## 3 Comparison of Premiums  
In Section 2, we have derived the premium under two different approximation levels. We wish to see if the more accurate fourth-order approximation entails a higher premium, i.e., if the use of the usual second-order approximation results in an underestimated premium, which may be dangerous for the insurer. Though overestimation is similarly dangerous, here we take the viewpoint of the insurer, for which premium overestimation is the lesser problem, since it may result in a reduced number of insurance subscriptions. In the following, we limit ourselves to consider underestimation. In this section, we compare the two expressions and look for the conditions where that happens. Throughout this section, the number \(n\) of claims will be considered to be a fixed quantity; this limitation will be removed in Section 4.  
From the two expressions (11) and (12), we can form their ratio  
\[\begin{split} R&=\frac{P_{4}}{P_{2}}=\frac{\mathbb{E}[ X]+\frac{1}{2}\alpha\mathbb{V}[X]+\frac{1}{6}\alpha^{2}\mathbb{S}[X]\mathbb{V}^{3/2}[X]+ \frac{1}{24}\alpha^{3}\mathbb{K}[X]\mathbb{V}^{2}[X]}{\mathbb{E}[X]+\frac{1}{2 }\alpha\mathbb{V}[X]}\\ &=1+\frac{1}{12}\frac{4\alpha^{2}\mathbb{S}[X]\mathbb{V}^{3/2}[X] +\alpha^{3}\mathbb{K}[X]\mathbb{V}^{2}[X]}{2\mathbb{E}[X]+\alpha\mathbb{V}[X]} \end{split} \tag{17}\]  
We consider the case when setting the premium through the usual second-order approximation can lead to underestimating the premium; we focus on underestimation, since it is the more dangerous error of the two (under- vs overestimation), leading to the possibility that premiums do not cover losses. Going back to Equation (17), we see that we have underestimation when \(R>1\). Therefore,  
Figure 1: Ratio of the Pitacco and Babcock risk-aversion coefficient.  
whether the fourth-order premium is larger or not than its second-order counterpart depends just on the sign and relative values of the skewness and kurtosis of the loss. In Table 2, we report the general conditions for underestimation of the premium when we stop at the second-order approximation. In two out of four cases (same signs for skewness and kurtosis), the conclusions are general. In the other two cases, the possibility of underestimation depends on the specific distribution of losses. Since we must have \(R>1\) for underestimation, the underestimation condition can be formulated as  
\[4\alpha^{2}\mathbb{S}[X]\mathbb{V}^{3/2}[X]+\alpha^{3}\mathbb{K}[X]\mathbb{V }^{2}[X]>0\rightarrow\mathbb{S}[X]>-\frac{1}{4}\alpha\mathbb{K}[X]\sqrt{ \mathbb{V}[X]} \tag{18}\]  
Since the overall loss is determined by the accrual of losses over the single events, we can derive a condition on the statistics of any single event by recalling that  
\[X=\sum_{i=1}^{n}L_{i}, \tag{19}\]  
where \(L_{i}\) is the loss suffered during th \(i\)-th event and \(n\) is the overall number of events (and therefore claims), with \(L_{1}\),..., \(L_{n}\) being i.i.d. random variables.  
Due to the i.i.d. nature of the \(L_{i}\) variables, the skewness and the kurtosis of the random variable \(X\) may be derived from that of the loss in any single event (see [28]):  
\[\mathbb{S}[X] =\frac{1}{\sqrt{n}}\mathbb{S}[L_{i}] \tag{20}\] \[\mathbb{K}[X] =\frac{1}{n}\mathbb{K}[L_{i}].\]  
The underestimation condition can then be reformulated as follows, by replacing Equation (20) in Equation (18):  
\[\frac{1}{\sqrt{n}}\mathbb{S}[L_{i}]>-\frac{1}{4n}\alpha\mathbb{K}[L_{i}]\sqrt {n\mathbb{V}[L_{i}]}\rightarrow\mathbb{S}[L_{i}]>-\frac{1}{4}\alpha\mathbb{ K}[L_{i}]\sqrt{\mathbb{V}[L_{i}]} \tag{21}\]