Why the bootstrap fails on an extreme tail quantile

Question

A risk team wants a confidence interval for the 99.9% Value-at-Risk of a portfolio, an extreme lower-tail quantile, from a few thousand daily returns. They bootstrap: resample the returns with replacement, recompute the $99.9\%$ VaR each time, and take the percentile interval. It comes out surprisingly narrow.

Explain why the ordinary bootstrap is unreliable for this statistic and what resampling or modeling approaches are more trustworthy.

Accepted Answer

The core mistake: the bootstrap struggles with extreme order statistics. The 99.9\% VaR from a few thousand points depends on a mere handful of the most extreme observations. The nonparametric bootstrap can only ever resample the values it has already seen: it cannot produce a loss worse than the worst observed loss, and the deep-tail quantile it computes just hops among the same few extreme data points. So the bootstrap distribution of the tail VaR is lumpy and artificially narrow; it captures resampling variability among the observed extremes but not the true uncertainty about losses beyond…

Why the bootstrap fails on an extreme tail quantile

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