VaR Backtest (Kupiec, Christoffersen): Examples
A VaR model can pass the frequency test and still be dangerously wrong. That is what the fourth scenario demonstrates: the right breach count, but breaches bunched in time, which the Christoffersen test catches and Kupiec does not. All scenarios use 250 daily observations and a 95% VaR, expecting about 12.5 exceptions. Kupiec checks whether the observed count matches; Christoffersen checks whether breaches are independent over time. A likelihood-ratio statistic above 3.84 (the chi-squared critical value at one degree of freedom) means reject.
Worked Examples
See the inputs and outcome together
Each scenario keeps the starting point, the outcome, and the actual lesson in one place so the page reads like a decision notebook, not a data dump.
- 1
Too few breaches: model too conservative
Only 5 exceptions in 250 days when about 12.5 were expected, spread evenly through the year. The VaR is set too high.
Observed rate 2.0% vs expected 5.0%, Kupiec LR 6.07 (p = 0.014), Christoffersen LR 0, joint p = 0.048.
Observations
250 days
Confidence level
95%
Exceptions
5 (spread out)
Kupiec rejects: a 2 percent breach rate is too low for a 95 percent VaR, meaning the model overstates risk and ties up capital. Christoffersen is fine because the few breaches are independent. An over-conservative VaR fails the coverage test just as a reckless one does.
- 2
Right count, independent: the model passes
13 exceptions, close to the expected 12.5, spread evenly across the year. This is what a correctly calibrated model looks like.
Kupiec LR 0.02 (p = 0.885), Christoffersen LR 0 (p = 1.0), joint p = 0.990.
Observations
250 days
Confidence level
95%
Exceptions
13 (spread out)
Both tests pass clearly. The breach count matches expectation and the breaches do not cluster, so the joint test is satisfied. This is the only outcome that should let a VaR model into production unchanged.
- 3
Far too many breaches: model too aggressive
25 exceptions in 250 days, double the expected count, spread out. The VaR is set too low and understates risk.
Observed rate 10.0% vs expected 5.0%, Kupiec LR 10.33 (p = 0.001), joint p = 0.006.
Observations
250 days
Confidence level
95%
Exceptions
25 (spread out)
A 10 percent breach rate is twice what a 95 percent VaR should allow, and Kupiec rejects hard. This is the dangerous failure: the model is systematically understating losses, the kind regulators penalize through a higher capital multiplier.
- 4
Right count, but breaches cluster
12 exceptions, almost exactly the expected count, but bunched into two consecutive bursts rather than spread out. The hidden failure mode.
Kupiec LR 0.02 (p = 0.884, passes), but Christoffersen LR 62.3 (p ≈ 0), joint p ≈ 0.
Observations
250 days
Confidence level
95%
Exceptions
12 (two clustered bursts)
This is why you run both tests. The breach count is perfect and Kupiec passes, but the breaches cluster, so Christoffersen rejects overwhelmingly. Clustered breaches mean the model misses volatility regimes, taking all its losses at once, exactly when capital is scarcest.
Patterns
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Sources & References
- Techniques for Verifying the Accuracy of Risk Measurement Models — Kupiec, P., Journal of Derivatives (1995)
- Evaluating Interval Forecasts — Christoffersen, P. F., International Economic Review (1998)
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