Choosing the Kupiec Test Level for VaR Backtesting

Summary

The document addresses which probability level to use when applying the Kupiec test to historical-simulation Value at Risk forecasts. Its answer is to match the test’s expected exception probability to the VaR tail probability being evaluated. Thus, the backtest should use the exceedance rate associated with that VaR estimate, rather than an unrelated test significance level. The test compares observed exceptions in paired profit-and-loss and VaR observations with the number expected under the selected VaR confidence level.

The question describes evaluating VaR estimates at two tail probabilities and reports that using a mismatched level leads to rejection across the stocks considered. The response clarifies the parameter choice but does not analyze those data or establish whether the rejections indicate poor VaR calibration. It also does not discuss sample size, test power, dependence across daily exceptions, or complementary conditional coverage diagnostics, all of which can matter when interpreting backtest results.

Key ideas

• The Kupiec test’s expected exception probability should match the tail probability of the VaR forecast under evaluation.
• Observed exceptions are assessed against the rate implied by the selected VaR confidence level.
• Using a mismatched probability can make the backtest comparison invalid.
• A test rejection alone does not explain why VaR forecasts failed or whether exceptions cluster over time.

Tags

# Backtesting Value at Risk. With kupiec test


# Backtesting Value at Risk. With kupiec test












I have a certain problem with backtesting calculated earlier Value at Risk.

I've got calculated daily VaR with historical simulation method for stocks. I've used two values of alpha 0.05 and 0.1. Now, I want to backtest this values with Kupiec test. What alpha should I use for Kupiec test? When I'm using 0.05 alpha for VaR calculated with 0.1, I have to reject null hypothesis for all stocks. Is it normal? Or I should use 0.1 alpha for Kupiec test in this case for comparison between these two models.

## Answer by AK88 (score 1, accepted)

https://quant.stackexchange.com/a/34649

When backtesting VaR results using Kupiec test you should choose the respective significance levels as your VaRs. Thus, using 0.05 alpha for VaR with 99% confidence level is not correct. From Dowd:

> To implement the Kupiec test, we require data on n, p and x. The first are easily found from the sample size and VaR confidence level, and we can derive x from a set of paired observations of P/L and VaR each period.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.