Exercise: Kupiec POF test by hand

A bank runs a $5\%$-VaR model over $T = 500$ trading days and observes $N = 40$ exceedances.

Tasks

1. Compute the expected exceedance count under the null.

2. Compute the Kupiec POF likelihood ratio statistic.

3. At the $5\%$ significance level, does the test reject? What about at $1\%$? ($\chi^2_{1, 0.95} = 3.841$, $\chi^2_{1, 0.99} = 6.635$.)

4. Repeat with $N = 35$ and $N = 45$ to see how sensitive the rejection decision is.

5. Under the Basel traffic-light system (defined for $\alpha = 1\%$ over 250 days), the bank's scenario above doesn't fit directly. Propose an analogous table for $\alpha = 5\%$, $T = 500$: what exceedance counts would map to green, yellow, and red zones?

Hint

The asymptotic Chi-squared approximation is reliable when $N, T-N \ge 5$.