Block Maxima and the GEV Distribution

If you want to know how bad the worst day of a typical year could get, fitting a normal distribution to all daily returns and reading off a tail quantile badly understates the risk — normal tails are too thin for market crashes. Block maxima takes a different approach: chop the return history into blocks (say, one calendar year each), keep only the single worst loss from each block, and throw everything else away. What's left is a much smaller sample of "worst-of-year" numbers, and extreme value theory says that, under mild conditions, those maxima converge to one specific family regardless of what the underlying daily distribution looks like: the Generalized Extreme Value (GEV) distribution,

$$G(x) = \exp\!\left\{ -\left[1 + \xi \left(\frac{x-\mu}{\sigma}\right)\right]^{-1/\xi} \right\} .$$

In plain English: $\mu$ and $\sigma$ locate and scale the typical worst-of-block loss, and the shape parameter $\xi$ controls how heavy the tail of those maxima is — $\xi > 0$ means fat tails (crash risk that doesn't taper off quickly), $\xi = 0$ is the thin-tailed Gumbel case, and $\xi < 0$ means a bounded worst case.

Worked example

Ten years of daily losses give ten annual worst-day losses: roughly 2%, 2.5%, 3%, 2%, 2.2%, 8% (a crisis year), 2.8%, 2.1%, 3.5%, 2.4%. Fitting a GEV to just these ten numbers (rather than all ~2,500 daily returns) estimates a shape parameter $\xi$ noticeably above zero, because the 8% year sits far out relative to the others — signaling that the true worst-case tail is fatter than the other nine "normal" years would suggest on their own.

Only using the block maximum discards a lot of data, which is block maxima's main cost: nine other loss numbers per block are thrown away even if several were nearly as extreme as the block's official maximum.