Skip to main content
AI Fin Hub
Menu

Worked examples, in Risk & Portfolio Construction

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.

4 examples. Published May 26, 2026. By AI Fin Hub Research.

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. 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. 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. 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. 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

Kupiec rejects both too-few and too-many breaches; an over-conservative VaR fails just like a reckless one.
The chi-squared critical value near 3.84 for one degree of freedom is the line between pass and reject.
The right breach count can still fail Christoffersen if the breaches cluster in time.
Clustered breaches are the worst outcome operationally: the model takes its losses all at once during a volatility regime it failed to see.

Run the numbers next

  • Calculators Risk-Adjusted Returns Calculator

    Paste a returns CSV. Sharpe, Sortino, Calmar, Omega, alpha, beta, tracking error, information ratio, max drawdown, and tail moments — plus.

  • Calculators Returns Distribution Analyzer

    Paste a returns CSV. Histogram, normal QQ plot, skewness, excess kurtosis, Jarque-Bera test, tail-weight index. See why Sharpe alone misleads.

Sources

  1. Techniques for Verifying the Accuracy of Risk Measurement Models , Kupiec, P., Journal of Derivatives (1995)
  2. Evaluating Interval Forecasts , Christoffersen, P. F., International Economic Review (1998)

Keep the topic connected

  • Learn Value at Risk (VaR)

    Value at Risk: the loss threshold you'll exceed with probability α. Why historical VaR is brittle and what it doesn't tell you about the tail.

  • Learn Expected Shortfall (CVaR)

    Expected shortfall: the average loss given a VaR breach. Why regulators are migrating from VaR and what ES catches that VaR misses.

  • Learn Volatility

    Volatility as the standard deviation of returns: realized vs implied, the annualization gotcha, and why volatility-of-volatility matters.