Extreme Value Theory Kit

The worst drawdown or the worst single-day loss seen so far is one observed point, not a bound. It says nothing about how much worse a genuinely extreme event could plausibly be, because a sample of a few thousand observations has, almost by definition, not yet sampled the far tail. Extreme value theory exists for exactly this question: under very mild conditions, the distribution of values above a high threshold converges to a Generalized Pareto Distribution (GPD) regardless of what the bulk of the data looks like, and the distribution of block maxima converges to a Generalized Extreme Value distribution (GEV). Both give a principled way to extrapolate a return level - "the loss expected to be exceeded once every N observations" - beyond anything actually observed. Two independent fitting routes are provided so they can be cross-checked against each other. The peaks-over-threshold route extracts every excess above a chosen threshold and fits the GPD shape and scale by the method of moments (Hosking & Wallis 1987) - a closed-form estimator that is more stable on short tail samples than numerical maximum likelihood, at some cost in asymptotic efficiency. The block-maxima route splits the series into equal windows, takes each window's maximum, and fits the GEV shape, scale and location by probability-weighted moments (Hosking 1990) - again closed-form, no iterative solver, and no matrix inversion. Both routes report a signed shape parameter xi: positive means a genuinely heavy, unbounded tail (the Pareto-type case relevant to most return series); zero is the exponential-tail special case; negative means a bounded tail. A naive normal-distribution assumption implicitly forces xi to negative infinity and will always understate tail risk versus a properly fitted GPD or GEV when the true tail is heavy. Notes and limitations - Method-of-moments GPD fitting is a closed-form approximation, not maximum likelihood - it is more robust on short tail samples but less statistically efficient than an iterative MLE fit, which this header does not implement. - The PWM GEV fit uses Hosking's rational approximation to convert the PWM ratio into the shape parameter xi rather than solving the exact PWM-to-GEV system; this is standard practice and accurate for the shape range normal financial tails fall in, but is itself an approximation, not exact. - Both routes assume the underlying data is stationary and the exceedances (or block maxima) are approximately independent. Strongly autocorrelated series (e.g. volatility clustering) violate this and will bias the fit - declustering the exceedances first is a standard EVT refinement this header does not implement. - Return levels are extrapolations beyond the observed data by construction. They carry genuine statistical uncertainty this header does not quantify (no confidence interval is returned) - treat them as estimates, not limits. - EvtGpdFitMoments and EvtGevFitPWM both refuse (return false) below a minimum sample size rather than returning an unreliable fit silently.

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