## 2 The Expected Utility Principle for Premium Computation
When we replace those approximate expressions into Equation (1), omitting the argument \(w-\mathbb{E}[X]\) of the utility function for the sake of simplicity, we obtain  
\[(\mathbb{E}[X]-P)u^{i}=\frac{1}{2}u^{ii}\mathbb{V}[X]-\frac{1}{6}u^{iii} \mathbb{S}[X]\mathbb{V}^{3/2}[X]+\frac{1}{24}u^{ii}\mathbb{K}[X]\mathbb{V}^{2 }[X], \tag{3}\]  
where we have introduced the third- and fourth-order statistics (skewness and kurtosis)  
\[ \begin{split}\mathbb{S}[X]&=\mathbb{E}\left[\frac {(X-\mathbb{E}[X])^{3}}{\mathbb{V}^{3/2}[X]}\right]\\ \mathbb{K}[X]&=\mathbb{E}\left[\frac{(X-\mathbb{E}[X] )^{4}}{\mathbb{V}^{2}[X]}\right]\end{split}  \tag{4}\]  
Equation (3) can be solved for the premium (which we label as \(P_{4}:=P\), to make it clear that we stop at the fourth-order term of [PERSON]'s expansion)  
\[P_{4}=\mathbb{E}[X]-\frac{1}{2}\frac{u^{ii}}{u^{i}}\mathbb{V}[X]+\frac{1}{6} \frac{u^{iii}}{u^{i}}\mathbb{S}[X]\mathbb{V}^{3/2}[X]-\frac{1}{24}\frac{u^{iv} }{u^{i}}\mathbb{K}[X]\mathbb{V}^{2}[X], \tag{5}\]  
This is to be compared with the standard second-order approximation, which would give us  
\[P_{2}=\mathbb{E}[X]-\frac{1}{2}\frac{u^{ii}}{u^{i}}\mathbb{V}[X], \tag{6}\]  
However, both expressions depend on the choice of the utility function. We can now introduce the Arrow-Pratt measure of Absolute Risk Aversion (ARA) [20]:  
\[A(x):=-\frac{u^{ii}(x)}{u^{i}(x)}, \tag{7}\]where the normalization by \(u^{i}(x)\) makes that measure of risk aversion independent of the unit of measurement adopted so that \(A(x)\) is a dimensionless quantity (hence the _absolute_ qualification). The second-order premium is then  
\[P_{2}=\mathbb{E}[X]+\frac{1}{2}A(X)\mathbb{V}[X], \tag{8}\]  
A popular choice for the utility function is the exponential form  
\[u(x)=1-e^{-\alpha x}, \tag{9}\]  
which results in the risk-aversion measure being constant \(A(x)\equiv\alpha\), where \(\alpha\) is called the risk-aversion coefficient. The exponential function is the only one possessing this Constant Absolute Risk Aversion (CARA) property. Due to the CARA property, the exponential utility function has been extensively employed in the literature, see, e.g., [8; 10; 21; 22]. In addition to the immediate simplification of the second-order premium, we can also recognize that the ratios of derivatives of the utility functions involved in the fourth-order premium become  
\[ \begin{split}\frac{u^{ii}}{u^{i}}&=\frac{-\alpha^{ 2}e^{-\alpha x}}{\alpha e^{-\alpha x}}=-\alpha\\ \frac{u^{iii}}{u^{i}}&=\frac{\alpha^{3}e^{-\alpha x }}{\alpha e^{-\alpha x}}=\alpha^{2}\\ \frac{u^{iv}}{u^{i}}&=\frac{-\alpha^{4}e^{-\alpha x }}{\alpha e^{-\alpha x}}-\alpha^{3},\end{split}  \tag{10}\]  
so that the premium can then be rewritten as  
\[P_{4}=\mathbb{E}[X]+\frac{1}{2}\alpha\mathbb{V}[X]+\frac{1}{6}\alpha^{2} \mathbb{S}[X]\mathbb{V}^{3/2}[X]+\frac{1}{24}\alpha^{3}\mathbb{K}[X]\mathbb{ V}^{2}[X] \tag{11}\]  
for the fourth-order case and  
\[P_{2}=\mathbb{E}[X]+\frac{1}{2}\alpha\mathbb{V}[X] \tag{12}\]  
for the second-order case. As to the proper value to assign to the risk-aversion coefficient, it should be chosen to reflect the individual sensitivity towards risk: the higher is \(\alpha\), the more importance is attributed to risk. However, some proposals have appeared in the literature to assign sensible values to \(\alpha\). [PERSON] and [PERSON] have considered the range of values \(\alpha\in[0.5,4]\)[23]. [PERSON] et al. [24] and [PERSON] [25] have considered several alternatives for risk-aversion coefficient. [PERSON] et al. [26] proposed to set it as the inverse of the expected loss  
\[\alpha_{Pit}=\frac{1}{\mathbb{E}[X]}. \tag{13}\]  
[PERSON] et al. proposed to set it as proportional to the inverse of the expected loss through the following formula [27]:  
\[\alpha_{Bab}=\frac{\ln\frac{1+2\eta}{1-2\eta}}{\mathbb{E}[X]}, \tag{14}\]  
where \(\eta\in[0,0.5]\) is the probability premium, i.e., the event probability in excess of 0.5 so that an individual gets the same utility as the status quo (i.e., it sees its utility unchanged by the event).