Extreme value theory and tail models

Definition Let $L_1, L_2, \dots$ be independent losses, positive for adverse outcomes, with common CDF $F$ under the physical measure $\mathbb{P}$, and let $x_F = \sup\{x : F(x) < 1\} \le \infty$ be the right endpoint of $F$. Extreme value theory studies the limit laws of the largest losses, such as the maximum $M_n = \max(L_1, \dots, L_n)$, whose CDF is $\mathbb{P}(M_n \le x) = F(x)^n$ because all $n$ losses must be at most $x$. The distribution $F$ is in the maximum domain of attraction of the generalized extreme value law $G_{\xi}$ with shape $\xi \in \mathbb{R}$, written $F \in \operatorname{MDA}(G_{\xi})$, if there are constants $a_n > 0$ and $b_n \in \mathbb{R}$ with

for every real $x$. The generalized extreme value CDF is

where $1 + \xi x > 0$ and $\xi \ne 0$, and its limit as $\xi \to 0$ is the Gumbel CDF $G_0(x) = \exp(-e^{-x})$ for every real $x$. Off its support, $G_{\xi}(x)$ equals $0$ for $x \le -1/\xi$ when $\xi > 0$ and $1$ for $x \ge -1/\xi$ when $\xi < 0$, and it is continuous everywhere. By the Fisher-Tippett-Gnedenko theorem, which is stated here without proof, every nondegenerate limit of normalized maxima has the form $G_{\xi}((x - \mu)/s)$ for some $\xi$, $\mu$ and $s > 0$, and the location $\mu$ and scale $s$ can be absorbed into $a_n$ and $b_n$. A domain with $\xi > 0$ forces $x_F = \infty$, one with $\xi < 0$ forces $x_F < \infty$, and $\xi = 0$ allows either. The normalization is part of the condition, since $M_n$ itself increases to $x_F$. Not every distribution has a domain of attraction, and the Poisson and geometric distributions, for example, belong to none.

Definition Fix a threshold $u < x_F$, so that the exceedance probability $p_u = \mathbb{P}(L > u) = 1 - F(u)$ is positive. The excess over $u$ is $Y = L - u$ on the event $\{L > u\}$, and its CDF under the conditional probability given $L > u$ is the excess distribution function

for $0 \le y < x_F - u$, with $F_u(y) = 1$ for $y \ge x_F - u$ when $x_F$ is finite. Since the event $\{L > u + y\}$ lies inside $\{L > u\}$, the definition of conditional probability gives the exact factorization

for every $y \ge 0$. When $\mathbb{E}[\max(L, 0)] < \infty$, the mean excess $e(u) = \mathbb{E}[L - u \mid L > u]$ is finite. In a sample, the excesses are the amounts by which observed losses pass $u$, and the fraction of observations above $u$ estimates $p_u$. Fitting a tail to them is beyond this lesson.

Definition For a shape $\xi \in \mathbb{R}$ and a scale $\beta > 0$, let $H_{\xi,\beta}$ be the generalized Pareto CDF of the next card, extended by $1$ beyond its endpoint when $\xi < 0$. The Pickands-Balkema-de Haan theorem, stated here without proof, says that $F \in \operatorname{MDA}(G_{\xi})$ if and only if there is a positive measurable function $\beta(u)$ with

as $u \uparrow x_F$. The shape is the same $\xi$ as in the domain of attraction. The convergence is of an absolute difference of CDFs, uniformly over the possible excesses. It gives no rate, it does not make the relative error small for a tail probability that is itself small, and it does not say that any finite threshold has an exactly generalized Pareto excess law. The theorem concerns one distribution $F$. Applying it to a sample also needs independent, identically distributed losses, or a separate treatment of clustering and changing volatility.

Definition The generalized Pareto distribution with shape $\xi \in \mathbb{R}$ and scale $\beta > 0$ is the law of a random variable $Y$ with survival function

for $\xi \ne 0$ and $\mathbb{P}(Y > y) = e^{-y/\beta}$ for $\xi = 0$, at every $y \ge 0$ with $1 + \xi y/\beta > 0$, and with survival $0$ beyond that range. The scale $\beta$ is in loss units, here currency units, and the shape $\xi$ is dimensionless. The case $\xi = 0$ is the limit of the others, because the logarithm of the survival, $-\xi^{-1}\ln(1 + \xi y/\beta)$, tends to $-y/\beta$. For $\xi > 0$ the support is $[0, \infty)$ and the survival decays like the power $y^{-1/\xi}$, for $\xi = 0$ it decays exponentially, and for $\xi < 0$ the support is $[0, -\beta/\xi]$. For $\xi > 0$ the number $1/\xi$ is the tail index, and some texts give that name to $\xi$ itself. The moment $\mathbb{E}[Y^k]$ of order $k > 0$ is finite exactly when $\xi < 1/k$, and every moment is finite when $\xi \le 0$. In particular

where the variance needs $\xi < \tfrac12$. More generally, a result of Gnedenko stated here says that $F \in \operatorname{MDA}(G_{\xi})$ with $\xi > 0$ exactly when $1 - F(x) = x^{-1/\xi}\ell(x)$ for a slowly varying $\ell$, meaning $\ell(tx)/\ell(x) \to 1$ as $x \to \infty$ for every $t > 0$. Then the moments of $\max(L, 0)$ of order below $1/\xi$ are finite and those of order above $1/\xi$ are infinite.

Definition Assume a tail model for a loss $L$ with CDF $F$, in which the exceedance probability $p_u = \mathbb{P}(L > u)$ is positive and the excess law over $u$ is exactly $H_{\xi,\beta}$, with nothing assumed below $u$. At a confidence level $\alpha \in (0, 1)$, value at risk is the lower quantile and expected shortfall is the quantile integral,

as in Expected shortfall and coherent risk. For $x \ge u$ in the support, the factorization gives $\mathbb{P}(L > x) = p_u(1 + \xi(x - u)/\beta)^{-1/\xi}$, which is continuous and strictly decreasing. If $1 - p_u < \alpha < 1$, setting it equal to $1 - \alpha$ and solving gives $q_{\alpha} = \operatorname{VaR}_{\alpha}$ as

for $\xi \ne 0$, and $q_{\alpha} = u + \beta \ln r$ for $\xi = 0$, the limit of the first formula. The condition $\alpha > 1 - p_u$ makes $r > 1$ and so $q_{\alpha} > u$, and when $\xi < 0$ it keeps $q_{\alpha}$ below the endpoint $u - \beta/\xi$. Given $L > q_{\alpha}$, the excess over $q_{\alpha}$ is again generalized Pareto with shape $\xi$ and scale $\beta + \xi(q_{\alpha} - u)$, as the threshold stability card shows. The tail is continuous, so $\operatorname{ES}_{\alpha} = \mathbb{E}[L \mid L > q_{\alpha}]$, and for $\xi < 1$

For $\xi \ge 1$ the tail has an infinite mean and $\operatorname{ES}_{\alpha} = \infty$. These formulas are exact under the assumed tail. Using them on data is an extrapolation, since $1 - \alpha$ can be far smaller than any frequency the sample can measure, and the uncertainty in the estimates of $\xi$, $\beta$ and $p_u$ usually dominates the result.

Definition Suppose the excess over $u$ is exactly $H_{\xi,\beta}$, and take a higher threshold $v \ge u$, with $v - u < -\beta/\xi$ when $\xi < 0$. Dividing survival functions,

so the excess over $v$ is generalized Pareto with the same shape and a scale that is linear in the threshold, and $p_v = p_u(1 + \xi(v - u)/\beta)^{-1/\xi}$. For $\xi = 0$ the scale stays $\beta$. The tail of $L$ above $v$ is the same function in both descriptions, so the quantile formula with $v$, $p_v$ and $\beta(v)$ returns the same $q_{\alpha}$ for every $\alpha$ with $1 - \alpha < p_v$. For $\xi < 1$ the mean excess $e(v) = \beta(v)/(1 - \xi)$ is linear in $v$ with slope $\xi/(1 - \xi)$. This exact stability belongs to the assumed law. With data, a higher threshold leaves fewer excesses and noisier estimates, while a lower one supplies more excesses whose law may be further from generalized Pareto. Estimated shapes that stay roughly constant across a range of thresholds support a choice, but the stability of the assumed law does not show that estimates from sparse data are stable.