas which Levy Alpha that Student T degrees freedom = 2 is closest to? Is by any chance the Student degrees of freedom =2 Stable/Self Similar?

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Sun Nov 16, 2008 10:47 am

Hi Fat Tail - welcome!
(a) MLE should be fine, with a good optimizer - not sure, but maybe Hill estimation would be over-doing it? Your idea about using the ETL/VAR ratio seems clever, but may be crude because it uses only the tails of the distribution - a bit like using method of moments, which only uses summary stats
(b) Student t is not stable, for d.f. > 1 (I think) so sum of n independent Student t's is not another Student t.
(c) Sort of answered in (b)...But, why are you interested in Student t with d.f. 2, since variance is infinite? In fact, what is the point using d.f. less than 4, since kurtosis is infinite? Are you ignoring the volatility clustering aspect, which is very important for VaR assessment?

Cheers, Carol

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Sun Nov 16, 2008 5:43 pm

Hi,

Looking forward to the new book on VAR

On (b) a Student t distribution has power law tails in the extreme of distribution with power = df. So if you add lots independent distributions that each have the same power law .....is the power law maintained or does it decay away back to say normal in the end if there are enough of them. (the extreme case of df=2 seems interesting in this context as presumably if the individual distributions have infinite variance then presumably sum of these will also have infinite variance)

On (c) not sure why it matters practically that we have finite kurtosis. A lot of studies find a power/tail index in financial data of between 2and 5 so interested to understand extreme end of that.

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Mon Nov 17, 2008 8:41 pm

RE (b): I did a quick test via monte carlo for df 4 and greater then tails decay away relatively quickly to normal under simple addition of multiple iid runs. For df 3 it decayed away but quite slowly and at least in my test did not get back to normal for percentiles I was looking at. For df 2 it did not decay away at all back to normal in the tails that I looked at, and bevhaviour was somewhat similar by observation to adding L stable distributions (with about alpha approx = 1.85).

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Mon Nov 17, 2008 8:49 pm

Hiya

Just to say - I am reading your posts but have imminent deadline for Vol Iv to GO TO PRESS (at last!). Since your questions/comments require quite a bit of thought, I'll have to reply later - as soon as I can though!

Cheers C.

PS Did you know that 'Fat tail' is a bit of a misnomer? Leptokurtic distributions often have long, THIN tails.... C x

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Thu Nov 20, 2008 11:03 pm

So Leptokurtic means high in the middle rather than fat in the tails....we like fat tails, high in the middle is just incidental. Look forward to the book.
Hopefully it has some insights into the T distribution that has a lot going for it but is held back from wider use by, in it middle states, not being stable and self similar which are features that are attractive mainly for tractability and communication reasons rather than anything more fundemental - i guess.

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Fri Nov 21, 2008 9:02 am

I am interested in your tests of speed of convergence in CLT. How may steps until normality for 4 d.f.? When d.f. = 2 there is infinite variance, and finite variance is a condition of i.i.d. variables for CLT, I guess.

I use Student t quite a lot in Vol IV - Chapter 2 has Student t VaR and Student t ETL, and Student t [b:1t2rq36f]mixture[/b:1t2rq36f] VaR and ETL formulae too. I like mixtures a lot, as you'll see. What I don't do in Vol IV, but will do in Vol II 2nd edition, is skewed Student t. That has much to be said for it.

Cheers, C

PS Are you familiar with Sato processes, and the stable distributions you get as limit law distributions from non-independent but i.d. variables? There is a long (half a page) footnote on this in Vol III, p229, with references there and in the text.

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Sat Nov 22, 2008 8:24 pm

Last thing i was looking at df=2 did seem to be converging a little bit will do some more tests......

In terms of tail index/tail exponent estimators. Simple Hill estimator seems to be the most stable in tests I am doing versus theoretical data - tests that require solving two parameters are more volatile in terms of calculating the exponent part - i guess that obvious. Obviuosly using Hill estimator as a direct input into degrees of freedom in T distribution is biased.

What was quite insightful to do was to take a t distribution (non random) and calculate Hill estimator on it for different df and cut off points and actually that potentailly gives you a look up table/matrix to map the Hill estimator back to a degrees of freedom estimator - or perhaps can be done more directly.

Mainly looking at theorectical data now but hoping this sort of t distribution tail factor estimator maybe (i hope) less sensitive to cut off points than directly using Hill.

It also seems tail factor = 1 /df is a more logical scale to work in than df directly.

I will get volume 3 to look at Sato process. tks

coalexander
Posts: 815
Joined: Sun Sep 28, 2008 10:30 pm

Re: Risk using Student T

Postby coalexander » Sun Nov 23, 2008 7:15 pm

Hiya

Problem with any tail estimation seems to be the sensitivity to choice of threshold - there's quit e alot of research on this for GPD parameter estimation - e.g. Roncalli and his guys at Credit Suisse.

Don't buy Vol III just for Sato processes - there is only a large footnote - better look at Eberlien's paper with Madan?

Deep in Vol IV proofs - should be signing them off tomorrow !

Cheers, Carol

FatTail
Posts: 20
Joined: Sun Nov 16, 2008 10:07 am

Re: Risk using Student T

Postby FatTail » Sun Nov 30, 2008 10:37 am

For estimating tail estimator applying the enhancement on Hill proposed by Huisman et al in "Fat tails in small samples"improves things quite a lot. It seems people use this also in OP risk sometimes when trying to anchor the tail shape paramter for use with a GPD.
However if you throw a normal distribution at it you still get a small positive shape parameter and that needs to be adjusted back to zero for consistent use with t distribution - can do in an ad hoc way but good to find a more justifiable way.


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