There are many ways of calculating OpVar and without embroidering semantics the process really falls into one of two camps; parametric or non-parametric. Most credible operational risk systems used to calculate capital, approaches discussed on the internet, the capital articles on this site and the quantification work I have been part of seem to land in the parametric world.
So what about the non-parametric techniques?
+ Bank of Japan
[Makes mention to such statistical analysis, even though it was a while back now.]
Current studies have revealed that validity of VaR could be robust with some kind of non-parametric or distribution-free estimator such as Harrell-Davis estimator, which has been recognized as very strong order statistics.
Bank of Japan
As we all know, operational risk isn`t quite a science of precision or perhaps more accurately the discipline isn`t truly a coherent risk measure because it lacks subadditivity [For two random uncorrelated losses A and B and a risk measure denoted by p() ... For all A and B, p(A+B)<= P(A)+P(B) which implies that aggregating individual risks does not increase the overall exposure. Operational risk fails this rule.] we`ll still keep an open mind and investigate the equivalence of this line of thought.
The process the Bank of Japan is leaning towards is similar to the L-Estimator and it computes linear combinations of the ordered statistics by estimating the quantile as a weighted average of one or more losses.
+ Parametric
Value at Risk is a measure that is driven in terms of a quantile of a given distribution, or to be precise where loss has the potential of exceeding a specified probability. In the parametric operational risk world this is accomplished by measuring losses, grouping loss data so that it may be aligned to a probability distribution family, remove any outliers, describe the curve mean, skew, variance, model the function of choice, take estimate parameters, evaluate quality of fit and finally carry out goodness of fit test. The tails of the curve can either be described through scenarios or semi-parametric methods such extreme value theory.
+ Non-Parametric
Alternatively non-parametric methods make no explicit assumptions about the distribution itself except the quantile is derived directly from the data which is assumed to be an independent and identically distributed set of losses. The concept works by taking a point estimate from the ordered statistic of the sample and at the smallest value in the sample outwards. For example; if we have 100 losses and we are looking for VaR at 99%, one would order the losses and estimate this value as the second to last statistic. This process is often dubbed the Upper Empirical Cumulative distribution Value (UECV) however it obviously can sustain high variability with subsequent measurements, so we have to elaborate on the process.
Theoretically, if these variants or `marginals` are tracked, captured and plotted they should potentially yield a curve, albeit a jagged one depending on the number of marginals we capture. This curve can be smoothed and the risk analytics based on the differential application of the smoothing outcome. The drawbacks unfortunately are that many measurements need to be taken for confidence to increase and data may be withholding. A solution to the problem is to calculate the marginal VaR as a weighted average over a range of quantile positions which is the L-Estimator process. UECV is actually an L-Estimator, the last one in the series and it places entire weight on the final ordered statistic.
The process being driven at here however is slightly different as it is carried out across the whole distribution function, spreading each sample observation over an interval with a `kernel`. As we know the kernel (plotted marginals through the process) is actually a symmetric probability density function with its own shape, skew and kurtosis.
For a full description of how this can be applied to market risk please follow this link, the article is an Algo Research Quarterly paper and outlines the process in far more depth.
Algo Paper
For what it`s worth I wasn`t able to locate any L-Estimator references in the context of operational risk on the internet, except The Bank of Japan`s comments. Most operational risk analysts that have been around a while will agree that there are some bizarre attempts to measure this risk classification, particularly in the light of capital allocation and I was certainly expecting some mention on this in the broader network, but none was to be found.
+ Critique
On the up side though, curve fitting errors are less likely to be a problem and the attachment of external data should be quite straight forward assuming it is scaled correctly, but that has always been the case. Correlation of risk factors can of course be asserted through snapshot stages of the kernel and that in itself is an interesting thought that might incite some further contemplation. The process could also assist with the management of events assuming such investigations are carried out in an open reported format and that is an important function of any operational risk capital system.
My critique of it though is that while an OpVar number is returned with very small samples, accuracy is dependent on the completeness of these samples, so we aren`t really escaping the data problem. How many samples are required for completeness to be accepted is going to be one question that will need to be answered and documented. There also seems to be a real detachment to the frequency of events and these will need to be modeled and integrated separately. Then there is the concern that this approach may have difficulty actually separating tails. Combining them by ignoring (automatically including) them may seem to resolve some problems but it introduces others, particularly where the zone between expected loss and unexpected loss begins. This may seem trivial but a definition of this zone is critical to represent how capital is applied against loss; that is some expected losses may have their own reserves/budgets or may actually be costed into the business or products. Understanding whether these budgets are healthy, suitable and sustainable is part of the whole capital process and necessary for Basel II accreditation.
On a closing note, we accept that Harrel-Davis appears on the surface as a `tidy technique`, it is debatable whether the regulator would endorse such an approach and that alone might steal any enthusiasm away from further advancements in this arena. Perhaps a good use for it though, is as a test or second opinion on an existing capital model and that in itself has value.
