A common choice of distribution for operational risk modelling is lognormal. This distribution has a fairly heavy tail and is mathematically and conceptually simple. Often a separate lognormal will be used to model the body and the tail of each risk cell, so that they can be parameterised separately. The body distribution can be chosen to closely match the common losses that occur in the internal data for that risk cell, which the tail can chosen to match the expectation of extreme losses. The necessity of this is questionable, since the high quantiles computed in op risk modelling are overwhelmingly driven by the tail. In most cases, if model simplicity is desired, a single distribution using the tail parameters is sufficient.
Once the loss severity distribution has been estimated for each risk cell, along with a corresponding frequency distribution (typically Poisson, although op risk models are not sensitive to this choice), then Monte Carlo simulation is used to sum them and produce the bank’s total or aggregate loss distribution. The modeller may then be surprised to discover that the median aggregate loss is far higher than the bank’s typical annual loss! After all, the median loss from the body distribution matches that in the internal data for each risk cell, so shouldn’t the same be true of the aggregate distribution?
The explanation here lies in the heavy tailed nature of the lognormal distribution. In particular, the lognormal has the property that for large \(N\), the sum of \(N\) randomly generated losses approaches the maximum loss, i.e.
\[\sum_{i=1}^N x_i \to \max(\{x_i\}).\]
This means that, surprisingly, even a typical loss in the aggregate distribution (say, the median) is primarily constituted by a small number of tail losses from the individual distributions, and not losses near the medians! Thus, the medians of the individual distributions end up being pretty much irrelevant.
Of course, op risk models are typically constructed to estimate the 99.9% quantile, not the median (which banks have data for anyway). However, it is instructive to note that it is very difficult to model op risk in a way that accurately reproduces both the extreme losses, and the more common losses, due to the heavy tailed nature of the distributions.
This illustration shows how even relatively simple financial modelling problems can result in technical complexities that are very difficult for finance profressionals to understand and resolve. This is a great example of how our quantitative analysis consulting services can deliver great value and impress the regulators!