distrbution for loss severity?

[ajsa]

Hi David,

Could you expalin which distribution(s) can be used to model loss severity? I think it does not mean EVT since it should cover HFLS as well. I saw Weibull can model it in a practice exam question

BTW is it true that Possion distribution assumes a correlation among occurances?

Thanks.

 

[David Harper CFA FRM]

Hi ajsa,

Fabozzi (Operational risk) lists the following severity loss distributions:

Parametric Approach: Continuous Loss Distributions
Exponential Distribution
Lognormal Distribution
Weibull Distribution
Gamma Distribution
Beta Distribution
Pareto Distribution

I'd add the following thoughts:

* In FRM, we say: frequency tends to be discrete, severity continuous (the above are all continuous, i think)
* the other obvious option is a (non-parametric) EMPIRICAL OBSERVATION
* e.g., in our LDA case study, right? they combine piece-wise empirical distributions (to $50 MM loss) with parametric EVT (> $50 MM loss)
* Yes, EVT (e.g., POT) is often used to model the extreme tail in loss severity -- that's what they do in LDA case.

Poisson assumes independence; in LDA case they capture loss freq. with copula function

David

 

[ajsa]

Hi David,

1. For these distributions Fabozzi lists, are they all fat tailed? at least Lognormal is not, right? so does it mean they need to be combined with EVT for severity loss distribution? If so, why normal distribution cannot be used? Because it is not skewed?

2. In GEV, if the shape parameter is smaller than 0, it is called Weibull. Is it the same as the Weibull Distribution? Does GEV only cover the cases that the shape parameter is greater than 0 (Frchet)?

3. Is the Pareto Distribution listed same as GPD which however is for extreme tail..?

thanks!!

 

[ajsa]

Hi David,

so if there is correlation between loss events (serial correlation?), Poisson should not be used for frequency, right?

thanks.

 

[David Harper CFA FRM]

Hi ajsa,

1. I don't have time to check skew & kurtosis for all of the above distributions, but eyeballing them, i think it's safe to say the are all capable of either (i) positive skew (S>0) and/or (ii) heavy-tail (K > 3). Fabozzi's emphasis for OpRisk distributions is positive skew; and then heavy-tail may be appropriate also. The lognormal is moderately heavy tailed; see kurtosis http://en.wikipedia.org/wiki/Log-normal_distribution .
In regard to normal, it would not be included b/c it exhibits neither +skew nor heavy tail; but also, please note most (all?) of the above distributions are NON-ZERO; i.e., unlike market risk, you tend to want a non zero distribution for oprisk. So, i would say oprisk severity distribution characteristics incl.: (i) nonzero, (ii) skew and/or (iii) heavy tail

2. I think that this (unpopular) case of GEV is indeed (merely) a re-parameterized case of the "regular" Weibull, but am < 100% certain. GEV has three cases, but only Frechet really matters to us b/c it's the only heavy-tailed case (i.e., shape/tail index > 0)

3. Yes, it is (essentially) the same, it is re-parameterized version of same

4. No, LDA case study employs Poisson for frequency (Poisson connotes freq, not loss severity) but handles the correlation via copula. So, Poisson is not incompatible with a dependence assumption. (but that's the point of a copula)

David

 

[ajsa]

Thanks David.. I wonder if the correlation in frequency is serial correlation?

Thanks.

 

[David Harper CFA FRM]

Hi asja,

Well, I was sloppy to write "the LDA case handles correlation"
correlation is the limited (read: makes assumptions, has weaknesses) and specific type of LINEAR dependency
dependency is a broader more flexible term: copulas can handle non-linear *dependencies*

So, in regard to DB LDA:
they used Poisson to characterize FREQUENCY of losses within ET/BL cells;
the Poisson itself assumes i.i.d. (i.e., no correlation between events in time)

okay, but in regard to DEPENDENCIES (not correlations!) beween ET/BL cells, they did use copula function
...so this is only analogous to correlation but it's a non-linear dependence: copula function = glue = copula (Poisson Distribution1, Poisson Distribution2)

so: dependence is big wide bucket of many things, correlation is specific, linear, challenged

David

 

[ajsa]

Thank you David! That makes a lot sense now!

I also remember I read somewhere that if the loss events have positive correlation (I think this is the real serial correlation case), frequency should use negative binomial distribution.. could you confirm?

thanks.

 

[David Harper CFA FRM]

Hi asja - Right, in the LDA case, they considered negative bionomial instead of Poisson b/c it's sort of a super-class of Poisson (it does not require constant lambda), so it can characterize a relaxation of the "independence" in events ... i think it may be fair paraphrase to say: they hypothesized + correlation and therefore neg binomial is more appropriate, but as so often is the case, settled on Poisson for convenience/ease...

 

[ajsa]

Hi David,

I think serial correlation is *error terms* from different time periods are correlated. So *generally speaking*, what do we call the the situation where the variable values themselves from different time periods are correlated?
Thank you!

 

[David Harper CFA FRM]

Hi asja

Auto or serial correlation (technically) refers to the violation of an CLRM assumption: covariance(error, error) = 0; and so, refers to correlated errors in a regression. I agree this connotes time series but can also be more general (i.e., time does not need to be the X axis).

Generally, as above, we use correlation if mean the limited, linear case. But, best (IMO) is "dependence," as used in the DB LDA case, because it encompasses any association (incl. nonlinear) (So i suppose "associated" isn't bad either).

David