It is with the statistical limitations of backtesting in mind that the Committee is introducing a framework for the supervisory interpretation of backtesting results that encompasses a range of possible responses, depending on the strength of the signal generated from the backtest. These responses are classified into three zones, distinguished by colours into a hierarchy of responses. The green zone corresponds to backtesting results that do not themselves suggest a problem with the quality or accuracy of a bank's model. The yellow zone encompasses results that do raise questions in this regard, but where such a conclusion is not definitive. The red zone indicates a backtesting result that almost certainly indicates a problem with a bank's risk model.
The Committee has agreed to standards regarding the definitions of these zones in respect of the number of exceptions generated in the backtesting program, and these are set forth below. To place these definitions in proper perspective, however, it is useful to examine the probabilities of obtaining various numbers of exceptions under different assumptions about the accuracy of a bank's risk measurement model.
(b) Statistical considerations in defining the zones
Three zones have been delineated and their boundaries chosen in order to balance two types of statistical error: (1) the possibility that an accurate risk model would be classified as inaccurate on the basis of its backtesting result, and (2) the possibility that an inaccurate model would not be classified that way based on its backtesting result.
Table 1 reports the probabilities of obtaining a particular number of exceptions from a sample of 250 independent observations under several assumptions about the actual percentage of outcomes that the model captures (that is, these are binomial probabilities). For example, the left-hand portion of Table 1 reports probabilities associated with an accurate model (that is, a true coverage level of 99%). Under these assumptions, the column labelled "exact" reports that exactly five exceptions can be expected in 6.7% of the samples.
The right-hand portion of Table 1 reports probabilities associated with several possible inaccurate models, namely models whose true levels of coverage are 98%, 97%, 96%, and 95%, respectively. Thus, the column labelled "exact" under an assumed coverage level of 97% shows that five exceptions would then be expected in 10.9% of the samples.
Table 1 also reports several important error probabilities. For the assumption that the model covers 99% of outcomes (the desired level of coverage), the table reports the probability that selecting a given number of exceptions as a threshold for rejecting the accuracy of the model will result in an erroneous rejection of an accurate model ("type 1" error). For example, if the threshold is set as low as one exception, then accurate models will be rejected fully 91.9% of the time, because they will escape rejection only in the 8.1% of cases where they generate zero exceptions. As the threshold number of exceptions is increased, the probability of making this type of error declines.
Under the assumptions that the model's true level of coverage is not 99%, Table 1 reports the probability that selecting a given number of exceptions as a threshold for rejecting the accuracy of the model will result in an erroneous acceptance of a model with the assumed (inaccurate) level of coverage ("type 2" error). For example, if the model's actual level of coverage is 97%, and the threshold for rejection is set at seven or more exceptions, the table indicates that this model would be erroneously accepted 37.5% of the time.
In interpreting the information in Table 1, it is also important to understand that although the alternative models appear close to the desired standard in probability terms (97% is close to 99%), the difference between these models in terms of the size of the risk measures generated can be substantial. That is, a bank's risk measure could be substantially less than that of an accurate model and still cover 97% of the trading outcomes. For example, in the case of normally distributed trading outcomes, the 97th percentile corresponds to 1.88 standard deviations, while the 99th percentile corresponds to 2.33 standard deviations, an increase of nearly 25%. Thus, the supervisory desire to distinguish between models providing 99% coverage, and those providing say, 97% coverage, is a very real one.
(c) Definition of the green, yellow, and red zones
The results in Table 1 also demonstrate some of the statistical limitations of backtesting. In particular, there is no threshold number of exceptions that yields both a low probability of erroneously rejecting an accurate model and a low probability of erroneously accepting all of the relevant inaccurate models. It is for this reason that the Committee has rejected an approach that contains only a single threshold.
Given these limitations, the Committee has classified outcomes into three categories. In the first category, the test results are consistent with an accurate model, and the possibility of erroneously accepting an inaccurate model is low (green zone). At the other extreme, the test results are extremely unlikely to have resulted from an accurate model, and the probability of erroneously rejecting an accurate model on this basis is remote (red zone). In between these two cases, however, is a zone where the backtesting results could be consistent with either accurate or inaccurate models, and the supervisor should encourage a bank to present additional information about its model before taking action (yellow zone).
Table 2 sets out the Committee's agreed boundaries for these zones and the presumptive supervisory response for each backtesting outcome, based on a sample of 250 observations. For other sample sizes, the boundaries should be deduced by calculating the binomial probabilities associated with true coverage of 99%, as in Table 1. The yellow zone begins at the point such that the probability of obtaining that number or fewer exceptions equals or exceeds 95%. Table 2 reports these cumulative probabilities for each number of exceptions. For 250 observations, it can be seen that five or fewer exceptions will be obtained 95.88% of the time when the true level of coverage is 99%. Thus, the yellow zone begins at five exceptions.
Similarly, the beginning of the red zone is defined as the point such that the probability of obtaining that number or fewer exceptions equals or exceeds 99.99%. Table 2 shows that for a sample of 250 observations and a true coverage level of 99%, this occurs with ten exceptions.
(d) The green zone
The green zone needs little explanation. Since a model that truly provides 99% coverage would be quite likely to produce as many as four exceptions in a sample of 250 outcomes, there is little reason for concern raised by backtesting results that fall in this range. This is reinforced by the results in Table 1, which indicate that accepting outcomes in this range leads to only a small chance of erroneously accepting an inaccurate model.
(e) The yellow zone
The range from five to nine exceptions constitutes the yellow zone. Outcomes in this range are plausible for both accurate and inaccurate models, although Table 1 suggests that they are generally more likely for inaccurate models than for accurate models. Moreover, the results in Table 1 indicate that the presumption that the model is inaccurate should grow as the number of exceptions increases in the range from five to nine.
The Committee has agreed that, within the yellow zone, the number of exceptions should generally guide the size of potential supervisory increases in a firm's capital requirement. Table 2 sets out the Committee's agreed guidelines for increases in the multiplication factor applicable to the internal models capital requirement, resulting from backtesting results in the yellow zone.
These guidelines help in maintaining the appropriate structure of incentives applicable to the internal models approach. In particular, the potential supervisory penalty increases with the number of exceptions. The results in Table 1 generally support the notion that nine exceptions is a more troubling result than five exceptions, and these steps are meant to reflect that.
These particular values reflect the general idea that the increase in the multiplication factor should be sufficient to return the model to a 99th percentile standard. For example, five exceptions in a sample of 250 implies only 98% coverage. Thus, the increase in the multiplication factor should be sufficient to transform a model with 98% coverage into one with 99% coverage. Needless to say, precise calculations of this sort require additional statistical assumptions that are not likely to hold in all cases. For example, if the distribution of trading outcomes is assumed to be normal, then the ratio of the 99th percentile to the 98th percentile is approximately 1.14, and the increase needed in the multiplication factor is therefore approximately 0.40 for a scaling factor of 3. If the actual distribution is not normal, but instead has "fat tails", then larger increases may be required to reach the 99th percentile standard. The concern about fat tails was also an important factor in the choice of the specific increments set out in Table 2.
It is important to stress, however, that these increases are not meant to be purely automatic. The results in Table 1 indicate that results in the yellow zone do not always imply an inaccurate model, and the Committee has no interest in penalising banks solely for bad luck. Nevertheless, to keep the incentives aligned properly, backtesting results in the yellow zone should generally be presumed to imply an increase in the multiplication factor unless the bank can demonstrate that such an increase is not warranted.
In other words, the burden of proof in these situations should not be on the supervisor to prove that a problem exists, but rather should be on the bank to prove that their model is fundamentally sound. In such a situation, there are many different types of additional information that might be relevant to an assessment of the bank's model.
For example, it would then be particularly valuable to see the results of backtests covering disaggregated subsets of the bank's overall trading activities. Many banks that engage in regular backtesting programs break up their overall trading portfolio into trading units organised around risk factors or product categories. Disaggregating in this fashion could allow the tracking of a problem that surfaced at the aggregate level back to its source at the level of a specific trading unit or risk model.
Banks should also document all of the exceptions generated from their ongoing backtesting program, including an explanation for the exception. This documentation is important to determining an appropriate supervisory response to a backtesting result in the yellow zone. Banks may also implement backtesting for confidence intervals other than the 99th percentile, or may perform other statistical tests not considered here. Naturally, this information could also prove very helpful in assessing their model.
In practice, there are several possible explanations for a backtesting exception, some of which go to the basic integrity of the model, some of which suggest an under-specified or low-quality model, and some of which suggest either bad luck or poor intra-day trading results. Classifying the exceptions generated by a bank's model into these categories can be a very useful exercise.
Basic integrity of the model
1) The bank's systems simply are not capturing the risk of the positions themselves (e.g., the positions of an overseas office are being reported incorrectly).
2) Model volatilities and/or correlations were calculated incorrectly (e.g., the computer is dividing by 250 when it should be dividing by 225).
Model's accuracy could be improved
3) The risk measurement model is not assessing the risk of some instruments with sufficient precision (e.g., too few maturity buckets or an omitted spread).
Bad luck or markets moved in fashion unanticipated by the model
4) Random chance (a very low probability event).
5) Markets moved by more than the model predicted was likely (i.e., volatility was significantly higher than expected).
6) Markets did not move together as expected (i.e., correlations were significantly different than what was assumed by the model).
7) There was a large (and money-losing) change in the bank's positions or some other income event between the end of the first day (when the risk estimate was calculated) and the end of the second day (when trading results were tabulated).
In general, problems relating to the basic integrity of the risk measurement model are potentially the most serious. If there are exceptions attributed to this category for a particular trading unit, the plus should apply. In addition, the model may be in need of substantial review and/or adjustment, and the supervisor would be expected to take appropriate action to ensure that this occurs.
The second category of problem (lack of model precision) is one that can be expected to occur at least part of the time with most risk measurement models. No model can hope to achieve infinite precision, and thus all models involve some amount of approximation. If, however, a particular bank's model appears more prone to this type of problem than others, the supervisor should impose the plus factor and also consider what other incentives are needed to spur improvements.
The third category of problems (markets moved in a fashion unanticipated by the model) should also be expected to occur at least some of the time with value-at-risk models. In particular, even an accurate model is not expected to cover 100% of trading outcomes. Some exceptions are surely the random 1% that the model can be expected not to cover. In other cases, the behaviour of the markets may shift so that previous estimates of volatility and correlation are less appropriate. No value-at-risk model will be immune from this type of problem; it is inherent in the reliance on past market behaviour as a means of gauging the risk of future market movements.
Finally, depending on the definition of trading outcomes employed for the purpose of backtesting, exceptions could also be generated by intra-day trading results or an unusual event in trading income other than from positioning. Although exceptions for these reasons would not necessarily suggest a problem with the bank's value-at-risk model, they could still be cause for supervisory concern and the imposition of the plus should be considered.
The extent to which a trading outcome exceeds the risk measure is another relevant piece of information. All else equal, exceptions generated by trading outcomes far in excess of the risk measure are a matter of greater concern than are outcomes only slightly larger than the risk measure.
In deciding whether or not to apply increases in a bank's capital requirement, it is envisioned that the supervisor could weigh these factors as well as others, including an appraisal of the bank's compliance with applicable qualitative standards of risk management. Based on the additional information provided by the bank, the supervisor will decide on the appropriate course of action.
In general, the imposition of a higher capital requirement for outcomes in the yellow zone is an appropriate response when the supervisor believes the reason for being in the yellow zone is a correctable problem in a bank's model. This can be contrasted with the case of an unexpected bout of high market volatility, which nearly all models may fail to predict. While these episodes may be stressful, they do not necessarily indicate that a bank's risk model is in need of redesign. Finally, in the case of severe problems with the basic integrity of the model, the supervisor should consider whether to disallow the use of the model for capital purposes altogether.
(f) The red zone
Finally, in contrast to the yellow zone where the supervisor may exercise judgement in interpreting the backtesting results, outcomes in the red zone (ten or more exceptions) should generally lead to an automatic presumption that a problem exists with a bank's model. This is because it is extremely unlikely that an accurate model would independently generate ten or more exceptions from a sample of 250 trading outcomes.
In general, therefore, if a bank's model falls into the red zone, the supervisor should automatically increase the multiplication factor applicable to a firm's model by one (from three to four). Needless to say, the supervisor should also begin investigating the reasons why the bank's model produced such a large number of misses, and should require the bank to begin work on improving its model immediately.
Although ten exceptions is a very high number for 250 observations, there will on very rare occasions be a valid reason why an accurate model will produce so many exceptions. In particular, when financial markets are subjected to a major regime shift, many volatilities and correlations can be expected to shift as well, perhaps substantially. Unless a bank is prepared to update its volatility and correlation estimates instantaneously, such a regime shift could generate a number of exceptions in a short period of time. In essence, however, these exceptions would all be occurring for the same reason, and therefore the appropriate supervisory reaction might not be the same as if there were ten exceptions, but each from a separate incident. For example, one possible supervisory response in this instance would be to simply require the bank's model to take account of the regime shift as quickly as it can while maintaining the integrity of its procedures for updating the model.
It should be stressed, however, that the Committee believes that this exception should be allowed only under the most extraordinary circumstances, and that it is committed to an automatic and non-discretionary increase in a bank's capital requirement for backtesting results that fall into the red zone.