Peaks Over Threshold and the Generalised Pareto Distribution

Fit a normal distribution to a stock's daily returns and it will describe the middle of the data beautifully, and lie to you about the tails. Most of your 1,000 observations are ordinary days clustered near zero, and a curve-fit process naturally spends its effort matching that crowded middle, not the dozen crash days that determine your actual risk. If you care about a 1-in-1,000-day loss, fitting one distribution to all 1,000 days and reading off the tail is asking the wrong 988 observations to vote on an answer that should be decided by the worst 12. Peaks Over Threshold (POT) fixes this by throwing away everything except the extremes and fitting a distribution only to those.

The analogy

Imagine you want to know how bad the worst floods on a river can get. You would not average every day the river was at any height, most days it is at ordinary levels and tells you nothing about flood risk. Instead a hydrologist picks a high threshold, "flood stage," and studies only the days the river crossed it: by how much did it exceed flood stage each time? That collection of exceedance amounts, not the whole river-height history, is what tells you about genuine flood risk. POT is exactly this idea applied to portfolio losses.

Building the method

Pick a high threshold $u$ (say, the 95th percentile of historical losses). For every loss $X$ that exceeds $u$, record the exceedance, $Y = X - u$, how much past the threshold it went. A remarkable mathematical result (the Pickands–Balkema–de Haan theorem) says that, for a threshold high enough, the distribution of these exceedances is well approximated by the Generalised Pareto Distribution (GPD), almost regardless of what the original return distribution looked like:

$F(y) = 1 - \left(1 + \xi \frac{y}{\beta}\right)^{-1/\xi}.$

Read the two parameters in words. $\beta$ is a scale parameter, it stretches or compresses the exceedances, roughly "how big is a typical overshoot." $\xi$ (xi) is the shape parameter and is the one that matters most: $\xi > 0$ means a genuinely fat tail (extreme exceedances remain plausible no matter how far out you look, the classic signature of financial returns); $\xi = 0$ recovers a thin, exponential-like tail; $\xi < 0$ means the tail is bounded, exceedances cannot go past some finite ceiling. Fitting the GPD means estimating $\beta$ and $\xi$ from just the handful of exceedance values, not the whole dataset.

Once fitted, the model gives a formula for VaR at any confidence level beyond the data you have actually observed, extrapolating the fitted tail shape rather than guessing:

$\text{VaR}_\alpha = u + \frac{\beta}{\xi}\left[\left(\frac{n}{n_u}(1-\alpha)\right)^{-\xi} - 1\right],$

where $n$ is the total sample size and $n_u$ is the number of exceedances above $u$. In words: start at the threshold, then travel further out using the fitted shape and scale, scaled by how rare the confidence level is relative to how often you actually crossed the threshold.

A normal curve, shown here, has thin tails that vanish almost to nothing a few standard deviations out. Real return tails, and the GPD fitted to POT exceedances, decay far more slowly: more probability mass sits out past where the normal curve says there should be almost none. That gap is exactly the risk a naive normal-distribution VaR model misses.

Worked example 1: choosing exceedances and eyeballing the shape

A trading desk has 1,000 days of daily losses (positive = loss, in percent). The 95th percentile is $u = 2.1\%$, leaving $n_u = 50$ exceedance days. Suppose those 50 exceedances range mostly between 0.1% and 1.5% past the threshold, but three of them are 4%, 6%, and 9% past it, unusually large relative to the rest. That handful of outsized exceedances is the empirical signal of a positive shape parameter $\xi$: the tail is not tapering off evenly, which is exactly the pattern a fitted $\xi > 0$ (say $\xi \approx 0.25$) represents, and what an unbounded, fat GPD tail predicts should keep happening.

Worked example 2: extrapolating beyond the data

Suppose fitting gives $\hat{\xi} = 0.25$, $\hat{\beta} = 0.9\%$, threshold $u = 2.1\%$, $n = 1000$, $n_u = 50$. To get the 99.9% VaR (a loss level exceeded on average once every 1,000 days, a level with almost no direct observations in this sample):

$\text{VaR}_{0.999} = 2.1 + \frac{0.9}{0.25}\left[\left(\frac{1000}{50}(0.001)\right)^{-0.25} - 1\right] = 2.1 + 3.6\left[(0.02)^{-0.25} - 1\right].$

$(0.02)^{-0.25} \approx 2.66$, so $\text{VaR}_{0.999} \approx 2.1 + 3.6 \times 1.66 \approx 2.1 + 5.98 \approx 8.1\%$. The empirical data alone (only 1 day in 1,000 at that level, if that) could not reliably estimate this quantile; the fitted GPD extrapolates from the shape of the 50 exceedances it did see.

What this means in practice

POT/GPD is standard for setting extreme risk limits and pricing deep out-of-the-money tail hedges, anywhere a desk needs a number for a loss rarer than its own data. It is preferred over fitting a normal or Student-t to the whole dataset because it lets the tail behave differently from the body.