## 4 Insurer's Risk  
As in any total coverage insurance scheme, the insured is held indemnified against any damage by just paying the premium. On the other hand, the insurer must cover all the losses the insured suffers. Since the premium is a fixed quantity and the loss is a random variable, the insurer incurs the risk of paying more than what is cashed in. Of course, this risk is different for the two approximation levels we have considered so far. It is larger for the second-order approximation as long as its premium is lower than the fourth-order approximation. In this section, we quantify that risk by computing the Value-at-Risk (VaR), also considering the number \(n\) of events as a random variable. The Value-at-Risk is a well-known measure of risk, defined first in the financial world and then extended to other fields. For example, it has been employed in the ICT sector in [3, 32, 33]. In our case, the Value-at-Risk at the confidence level \(\theta\) is the smallest number \(x_{\theta}\) such that the probability that the loss \(X\) exceeds \(x_{\theta}\) is not larger than \(\theta\)[34]:  
\[\text{VaR}_{\theta}=\inf\{x_{\theta}\in\mathbb{R}:\mathbb{P}[X>x_{\theta}] \leq 1-\theta\}. \tag{45}\]  
We can now compute the VaR for the four distributions we have considered so far. For the sake of simplicity, we use the notation \(P_{i}\), \(i=\{2,4\}\) to represent the insurance premium. In the following, the number of claims \(n\) is considered to follow a Poisson random variable with parameter \(\lambda T\), where \(\lambda\) refers to the frequency of the claims, and \(T\) is the length of the observation period. The losses taking place over the sequence of events are assumed to be i.i.d. random variables. **Lognormal case**: For the case of the lognormal distribution we use [PERSON]'s method as employed in [35] to estimate the queue of lognormal sums. They assume that the sum of lognormal variables is still lognormal. So that the overall loss \(X=\sum_{i=1}^{n}L_{i}\) follows a lognormal distribution, in particular \(X\sim e^{Z}\), with \(Z\) being a Gaussian variable with mean \(m_{Z}\equiv\mathbb{E}[X]\) and variance \(\sigma_{Z}^{2}\equiv\mathbb{V}[X]\), which we can compute using Wald identity:  
\[ \begin{split} m_{Z}&=\mathbb{E}[X]=\mathbb{E}[n] \mathbb{E}[L_{i}]=\lambda Te^{\mu+\frac{e^{2}}{2}}\\ \sigma_{Z}^{2}&=\mathbb{V}[X]=\mathbb{E}[n]\mathbb{V }[L_{i}]+\mathbb{E}^{2}[L_{i}]\mathbb{V}[n]=\lambda Te^{2\mu+2\sigma^{2}} \end{split}  \tag{46}\]  
The Value-at-Risk can therefore be determined as follows  
\[ \begin{split}\mathbb{P}(X>x_{\theta})&=1-\theta \Longrightarrow\mathbb{P}(e^{Z}>x_{\theta})=1-\theta\Longrightarrow\mathbb{ P}(Z>\ln(x_{\theta}))=1-\theta\\ \text{G}\left(\frac{\ln(x_{\theta})-m_{Z}}{\sqrt{2}\sigma_{Z} }\right)&=\theta\Longrightarrow x_{\theta}=e^{(\text{G}^{-1}( \theta)\sqrt{2}\sigma_{Z}+m_{z})}\\ \text{VaR}_{\theta}&=x_{\theta}=e^{(\text{G}^{-1}( \theta)\sqrt{\lambda T}e^{\mu+\sigma^{2}}+\lambda Te^{\mu+\frac{e^{2}}{T}})} \end{split}  \tag{47}\]  
where \(\text{G}(*)\) is the cumulative distribution function of a standard (zero-mean, unit-variance) Gaussian random variable. In Figure 6, we see how the extreme loss (i.e., the difference between the Value-at-Risk and the premium) grows with the expected number of claims \(\lambda T\). **Pareto case**: In the Pareto and GPD cases, we can use the Generalized Extreme Value Theory [36, 37, 38] to compute the VaR. As can be seen, for example, in [39], we can approximate the sum of \(n\) i.i.d. Pareto-distributed variables through a Generalized Extreme Value (GEV) variable, so that the overall loss \(X\) exhibits the probability density function  
\[f(\mu,\sigma,\eta)=\frac{1}{\sigma}r(x)^{\eta+1}e^{-r(x)} \tag{48}\]  
where  
\[r(x)= \begin{cases}\left(1+\eta\left(\frac{x-\mu}{\sigma}\right)\right)^{- \frac{1}{\theta}}\ \ \text{if}\ \ \eta\
eq 0\\ e^{-\frac{x-\mu}{\sigma}}\ \ \text{if}\ \ \eta=0\end{cases} \]and \(\mu\), \(\sigma\) and \(\eta\) are the location, scale and shape parameter of the GEV distribution.