## 2 The Expected Utility Principle for Premium Computation  
In this paper, we adopt the expected utility principle to compute the insurance premium. The principle is well-rooted in the literature. In this section, starting from its general definition, as described in [14], we derive the premium. We largely follow the derivation reported in [6; 19]. The list of symbols used throughout the paper are shown in Table 1 for convenience. Under the expected utility principle, the relevance of the loss suffered by the insured is evaluated through the utility function \(u:\mathbb{R}\rightarrow\mathbb{R}\), which is assumed to be a monotone non-decreasing function. In the absence of any event, the utility of the insured depends just on its assets \(w\), so that it is \(u(w)\). In the absence of an insurance policy, the occurrence of a damaging event would provoke a monetary loss \(X\), which would bring its utility down to \(u(w-X)\). Here, we do not make any assumption about the nature of the events leading to the loss, since what follows is derived under very general assumptions. An example of application of the expected utility principle to cloud storage is reported in [6] (where the loss is the compensation provided by the cloud service provider to its customers \begin{table}
\begin{tabular}{l l} \hline \hline
**Parameters/Variables** & **Meaning** \\ \hline \(\alpha\) & Risk-aversion coefficient \\ \(\Gamma(a,b)\) & Gamma distribution, with shape parameter \(a\) and scale parameter \(b\) \\ \(Gev(\mu,\sigma,\eta)\) & Generalized Extreme Value distribution, with location \((\mu)\), scale \((\sigma)\) and shape \((\eta)\) parameters \\ \(GPD(\beta,\xi)\) & Generalized Pareto distribution, with its scale \((\beta)\) and shape \((\xi)\) parameters \\ \(\eta\) & Probability premium \\ \(\lambda\) & Frequency of claims \\ \(L_{i}\) & Loss suffered during \(i\)-th event \\ \(Logn(\mu,\sigma^{2})\) & Lognormal distribution, with its mean-log \((\mu)\) and variance-log \((\sigma^{2})\) \\ \(p\) & Probability of an event \\ \(Par(a,h)\) & Pareto distribution, with its shape \((a)\) and scale \((b)\) parameters \\ \(T\) & Length of observation period \\ \(\theta\) & Confidence level \\ \(X\) & Random loss \\ \(x_{\theta}\) & Value at Risk \\ \(w\) & Wealth under no losses \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Variables and parameters used. when service quality falls below what is stated in Service Level Agreements). In Section 3.3, the pricing formulas are instantiated for some contexts. Since the insured has bought an insurance policy (assuming that the event falls fully under the coverage umbrella of the policy), it pays a (fixed) premium \(P\), so that its utility decreases to the fixed quantity \(u(w-P)\). The insured can then compare the two alternatives: buying an insurance policy and end up with utility \(u(w-P)\) or suffer the (random) monetary loss \(X\) and end up with utility \(u(w-X)\). A crucial element in deciding whether to buy an insurance policy is then the premium \(P\). Under the expected utility principle, the fair premium is defined as that for which the two alternatives are utility-equivalent (on average), i.e., that resulting from the following equilibrium equation:  
\[\mathbb{E}[u(w-X)]=u(w-P) \tag{1}\]  
We can solve that equation for \(P\) through an approximation provided by the Taylor series expansion for both sides, centered in \(w-\mathbb{E}[X]\):  
\[ \begin{split} u(w-P)&\simeq u(w-\mathbb{E}[X])+u^{ i}(w-\mathbb{E}[X])(\mathbb{E}[X]-P)\\ u(w-X)&\simeq u(w-\mathbb{E}[X])+u^{i}(w-\mathbb{E} [X])(\mathbb{E}[X]-X)+\frac{1}{2}u^{ii}(w-\mathbb{E}[X])(\mathbb{E}[X]-X)^{2}\\ &\qquad+\frac{1}{6}u^{iii}(w-\mathbb{E}[X])(\mathbb{E}[X]-X)^{3} +\frac{1}{24}u^{iv}(w-\mathbb{E}[X])(\mathbb{E}[X]-X)^{4}.\end{split}  \tag{2}\]  
It is to be noted that contrary to the standard treatment reported in [14] and applied, e.g., in [6], where the expansion stops at the second order, here we go through the fourth-order term. We aim to seek for a more accurate, though more complex, premium computation.