Statistical considerations in defining the backtesting zones
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| 16.9 | To place the definitions of three zones of the bank-wide backtesting 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.
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| 16.10 | Three zones have been delineated and their boundaries chosen in order to balance two types of statistical error:
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| | (1) | the possibility that an accurate risk model would be classified as inaccurate on the basis of its backtesting result, and
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| | (2) | the possibility that an inaccurate model would not be classified that way based on its backtesting result.
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| 16.11 | 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 (ie these are binomial probabilities). For example, the left- hand portion of Table 1 sets out 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.
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| Probabilities of exceptions from 250 independent observations | Table 1 | | Model is accurate | Model is inaccurate: possible alternative levels of coverage | | | Coverage = 99% | Coverage = 98% | Coverage = 97% | Coverage = 96% | Coverage = 95% | | Exact | Type 1 | Exact | Type 2 | Exact | Type 2 | Exact | Type 2 | Exact | Type 2 | | 0 | 8.1% | 100.0% | 0.6% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | | 1 | 20.5% | 91.9% | 3.3% | 0.6% | 0.4% | 0.0% | 0.0% | 0.0% | 0.0% | 0.0% | | 2 | 25.7% | 71.4% | 8.3% | 3.9% | 1.5% | 0.4% | 0.2% | 0.0% | 0.0% | 0.0% | | 3 | 21.5% | 45.7% | 14.0% | 12.2% | 3.8% | 1.9% | 0.7% | 0.2% | 0.1% | 0.0% | | 4 | 13.4% | 24.2% | 17.7% | 26.2% | 7.2% | 5.7% | 1.8% | 0.9% | 0.3% | 0.1% | | 5 | 6.7% | 10.8% | 17.7% | 43.9% | 10.9% | 12.8% | 3.6% | 2.7% | 0.9% | 0.5% | | 6 | 2.7% | 4.1% | 14.8% | 61.6% | 13.8% | 23.7% | 6.2% | 6.3% | 1.8% | 1.3% | | 7 | 1.0% | 1.4% | 10.5% | 76.4% | 14.9% | 37.5% | 9.0% | 12.5% | 3.4% | 3.1% | | 8 | 0.3% | 0.4% | 6.5% | 86.9% | 14.0% | 52.4% | 11.3% | 21.5% | 5.4% | 6.5% | | 9 | 0.1% | 0.1% | 3.6% | 93.4% | 11.6% | 66.3% | 12.7% | 32.8% | 7.6% | 11.9% | | 10 | 0.0% | 0.0% | 1.8% | 97.0% | 8.6% | 77.9% | 12.8% | 45.5% | 9.6% | 19.5% | | 11 | 0.0% | 0.0% | 0.8% | 98.7% | 5.8% | 86.6% | 11.6% | 58.3% | 11.1% | 29.1% | | 12 | 0.0% | 0.0% | 0.3% | 99.5% | 3.6% | 92.4% | 9.6% | 69.9% | 11.6% | 40.2% | | 13 | 0.0% | 0.0% | 0.1% | 99.8% | 2.0% | 96.0% | 7.3% | 79.5% | 11.2% | 51.8% | | 14 | 0.0% | 0.0% | 0.0% | 99.9% | 1.1% | 98.0% | 5.2% | 86.9% | 10.0% | 62.9% | | 15 | 0.0% | 0.0% | 0.0% | 100.0% | 0.5% | 99.1% | 3.4% | 92.1% | 8.2% | 72.9% |
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Notes to Table 1:The table reports both exact probabilities of obtaining a certain number of exceptions from a sample of 250 independent observations under several assumptions about the true level of coverage, as well as type 1 or type 2 error probabilities derived from these exact probabilities.
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The left-hand portion of the table pertains to the case where the model is accurate and its true level of coverage is 99%. Thus, the probability of any given observation being an exception is 1% (100% – 99% = 1%). The column labelled "exact" reports the probability of obtaining exactly the number of exceptions shown under this assumption in a sample of 250 independent observations. The column labelled "type 1" reports the probability that using a given number of exceptions as the cut-off for rejecting a model will imply erroneous rejection of an accurate model using a sample of 250 independent observations. For example, if the cut-off level is set at five or more exceptions, the type 1 column reports the probability of falsely rejecting an accurate model with 250 independent observations is 10.8%.
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The right-hand portion of the table pertains to models that are inaccurate. In particular, the table concentrates of four specific inaccurate models, namely models whose true levels of coverage are 98%, 97%, 96% and 95% respectively. For each inaccurate model, the exact column reports the probability of obtaining exactly the number of exceptions shown under this assumption in a sample of 250 independent observations. The type 2 columns report the probability that using a given number of exceptions as the cut-off for rejecting a model will imply erroneous acceptance of an inaccurate model with the assumed level of coverage using a sample of 250 independent observations. For example, if the cut-off level is set at five or more exceptions, the type 2 column for an assumed coverage level of 97% reports the probability of falsely accepting a model with only 97% coverage with 250 independent observations is 12.8%.
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| 16.12 | The right-hand portion of the table 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.
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| 16.13 | 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.
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| 16.14 | Under the assumptions that the model’s true level of coverage is not 99%, 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 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.
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| 16.15 | 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.
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| 16.16 | Given these limitations, the Committee has classified outcomes for the backtesting of the bank- wide model 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 (ie backtesting ”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 (ie backtesting ”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 SAMA encourage a bank to present additional information about its model before taking action (ie backtesting ”amber zone”).
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| 16.17 | Table 2 sets out the Committee’s agreed boundaries for these zones and the presumptive SAMA 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 backtesting amber 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 backtesting amber zone begins at five exceptions. Similarly, the beginning of the backtesting 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 10 exceptions.
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| Backtesting zone boundaries | Table 2 | | Backtesting zone | Number of exceptions | Backtesting-dependent multiplier (to be added to any qualitative add- on per [MAR 33.44]) | Cumulative probability | | Green | 0 | 1.50 | 8.11% | | | 1 | 1.50 | 28.58% | | | 2 | 1.50 | 54.32% | | | 3 | 1.50 | 75.81% | | | 4 | 1.50 | 89.22% | | Amber | 5 | 1.70 | 95.88% | | | 6 | 1.76 | 98.63% | | | 7 | 1.83 | 99.60% | | | 8 | 1.88 | 99.89% | | | 9 | 1.92 | 99.97% | | Red | 10 or more | 2.00 | 99.99% |
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Notes to Table 2: The table defines the backtesting green, amber and red zones that SAMA will use to assess backtesting results in conjunction with the internal models approach to market risk capital requirements. The boundaries shown in the table are based on a sample of 250 observations. For other sample sizes, the amber zone begins at the point where the cumulative probability equals or exceeds 95%, and the red zone begins at the point where the cumulative probability equals or exceeds 99.99%.
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The cumulative probability is simply the probability of obtaining a given number or fewer exceptions in a sample of 250 observations when the true coverage level is 99%. For example, the cumulative probability shown for four exceptions is the probability of obtaining between zero and four exceptions.
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Note that these cumulative probabilities and the type 1 error probabilities reported in Table 1 do not sum to one because the cumulative probability for a given number of exceptions includes the possibility of obtaining exactly that number of exceptions, as does the type 1 error probability. Thus, the sum of these two probabilities exceeds one by the amount of the probability of obtaining exactly that number of exceptions.
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| 16.18 | The backtesting 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.
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| 16.19 | The range from five to nine exceptions constitutes the backtesting amber 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.
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| 16.20 | 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 backtesting amber zone.
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| 16.21 | 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 imply 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 1.13 for a multiplier of 1. 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.
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