## 3 Comparison of Premiums
### An Application to Realistic Cases  
In the previous sections, we have derived two metrics (underestimation factor and premium estimator variance) that allow us to compare the insurance premium as computed at two different approximation levels: second and fourth order. In this section, we get a feeling of how the two approximation levels compare in realistic cases, i.e., when the loss distribution parameters take realistic values. For this purpose, we consider the parameter values obtained in [6; 30; 31], which we report in Table 5. In particular, the Generalized Pareto distribution describes the duration (in minutes) of a cloud service outage, where the amount of economic losses is proportional to the duration of the outage. The lognormal distribution describes the severity of the losses due to a cyber-attack by a hacker. The Gamma distribution describes the severity of the losses due to a natural catastrophe (as a hurricane). The Pareto distribution is one of the most used distributions in the literature to describe the severity of losses in many cases, for example, losses due to fire damages, car accidents, and ICT service failures. All the cases we consider here refer to a policy duration of one year, and assume full compensation. In this section, we adopt the Babcock method with \(\eta=0.4\) to determine the risk-aversion coefficient. As assumed in Section 3.2, for the sake of computing the premium uncertainty, the loss distribution parameters are considered to be random variables. We plug their estimators, assumed to be unbiased and with a standard deviation equal to 1% of their expected value (this value is taken as a reference for convenience, but it should be assessed for the specific case at hand), in the premium computation formula. To compare the two approximations, we first compute the underestimation factor through the percentage relative difference of the premiums, assuming that the distribution parameters are known exactly:  
\[\mathrm{DP}(n)=\frac{P_{4}(n)-P_{2}(n)}{P_{4}(n)}\cdot 100 \tag{41}\]  
As a second metric for comparison, we compute the coefficient of variation (i.e., the ratio of standard deviation to expected value) for the two approximation levels. We obviously wish that coefficient to be as low as possible. \begin{table}
\begin{tabular}{l l l} \hline \hline
**Loss Distribution** & **Parameter 1** & **Parameter 2** \\ \hline GPD & \(\xi=-0.64\) & \(\bar{\beta}=192.47\) \\ Lognormal & \(\bar{\mu}=6.83\) & \(\bar{\sigma}=0.87\) \\ Gamma & \(\bar{u}=0.78\) & \(\bar{b}=12.58\) \\ Pareto & \(\bar{u}=4.1\) & \(\bar{h}=12\) \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Parameters values  
In all cases, we can see that \(\mathbb{V}[P_{4}]>\mathbb{V}[P_{2}]\). Naturally, using \(\alpha_{Bab}\) (therefore also \(\alpha_{Pil}\)) the difference DP between the two approximation levels tends to zero  
\[\lim_{n\to\infty}\text{DP}(n)=\lim_{n\to\infty}\frac{\frac{1}{6}\ln^{2}(\frac{1 +2\alpha}{1-2\alpha})\frac{5\|\mathbb{V}[V^{2}\|_{2}]}{\mathbb{E}[\mathbb{I}_ {i}]}+\frac{1}{2\alpha\alpha^{2}}\ln^{3}(\frac{1+2\alpha}{1-2\alpha})\frac{5 \|\mathbb{V}[V^{2}\|_{2}]}{\mathbb{E}[\mathbb{I}_{i}]}}{n\mathbb{E}[\mathbb{I} _{i}]+\frac{1}{2}\ln^{2}(\frac{1+2\alpha}{1-2\alpha})\frac{5\|\mathbb{V}[V^{ 2}\|_{2}]}{\mathbb{E}[\mathbb{I}_{i}]}+\frac{1}{2\alpha\alpha}\ln^{3}(\frac{1 +2\alpha}{1-2\alpha})\frac{5\|\mathbb{V}[V^{2}\|_{2}]}{\mathbb{E}[\mathbb{I}_ {i}]}}=0. \tag{43}\]  
In fact, for large values of \(n\), \(P_{4}(n)\approx P_{2}(n)\). Instead, if \(\alpha\) is a fixed real constant, the relative difference DP is always positive and does not depend on the number of events \(n\):  
\[