A trading desk's 99% one-day VaR breached eight times last year on 250 trading days. The model claimed 2.5 expected breaches. The supervisor wants to know whether the model is broken or you got unlucky. Kupiec gives one answer, Christoffersen gives another, and Basel's traffic light tells you what your capital surcharge will be. Three frameworks; the exam tests all three.
A VaR model that nobody checks is a number with no meaning. Regulators require backtesting because risk models can drift away from reality silently: vol regimes shift, correlations move, fat tails appear. The exception-counting framework is the simplest possible check: a 99% one-day VaR should be exceeded 1% of the time. Count the exceptions; compare to expectation.
KEY: Backtesting gives you statistical evidence about whether a VaR model deserves to be trusted with capital decisions. Without it, the VaR number is hope, not measurement.
Kupiec's proportion-of-failures (POF) test is the standard exception-count check. The null hypothesis: the breach rate equals the model's stated p (e.g., 1% for a 99% VaR).
Common mistakes
- Using Basel zones as Kupiec cutoffs. The legacy traffic light uses green 0-4, yellow 5-9 and red 10 or more. The stated asymptotic 5% Kupiec test at 99% over 250 days accepts 1-6; zero or at least seven rejects. Always identify the requested test and confidence level.
- Treating Kupiec as sufficient. Counts alone miss clustering. Five consecutive exceptions can pass Kupiec on 250 days while producing evidence against independence. Calculate the independence statistic from transition counts rather than treating any cluster as automatic rejection.
- Double-counting the plus factor. In the legacy traffic light, the backtesting plus factor is the traffic-light add-on. Add it once to the base multiplier 3.0. Seven exceptions gives 3.65, not 0.65 or 4.30.
Bottom line
- Backtesting compares realized exceptions (breaches) to a VaR model's stated breach rate. Kupiec POF tests count; Christoffersen tests independence and conditional coverage.
- Type I error rejects a good model; Type II error fails to reject a bad model. The significance level sets the nominal Type I rate; small samples reduce power.
- Basel traffic light: green ≤4 / yellow 5-9 / red ≥10 breaches per 250 days at 99%. Multiplier rises through yellow and to 4.0 in red, with supervisor intervention.
- VaR mapping decomposes positions into primitive risk factors so a portfolio VaR is computed from a manageable factor covariance matrix.
Exam shortcut
When a question gives a breach count and asks for action, the Basel traffic light decides: ≤4 green, 5-9 yellow, 10+ red. If the question asks whether a model is rejected, run Kupiec and compare LR to 3.84 (1 df). When mapping a bond, the cash-flow approach is the most defensible answer; principal mapping is an exam trap that ignores key-rate risk.
The full lesson (about 3,321 words, 22 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
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