Generalized Pareto Distribution

Study guide: Exam ASTAM

The Generalized Pareto Distribution (GPD) G_{\xi,\beta} is the limiting distribution of the excess loss X - d, given X > d, over a high threshold d, for any loss distribution whose block maxima have a generalized extreme value limit. It is the basis of the peaks-over-threshold method, which fits the GPD to the largest losses only and uses it to estimate tail probabilities, Value at Risk and Expected Shortfall beyond the range of the data.

G_{\xi,\beta}(y) = 1 - \left(1 + \frac{\xi y}{\beta}\right)^{-1/\xi}, \quad \xi \neq 0

G_{0,\beta}(y) = 1 - e^{-y/\beta}

\Pr(X > x) \approx S_X(d)\left[1 + \frac{\xi(x - d)}{\beta}\right]^{-1/\xi}, \quad x > d

Q_\alpha = d + \frac{\beta}{\xi}\left[\left(\frac{S_X(d)}{1 - \alpha}\right)^{\xi} - 1\right]

ES_\alpha = \frac{Q_\alpha + \beta - \xi d}{1 - \xi}

• Symbols: Following Hardy and Saunders (the ASTAM study note on extreme value theory), \xi is the shape parameter and \beta > 0 the scale parameter. The support is y \ge 0 when \xi \ge 0 and 0 \le y \le -\beta/\xi when \xi < 0. Q_\alpha is the \alpha-Value at Risk and ES_\alpha = E[X \mid X > Q_\alpha] the Expected Shortfall, which Loss Models writes as \text{VaR}_p and \text{TVaR}_p for a continuous loss. S_X(d) is estimated by the proportion of the sample above the threshold.
• Pickands–Balkema–de Haan theorem: A distribution is in the maximum domain of attraction of the GEV with shape \xi if and only if its excess losses over d approach a GPD with the same \xi, and some scale \beta, as d rises to the top of its range. When \xi > 0 the GPD is a two-parameter Pareto with shape 1/\xi and scale \beta/\xi, when \xi = 0 it is exponential with mean \beta, and when \xi < 0 it is bounded.
• Linear mean excess: If Y is G_{\xi,\beta}, then Y - k given Y > k is G_{\xi,\,\beta + \xi k}, so for \xi < 1 the mean excess loss at d + k is (\beta + \xi k)/(1 - \xi), a straight line in k with slope \xi/(1-\xi). The mean is \beta/(1 - \xi) for \xi < 1, and moments exist only below order 1/\xi. The threshold is chosen where the empirical mean excess plot becomes roughly linear.
• Choosing the threshold: A higher threshold makes the GPD a better approximation but leaves fewer exceedances to fit it to, so the estimate of \xi trades bias against variance and often has a large standard error.

Peaks Over Threshold for a Liability Book Of 1,000 liability claims, 40 exceed a threshold of 500 (in $000s). A GPD fitted to the 40 excesses has \xi = 0.4 and \beta = 300. Estimate the probability that a claim exceeds 2,000, the 99% Value at Risk and the 99% Expected Shortfall.

Answer The threshold's survival probability is estimated as 40/1{,}000 = 0.04, and the GPD supplies the rest of the tail.

\begin{align*}
\Pr(X > 2{,}000) &= 0.04\left[1 + \frac{0.4(1{,}500)}{300}\right]^{-2.5} \\
&= 0.04\,(3)^{-2.5} \\
&= 0.002566 \\
Q_{0.99} &= 500 + \frac{300}{0.4}\left[\left(\frac{0.04}{0.01}\right)^{0.4} - 1\right] \\
&= 500 + 750(0.741101) \\
&= 1{,}055.83 \\
ES_{0.99} &= \frac{1{,}055.83 + 300 - 0.4(500)}{1 - 0.4} \\
&= 1{,}926.38
\end{align*}

The 99% Value at Risk is about $1.06 million, and the claims above it average about $1.93 million. The Expected Shortfall is 1.8 times the Value at Risk, because with \xi = 0.4 the mean excess keeps rising with the threshold.

Reading the GPD Off a Mean Excess Plot Above a threshold of 100, the empirical mean excess function of a set of losses is close to the line e(100 + k) = 50 + 0.5k. Estimate \xi and \beta for the excess losses over 100, and the probability that a loss above 100 also exceeds 200.

Answer A GPD's mean excess line has slope \xi/(1 - \xi) and intercept \beta/(1 - \xi).

\begin{align*}
\frac{\xi}{1 - \xi} &= 0.5 \\
\xi &= \tfrac{1}{3} \\
\beta &= 50\left(1 - \tfrac{1}{3}\right) \\
&= 33.33
\end{align*}

The excess over 100 must exceed 100, so the conditional probability is \left[1 + \tfrac{1}{3}(100)/33.33\right]^{-3} = 2^{-3} = 0.125. This GPD is a Pareto with shape 3 and scale 100, so the excess losses have a finite variance but no third moment. A maximum likelihood fit would refine these rough values, but the slope alone already shows a positive \xi.

How it appears on the exam

• ASTAM asks candidates to understand the derivation and characteristics of the GPD and to apply it to the estimation of tail risk measures and probabilities. A question gives a threshold, the number of observations above it out of the sample, and fitted values of \xi and \beta, then asks for a tail probability, a Value at Risk or an Expected Shortfall.
• The exponential case: When \xi = 0, the formulas take their limits, \Pr(X > x) = S_X(d)\,e^{-(x-d)/\beta}, Q_\alpha = d + \beta \ln\big(S_X(d)/(1 - \alpha)\big) and ES_\alpha = Q_\alpha + \beta, and the reading's exercises use this case too.
• Written comments: The reading's exercises ask for the trade-off in choosing a threshold, and for a comparison of GPD estimates with empirical ones, which cannot go beyond the largest observation.

Exam traps

• Two generalized Paretos: The exam tables from Loss Models list a different "generalized Pareto" with parameters \alpha, \theta and \tau, a member of the transformed beta family. The ASTAM objective that pairs the GPD with the GEV means the extreme value distribution G_{\xi,\beta}, and using the tables' entry for a peaks-over-threshold question gives the wrong answer.
• Conditional probabilities: G_{\xi,\beta} gives probabilities for losses that have already exceeded d. An unconditional tail probability needs the factor S_X(d), and a Value at Risk needs the tail level 1 - \alpha converted to (1 - \alpha)/S_X(d) above the threshold.
• The quantile must lie above the threshold: The formulas for Q_\alpha and ES_\alpha hold only when 1 - \alpha < S_X(d). A 90% Value at Risk with only 4% of the data above d cannot come from this tail fit.
• A shape of 1 or more: The mean of the GPD, and so the Expected Shortfall, is infinite when \xi \ge 1. A fitted \xi close to 1 makes the Expected Shortfall estimate very large and very uncertain.