Peaks Over Threshold and the Generalized Pareto Distribution

To price tail risk you need to know the distribution of losses beyond some painful level — how bad does a loss get, given that it already breached your risk limit? Fitting a single distribution to the whole dataset and reading off the tail wastes the fit's effort modeling the ordinary, boring middle of your data, which you don't care about. Peaks Over Threshold flips this: pick a high bar, keep only the amounts by which losses clear that bar, and fit a distribution to just those overflow amounts. There is a theorem guaranteeing what shape that distribution has to take, almost regardless of what your original data looked like.

An analogy: a dam and the water that spills over

Picture a dam wall at a fixed height. Most days no water reaches the top. On the days it does, what matters is not the water level itself but how much it overflows the wall — an inch, a foot, ten feet. If you record only the overflow amounts on the days there is any, and ignore every day the water stayed below the wall, the shape of those overflow amounts turns out to belong to one specific family of curves almost no matter what the underlying water-level distribution was. That family is the Generalized Pareto Distribution (GPD), and the "wall height" is your threshold.

The rule, one symbol at a time

Let $u$ be the threshold you chose. For any loss $X$ that exceeds $u$, define the excess $Y = X - u$ — how far past the wall it went. The Pickands–Balkema–de Haan theorem says that, for $u$ high enough, the distribution of $Y$ is approximately

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

read as: "the probability the excess is bigger than $y$ depends on two knobs, $\xi$ and $\sigma$, through this specific formula, no matter what distribution the original losses came from." Here $\xi$ (xi) is the shape parameter — the same tail-fatness idea as in the Hill estimator, with $\xi > 0$ meaning a fat, Pareto-like tail — and $\sigma$ (sigma) is a scale parameter, controlling how large a typical excess is in absolute units. In plain English: once you're past the threshold, the excess amounts follow one of a small, well-understood family of curves, and fitting that curve just means estimating two numbers, $\xi$ and $\sigma$, from your excess data.

Slide a threshold $u$ far into the right tail of a curve like the one above. Whatever shape the full distribution had, the theorem guarantees that just the spillover past $u$ settles into one predictable two-parameter family — that predictability is what makes the flood-wall analogy work.

Worked example 1: fitting by the method of moments

Suppose daily losses exceeding a threshold of $5m (out of a much larger dataset) give five excess amounts, in millions of dollars: $1, 3, 0.5, 2, 4.5$. The sample mean excess is $\bar{y} = (1+3+0.5+2+4.5)/5 = 11/5 = 2.2$, and the sample variance is $s^2 = \frac{1}{4}\sum(y_i - 2.2)^2 = \frac{1}{4}(1.44+0.64+2.89+0.04+5.29) = \frac{10.3}{4} = 2.575$ (in squared millions).

Method-of-moments formulas for the GPD give $\hat{\xi} = \frac{1}{2}\left(\frac{\bar{y}^2}{s^2} - 1\right)$ and $\hat{\sigma} = \frac{\bar{y}}{2}\left(\frac{\bar{y}^2}{s^2}+1\right)$. Compute $\bar{y}^2/s^2 = 4.84/2.575 = 1.880$. So $\hat{\xi} = \frac{1}{2}(1.880-1) = 0.440$, and $\hat{\sigma} = \frac{2.2}{2}(1.880+1) = 1.1 \times 2.880 = 3.168$. A shape of $0.44$ signals a fairly fat tail — excesses well beyond the threshold are not rare.

Worked example 2: pricing a 1-in-1000-day loss

Using $\hat{\xi} = 0.44$, $\hat{\sigma} = 3.168$ from above, and supposing losses exceed the $5m threshold on about 5% of days, find the loss level exceeded once every 1,000 days (probability $0.001$). Conditional on exceeding the threshold, we need $P(Y>y) = 0.001/0.05 = 0.02$. Solve $\left(1+\xi y/\sigma\right)^{-1/\xi} = 0.02$ for $y$: raise both sides to the power $-\xi$: $1+\xi y/\sigma = 0.02^{-\xi} = 0.02^{-0.44}$. Compute $0.02^{-0.44} = e^{-0.44 \ln 0.02} = e^{-0.44\times(-3.912)} = e^{1.721} = 5.59$. So $\xi y/\sigma = 4.59$, giving $y = 4.59 \times \sigma/\xi = 4.59 \times 3.168/0.44 = 33.05$ (in millions). The 1-in-1000-day loss is about $5m + $33.05m, roughly $38.05m.

What this means in practice

• Value-at-risk beyond the data. POT/GPD is the standard tool for extrapolating loss quantiles far beyond anything actually observed, used throughout tail-risk and stress-testing frameworks.
• Choosing the threshold. Too low a $u$ and the theorem's approximation hasn't kicked in yet; too high and there's too little excess data to fit two parameters reliably — the same trade-off as choosing $k$ for the Hill estimator.
• Complements the Hill estimator. Hill uses a fixed count of top points; POT uses a fixed threshold value. They often give similar shape estimates and are used as a cross-check on each other.