The Generalized Pareto Distribution and Peaks Over Threshold

You've fit a distribution to a year of daily returns, and it does a decent job in the middle. But a risk manager doesn't care about the middle — they care whether the model is right about the worst 1 percent, the part with the fewest data points and the highest stakes. Fitting one distribution to everything wastes effort on the bulk and often gets the tail wrong. The peaks-over-threshold method instead throws away everything except the tail and fits a distribution built specifically for it.

The analogy: only interviewing people who already cleared the bar

Imagine you only care about how much taller than 6'5" the tallest people in a population are — not the whole height distribution, just the overflow past a high bar. Set the bar, keep only the people who clear it, and measure how far each one clears it by. The generalized Pareto distribution (GPD) is the mathematical shape that this "overflow past a high bar" reliably takes, almost regardless of what the full height distribution looked like below the bar — a theorem (Pickands-Balkema-de Haan) guarantees this convergence for a very wide class of underlying distributions.

Writing it down

Pick a threshold $u$. For any loss $X$ that exceeds $u$, define the excess $Y = X - u$. The Pickands-Balkema-de Haan theorem says that for $u$ high enough, $Y$'s distribution is approximately

$$G(y) = 1 - \left(1 + \xi \frac{y}{\sigma}\right)^{-1/\xi}, \qquad y \ge 0,$$

with scale parameter $\sigma > 0$ and shape parameter $\xi$ (xi, pronounced "ksee" or "zai"). In words: $\xi$ controls how heavy the tail is. $\xi = 0$ gives an exponential tail (light, thin), $\xi > 0$ gives a genuine power-law tail (heavy — the case that matters for financial losses), and $\xi < 0$ gives a tail with a hard upper bound. Financial return tails almost always fit $\xi > 0$: losses beyond the threshold can, in principle, be arbitrarily large, with probability decaying slowly rather than being cut off.

Worked example 1: fitting the excess and pricing a deeper tail

Suppose you set a threshold $u$ of $2 million in daily losses, and fitting the excesses over that threshold gives $\xi = 0.25$ and a scale $\sigma$ of $1 million. What's the probability a loss exceeds $5 million, i.e. an excess of $y = 5 - 2 = 3$? Plug into the survival function $1 - G(y) = (1 + \xi y/\sigma)^{-1/\xi}$: $(1 + 0.25 \times 3/1)^{-1/0.25} = (1.75)^{-4}$. Computing $1.75^4 = 9.38$, so $(1.75)^{-4} \approx 0.1066$. That 10.66 percent is the probability of exceeding $5 million given you're already past $2 million — one clean formula covering a region with almost no raw data points to work from directly.

Worked example 2: the VaR-to-CVaR shortcut this gives you

The GPD also gives a closed form for expected shortfall once VaR is known: for $\xi < 1$,

$$\text{CVaR}_p = \frac{\text{VaR}_p}{1-\xi} + \frac{\sigma - \xi u}{1-\xi}.$$

Using the same fit ($\xi = 0.25$, scale $\sigma$ of $1 million, threshold $u$ of $2 million) and suppose the 99 percent VaR was separately computed as $4 million: $\text{CVaR} = \frac{4}{1-0.25} + \frac{1 - 0.25 \times 2}{1-0.25} = \frac{4}{0.75} + \frac{0.5}{0.75} = 5.333 + 0.667 = 6.0$. So the expected loss given that the 99th percentile is breached is $6 million — 50 percent worse than the VaR figure itself, a gap the GPD framework makes explicit and computable rather than leaving to guesswork.

What this means in practice

• Basel and internal risk frameworks use peaks-over-threshold GPD fitting specifically because the alternative — fitting one distribution to the whole return history — routinely underestimates tail risk, since the fit is dominated by thousands of ordinary days and only a handful of extreme ones.
• Threshold choice is a genuine bias-variance tradeoff: too low a threshold and the GPD approximation hasn't kicked in yet (bias); too high and there are too few exceedances left to fit reliably (variance). A "mean excess plot" — average excess as a function of threshold, which should be roughly linear once the GPD approximation is valid — is the standard diagnostic.
• The shape parameter $\xi$ is the single number worth reporting alongside VaR: it tells you whether the tail beyond your model's calibration range is thin, power-law, or bounded, which changes how much you should trust extrapolation past the data you've actually seen.