# Approach to add scenarios to OpRisk loss distribution

There is quite a lot of literature on OpRisk modelling. My question focuses on a loss distribution approach (LDA).

Let's look at a basic model. A Poisson-distributed $N$ and loss sizes $X_i$ and from the data I model $$L = \sum_{i=1}^N X_i.$$ Then the regulator wants me to add some forward looking scenarios. There is some literature on this e.g. A “Toy” Model for Operational Risk Quantification using Credibility Theory. They often use Bayes methods and the scenarios have to address the parameters of the distributions involved.

I thought about something easier: if my expert says: "I expect a loss of 10K to happen once every 3 years" then I could model this scenaro as a Poisson $N_1$ with intensity $1/3$ and loss severity with point mass 1 at 10 000: $$L_1 = \sum_{i=1}^{N_1} 10 000 = 10 000 N_1.$$ I could add this to my loss variable $L$: $L+L_1$ and incorporate many scenarios and still keep my model very tractable.

Is there something wrong or too simplifying in this approach? I have not seen it in papers. Is it just too easy for a paper? Thank you for any comments.

The problem is not the sum $L + L_1$ but the question whether your $L_1$ is really a good model for whatever you might be missing in $L$. I personally (and maybe also some regulators) would regard losses always equal to 10K and completely independent from everything else not to be a good model for the low frequency high severity events typically missing from historic data. So you need to elicit from your expert more and better information (which of course is a psychological not mathematical problem) about the tail and then you need to reconcile this with whatever you already have/know about the historical losses. But then simply adding a fat tailed distribution (maybe including some tail dependency to account for common underlying causes) on top might become onerous in terms of capital. This is where Bayes/Credibility stuff becomes relevant.

• Thank you for your thoughts. Among the many references for Bayes-methods to incorporate scenarios do you have one that is fit for purpose but not too compliaced to implement? Any thoughts on a simple Bayes approach? – Ric Feb 22 '18 at 16:25

Taken from my experience as a trader I would suggest there are two parameters that comprise OperationalRisk: 1) A distribution of the size of losses due to the event, 2) A distribution of the frequency of events.

I suspect a Poission distribution is fine to use to predict the frequency. Empirically this would tally with my experience. Secondly, with regard to size this is a little more complicated. If we categorise them as small medium and large let me give some reasonings:

Large losses can (and do) happen but they are less likely, because usually things are observed before they grow. For example an Interest Rate Swap with an erroneous trade booking will likely be picked up after the first missed cashflow and thereby not have time to accrue further losses. There are often more processes and human observance on larger trades which would give rise to the large losses.

Medium and small are more common. Small might arguably be less frequent since very small errors might never even be registered by either counterparty and thus result in an uncorrected and unobserved scenario. Medium might be the most frequent then, since they will be large enough to be ultimately noticed and they usually catch the large errors before they become large.

I do think that OpRisk is generally independent of market conditions due to its eclectic nature. There are a number of distributions that you might argue for (I'd pick the one from which it is easier to sample from). If you want to start at zero pdf for zero loss then maybe a scaled Gamma(2,2) or Beta(2,5). If you don't want to start at zero pdf (but assign probability to very small losses) then a truncated Normal or Cauchy. Since its a broad assumption anyway I'm not convinced the distribution matters that much as to afford stochastic nature to the scale of losses.

You should calibrate frequency with size though: more small losses or less frequent large losses will have same expectation but may other wise impact your Risk Statistics.

• Thank you for your remarks on the LDA approach (+1 from me) but my question was on how to incorporate scenarios ... you do not mention scenarios ... – Ric Feb 21 '18 at 11:24