Ashvel Capital's 99% one-day VaR model was breached on 7 of the last 250 days, against 2.5 expected. How unlikely is that if the model is right, and what do you change?

A correct 99% model would produce seven or more breaches only about 1.4% of the time, so the model understates Ashvel's tail risk. The expected count is 2.5. A likelihood-ratio test gives 5.5, above the 3.84 needed to reject at 5%. Four breaches in three weeks also says the model reacts too slowly to volatility. Change the volatility estimate to weight recent days, drop the normal assumption and add expected shortfall and a stressed window.

Treat each day as an independent 1% event, like a die with a hundred faces rolled 250 times, counting the ones. Two or three is normal; seven or more happens with a chance of about 1.4%. The count follows a binomial distributionThe number of successes in a fixed number of independent tries, each with the same chance. Here, the number of breach days in 250 days at 1% a day., with 2.5 expected. Five or more already happens only 10.8% of the time; ten or more is about 0.03%.

No, and the clustering is the more useful clue. Four breaches inside three weeks means the model's volatility estimate lagged the market: it was still using calm history when the market had already turned. A correct model's breaches should arrive scattered, not in bunches. Formal tests check both: the Kupiec test on the count, where 5.5 against a threshold of 3.84 rejects the model, and an independence test, such as Christoffersen's, on whether one breach makes the next more likely. Regulatory frameworks for banks also grade backtests by breach count, for example a traffic-light scheme where a handful of breaches in a year calls for review; confirm the current thresholds before quoting them.

Fix the two faults the evidence points to. First, the lag: weight recent days more heavily in the volatility estimate, so the model widens within days of a regime change rather than months. Second, the shape: replace the normal assumption with historical or fat-tailed returns, because a normal curve puts too little weight on large losses. Then add what VaR does not say: expected shortfall, the average loss on the days beyond VaR, and a stressed VaR computed on the worst year in the history, and set limits on those too.

Before any of that, check the plumbing. Breaches can come from comparing VaR with P&L that includes fees, new trades made during the day or a pricing error. Backtest against the P&L the positions held that morning would have made, and only then conclude the model is wrong. Seven real breaches still mean the book took more risk than the limits assumed, and the PMs' limits were set on a number that was too small.