**A.1 The generalised Pareto distribution (GPD)**

The GPD given by

$$\text{GPD}\left(\mathbf{x};\sigma,\xi,q\_{b}\right) = \begin{cases} 1 - \left[1 + \frac{\xi}{\sigma}\left(\mathbf{x} - q\_{b}\right)\right]^{\frac{-1}{\xi}} & \xi > 0\\ 1 - \exp\left(-\frac{\mathbf{x} - q\_{b}}{\sigma}\right) & \xi = \mathbf{0}, \end{cases} \tag{10}$$

with *x*≥ *qb*, thus taking *qb* as the so-called EVT threshold and with *σ* and *ξ* respectively scale and shape parameters. Note the Extreme Value Index (EVI) of the GPD distribution is given by *EVI* ¼ *ξ* and that heavy-tailed distributions have a positive EVI and larger EVI implies heavier tails. This follows (also) from the fact that for positive EVI the GPD distribution belongs to the Pareto-type class of distributions, having a distribution function of the form 1 � *F x*ð Þ¼ *<sup>x</sup>*�1*=ξℓF*ð Þ *<sup>x</sup>* , with *ℓF*ð Þ *x* a slowly varying function at infinity (see e.g. Embrechts et al., 1997). For Pareto-type, when the EVI > 1, the expected value does not exist, and when EVI > 0.5, the variance is infinite. Note also that the GPD distribution is regularly varying with index �1*=ξ* and therefore belongs to the class of sub-exponential

distributions. Note that the *<sup>γ</sup>*-th quantile of the GPD is *<sup>q</sup>*ð Þ¼ *<sup>γ</sup> GPD*�<sup>1</sup> *<sup>γ</sup>*, *<sup>σ</sup>*, *<sup>ξ</sup>*, *qb* � � <sup>¼</sup>

$$\left(q\_b + \frac{\sigma\left(\left(1-\gamma\right)^{-\zeta}-1\right)}{\xi}\right) \text{ when } \xi \neq 0 \text{ and } \text{GPD}^{-1}\left(\gamma, \sigma, \xi, q\_b\right) = q\_b - \sigma \ln\left(1-\gamma\right) \text{ when} = 0.$$
