Christoffersen's Conditional Coverage Test

A 99% VaR model should be breached roughly 1% of the trading days. A simple test — count the breaches and check if the rate is close to 1% — catches models with the wrong overall breach frequency, but it can be fooled: a model whose 1% of breaches all happen during one clustered crisis month is arguably far more dangerous than one whose breaches are spread evenly through the year, even though both have the identical unconditional breach rate.

Christoffersen's conditional coverage test fixes this by splitting the question in two. First, an unconditional coverage test checks whether the overall breach rate matches the target (the same idea as Kupiec's test). Second, an independence test checks whether a breach today makes a breach tomorrow more or less likely than the model's target rate would predict — it does this by looking at how breaches transition from one day to the next, essentially checking whether breach clustering exists. The two test statistics combine into a single chi-squared statistic; failing either component fails the overall test.

For example, a VaR model that breaches on exactly 5 of 250 days (2%, close to a 1% target after accounting for sampling noise) might still fail conditional coverage if all 5 breaches land in the same consecutive five-day stretch — the unconditional rate passes, but the independence component flags the clustering as evidence the model understates risk precisely when it matters most.

Regulators and risk desks favor conditional coverage over a simple breach count specifically because clustered VaR failures are the ones that bankrupt a book.