What Are We Hedging Against?

Paragraph 667 of the Basel II regulation (BCBS, 2004) states: „Given the continuing evolution of analytical approaches for operational risk, the Committee is not specifying the approach or distributional assumptions used to generate the operational risk measure for regulatory capital purposes. However, a bank must be able to demonstrate that its approach captures potentially severe ‘tail’ loss events. Whatever approach is used, a bank must demonstrate that its opera-tional risk measure meets a soundness standard comparable to that of the internal ratings-based approach for credit risk (i.e. comparable to a one year holding period and a 99.9th per-centile confidence interval).”

The above provision addresses three important aspects that must be considered at the out-set, before any further, more detailed assumptions regarding the operational risk model are formulated. The first of these is the problem of defining what, in essence, constitutes an unex-pected situation against which a financial institution should protect itself. The second concerns the level of protection formulated in statistical terms. The third issue arises directly from the two preceding ones. It is the question of how the concept of "rarity" is to be understood. This question will be addressed first.

Rare — meaning what exactly? The regulator formulated the concept of rarity on the time axis. This means that the intention is to protect against a situation occurring no more than once in some given period of time. This is a significant assumption, and one that carries conse-quences. A phenomenon can be described in various ways depending on the axis on which one chooses to place it. This is most clearly visible when a rare event is considered alongside other observations — those that are typical, and therefore not rare — that are placed on the same axis. These will constitute a reference point (sometimes of decisive importance, as will be demonstrated later) for the rare observation. It is thus apparent that the concept of rarity is inextricably linked to the concept of a typical event — average, everyday. It is precisely the definition of a typical event that determines the perception of what constitutes a rare one. Different types of non-rare (typical, average) observations will be placed: (a) on the time axis — where the non-rare observations are years in which the level of the phenomenon is typical and expected; and (b) on the value axis — where the non-rare observations are values of the phenomenon at an average, typical level. One may, for instance, have at one's disposal the following sets of observations: 10 years with typical values, or 300 observations of the phenom-enon with typical values. This represents a fundamental difference in the way of looking at the modeled phenomenon. This distinction affects the perception of what constitutes a rare (ex-treme) observation, on what its interpretation as rare depends, and how the probabilities as-signed to it are to be understood.

Once in 1,000 Years

Adopting such a widely used interpretation of the 99.9% quantile leads to a line of reasoning in which the view of the quantile as excessively conservative appears quite justified. For how can one reasonably expect to protect against a phenomenon occurring once every 1,000 years? What approach could be taken to at least attempt to approximate a hypothetical situation of such frequency and to determine the expected loss volume associated with it? It would seem that such an approach — supported by the regulator — undermines from the outset the very sense and rationality of the modeling effort undertaken.

Indeed, it is impossible to imagine a situation that is to occur 1,000 years hence — let alone its financial consequences, since by that time the very concepts of finance may no longer be in use. It would be equally difficult for a hypothetical modeler in the year 1000 AD to have pre-dicted a stock market crash in the year 2000. These are temporal distances that preclude any possibility of placing events separated by such intervals on a single axis. One can certainly not speak of any consistency in the causes of events so distant from one another — they do not belong to a homogeneous class, they do not share the iid (independent and identically distrib-uted) properties. Nor does this interpretation follow from the mechanics of the framework — that is, from the way in which the aggregate distribution is constructed (through the combina-tion of frequency and severity distributions), from which the quantile (99.9%) is read. It is therefore not an accurate interpretation.

Once in 1,000 Possible Scenarios

Drawing on the laws of statistics, the only thing that can be said about the 99.9% quantile reading from the aggregate annual loss distribution is that once in every 1,000 such readings from the distribution, a value may occur that corresponds to the 99.9% quantile. It should be noted that temporal distances (e.g., once every 1,000 years) are not in any way embedded in the construction of the aggregate loss distribution. This distribution is constructed by combin-ing the various possibilities of annual loss counts with the corresponding various possibilities of loss amounts for those counts. The aggregate loss distribution therefore represents rather a potential space of possibilities for annual aggregate losses — that is, what the scale (disper-sion) of possible occurrences of annual loss aggregates looks like. It would therefore seem that the most adequate interpretation of the 99.9% quantile is one that says the value at this quan-tile is the second-to-last of the 1,000 worst possible scenarios — a present-day scenario, one that could plausibly materialize within the coming year.

To illustrate the differences in interpretation and their consequences, let us consider an ex-ample in which the same question is approached with different answers arising from different understandings of the quantile.

The Correct Interpretation of the Quantile and the Concept of Rarity

The reasoning presented here regarding the correct interpretation of the 99.9% quantile as one in a thousand plausible annual scenarios (here and now — that is, within the coming year) brings greater precision to the concept of rarity that follows from such reasoning. Rarity should in this case be interpreted as an exceptional year in terms of operational loss values. This atypical year is, however, placed on an axis together with other annual periods that could materialize today — not on a linear, historical timeline. Rarity is therefore understood here as the exceptionality of a given scenario of events relative to the entire spectrum of other scenari-os, where the explicitly indicated annual period serves merely as an aggregating formula for the operational risk phenomenon — it is the aggregate of all losses within a scenario, that is, losses that may occur over the course of a year.

In the AMA method, only the actuarial construction was predefined, which required sepa-rate and independent modeling of two phenomena: the frequency of operational events and the severity of a single loss. The remaining assumptions were to be formulated independently by the bank applying the AMA method, as part of the development of its operational risk model. This constituted a significant complication.

Beyond describing the properties of the processes generating the individual components of the operational risk phenomenon (the selection of frequency and severity distributions), a cru-cial question — one that requires careful conceptual consideration — is the determination of the type of unexpected loss situation:

(a) a situation in which the threat and its scale are determined by a single high-severity event,

(b) a situation in which the threat and its scale are determined by the mass/volume of events, without a single dominant extreme event.

This is a fundamental reflection on the nature of the threat against which protection is sought. Resolving this dilemma requires a deep understanding of the mechanisms generating operational risk within a bank, including knowledge of their properties in atypical situations — which is rarely identical to the expected loss situation. Such an in-depth analysis is indispensa-ble for the correct formulation of the modeled phenomenon.

The Basel framework does not, however, address this question. The grounding of the meth-od in an actuarial foundation — with separate modeling of the two phenomena (frequency, severity) and the concept of their convolution — was intended to remove this dilemma from the path. The interplay between frequency and severity phenomena was meant to resolve, at the level of each institution applying the method, the ultimate shape of this extreme, danger-ous situation. The aggregate loss distribution arising from the combination of these two phe-nomena was intended to have the shape of that situation embedded in its definition, which in turn was to follow from the relative strength of these two components.