The traffic light is a binomial table with two probability cut-offs. Built by hand, it also shows how weak a count test is at catching a model that is merely wrong.
For a 99 per cent one-day value at risk tested on 250 days, ten exceptions put the model in the red zone: a correct model produces that many only 0.03 per cent of the time. Green runs from 0 to 4, amber (yellow in the Basel text) from 5 to 9. On 500 days the boundaries are not doubled: green ends at 8 and red starts at 15.
Worked in full in Financial Risk Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
An exception is a day on which the realised loss exceeds the VaR reported the evening before. If the model is right, each day is an independent coin with a 1 per cent chance of landing on an exception, so the count over n days follows a binomial distribution. The zones are drawn on that distribution with the two conventional cut-offs. The worked example is Halverton, the credit fund in Financial Risk Management, which records 6 exceptions in 500 days. Figures are illustrative.
| Input | Value |
|---|---|
| VaR confidence | 99% |
| Exception probability per day | 0.01 |
| Green zone ends below a cumulative probability of | 95% |
| Red zone starts at a cumulative probability of | 99.99% |
| Observations | 250 or 500 |
| Halverton, exceptions observed in 500 days | 6 |
For each possible count k, compute the probability that a correct model shows k or fewer exceptions. Green is every count whose cumulative probability is still below 95 per cent; red is every count from the point where it reaches 99.99 per cent; amber is between.
P(X ≤ k) = Σi=0..k C(n, i) × 0.01i × 0.99n−i
Excel: =BINOM.DIST(k, 250, 0.01, TRUE)
Green while P(X ≤ k) < 95%; red once P(X ≤ k) ≥ 99.99%
| Exceptions | Probability of exactly k | Cumulative | Zone |
|---|---|---|---|
| 0 | 8.11% | 8.11% | Green |
| 2 | 25.74% | 54.32% | Green |
| 4 | 13.41% | 89.22% | Green |
| 5 | 6.66% | 95.88% | Amber |
| 7 | 0.97% | 99.60% | Amber |
| 9 | 0.08% | 99.975% | Amber |
| 10 | 0.02% | 99.995% | Red |
Four exceptions sit at 89.22 per cent cumulative, still below 95, so they are green. Five take the cumulative to 95.88 per cent and open the amber zone. Nine leave it at 99.975 per cent, just short of 99.99; ten reach 99.995 per cent, and the model is red. The published table on 250 days, green to 4 and amber to 9, is exactly this construction.
Run the same formulas with n = 500. The expected count doubles to 5.0, but the spread of a binomial grows with the square root of n, not with n, so the boundaries move less than twice as far.
| Observations | Expected | Green | Amber | Red from |
|---|---|---|---|---|
| 250 | 2.5 | 0 to 4 | 5 to 9 | 10 |
| 500 | 5.0 | 0 to 8 | 9 to 14 | 15 |
Eight exceptions in 500 days sit at 93.29 per cent cumulative and are green; nine reach 96.89 per cent and are amber. Scaling the 250-day table in proportion would start amber at 10 and leave nine exceptions green, the error the companion workbook now corrects. Halverton's 6 in 500 is comfortably green: a correct model shows at least six 38.4 per cent of the time. The Kupiec likelihood ratio on the same count is 0.190, a p-value of 0.66.
Under the Basel 2.5 internal models approach the zone sets the multiplier applied to the 60-day average VaR and stressed VaR: a base of 3 plus a factor that rises through amber. Under the revised framework (FRTB) the same 0 to 4, 5 to 9 and 10 or more zones apply to the bank-wide 99 per cent VaR backtest, but the multiplier starts at 1.5 and its add-on reaches 0.50 in the red zone.
| Exceptions | Plus factor | Multiplier | Capital uplift |
|---|---|---|---|
| 5 | 0.40 | 3.40 | 13.3% |
| 6 | 0.50 | 3.50 | 16.7% |
| 7 | 0.65 | 3.65 | 21.7% |
| 8 | 0.75 | 3.75 | 25.0% |
| 9 | 0.85 | 3.85 | 28.3% |
| 10 or more | 1.00 | 4.00 | 33.3% |
Now turn the question round: how good is the light at catching a model that is wrong? Suppose the true exception rate is 2 per cent, so the reported VaR is really a 98 per cent figure. Over 250 days it lands green 43.9 per cent of the time and red only 3.0 per cent. At a true rate of 3 per cent it is still green 12.8 per cent of the time. One year of data separates a good model from a mediocre one poorly. The zones were designed to keep false alarms rare, and a correct model reaches amber or worse 10.78 per cent of the time; the price of that design is low power.
A count says nothing about timing. Six exceptions spread over two years and five of them in a single week produce the same count and the same green light. Report the largest number of exceptions in any five-day window beside the total, and run an independence test when the count alone looks fine.
The first mistake is reading zero exceptions as success. A correct 99 per cent model shows none in a year only 8.11 per cent of the time, and the Kupiec test rejects zero on 250 days with a p-value of 0.025: a model that never breaches is usually too conservative, which costs capital or risk budget every day. The second is scaling the 250-day table to a longer sample by proportion instead of rebuilding it from the binomial. The third is backtesting a VaR against a P&L that includes fees, carry or intraday trading, so that exceptions are counted against a number the model never tried to predict.
The one-day figure being tested here is built in how to scale a one-day VaR to ten days, and the site's VaR and expected shortfall model carries the same exception counter.
The zones are drawn by the binomial, not typed, in the free workbook for this case, so any sample length and confidence level can be read off directly.
About 2.5 in 250 trading days, since a 99 per cent measure promises a loss beyond it one day in a hundred. Zero is not a good result: a correct model shows no exception in a year only 8.11 per cent of the time, and the Kupiec test rejects a count of zero at the 5 per cent level, with a p-value of 0.025.
Under Basel 2.5, on 250 days the multiplier of 3 rises by 0.40 at five exceptions, 0.50 at six, 0.65 at seven, 0.75 at eight and 0.85 at nine, then by 1.00 in the red zone from ten. Seven exceptions lift it to 3.65, raising the multiplied VaR charge by 21.7 per cent. FRTB keeps the zones but starts from 1.5.
A likelihood ratio comparing the observed exception rate with the promised one, read against a chi-squared distribution with one degree of freedom. Six exceptions in 500 days give a statistic of 0.190 and a p-value of 0.66, so the count is consistent with a 99 per cent model. The test ignores when the exceptions happened.
The exception counter and its zones are worked in the value at risk and backtest workbook for chapters 4, 6 and 7 of Financial Risk Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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