## 3 Comparison of Premiums
### Premium Uncertainty  
So far, we have assumed that the insurer can compute the premium if it knows the distribution of losses. However, such knowledge is typically approximate, since the insurer will typically estimate the parameters of the loss model based on observations. Consequently, the premium computed according to Equations (11) or (12) will be just an approximation of the correct value. It is important to quantify the uncertainty associated with the premium since that is an additional source of error, which may even mask the error due to a lower-order approximation. In this section, we compute the uncertainty in the premium due to an approximate knowledge of the loss distribution parameters. Since the computation of the premium is carried out through a complex function, we resort to a Taylor series approximation of that function by considering it as a function of the loss distribution parameters. The expansion is performed around the expected values of those parameters. All the distributions considered so far rely on two parameters so that we can write the Taylor series expansion \(T(\cdot)\) of the premium by adopting the following generic notation  
\[T(x,y)=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\frac{(x-x_{0})^{n}(y-y_{0})^{m}} {n!m!}\frac{\partial^{n+m}f(x_{0},y_{0})}{\partial x^{n}\partial y^{m}} \tag{22}\]  
where \(x\) and \(y\) are the estimates of the two parameters, and \(x_{0}\) and \(y_{0}\) are their expected values. If we stop at the first order, we get  
\[T(x,y)\simeq f(x_{0},y_{0})+f_{x}(x_{0},y_{0})(x-x_{0})+f_{y}(x_{0},y_{0})(y-y _{0}) \tag{23}\] \begin{table}
\begin{tabular}{l c c c} \hline \hline
**Intensity of Losses Distribution** & **Skewness** & **Kurtosis** & **Underestimation** \\ \hline GPD & \(\frac{2(1+\xi)\sqrt{1-2\xi}}{1-3\xi}\), \(\xi<\frac{1}{3}\) & \(\frac{3(1-2\xi)(2\xi^{2}+2+3)}{(1-3\xi)(1-4\xi)}\), \(\xi<\frac{1}{4}\) & YES \\ Lognormal & \((e^{2\xi}+2)\sqrt{e^{2\xi}-1}\) & \(e^{4\xi^{2}}+2e^{3\xi^{2}}+3e^{2\xi^{2}}-3\) & YES \\ Gamma & \(\frac{2}{\sqrt{a}}\) & \(\frac{6}{a}+3\) & YES \\ Pareto & \(\frac{2(1+a)}{a-3}\sqrt{\frac{6-2}{a}}\), \(a>3\) & \(\frac{6(a^{2}+a^{2}-6a-2)}{a(a-3)(a-4)}+3\), \(a>4\) & YES \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Moments of common distributions and underestimation presence (the conditions on parameters \(\xi\) and \(a\) assure that skewness and kurtosis are not infinite). \begin{table}
\begin{tabular}{l c c} \hline \hline
**Distribution** & **Parameters** & **PDF** \\ \hline GDP & \(\beta\), \(\xi\) & \(f(x)=\frac{1}{\beta}\left(1+\frac{\xi x}{\beta}\right)^{-\frac{\xi+1}{4}}\) \\ Lognormal & \(\mu\), \(\sigma^{2}\) & \(f(x)=\frac{1}{\sqrt{2\pi nx}}e^{-\frac{\ln(x)-x}{2\sigma^{2}}}\) \\ Gamma & \(a\), \(b\) & \(f(x)=\frac{y^{\xi}}{\Gamma(a)}x^{a-1}e^{-bx}\) \\ Pareto & \(a\), \(h\) & \(f(x)=\alpha\frac{y^{\xi}}{x^{a+1}}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Some common distribution for lossesIn the following, the parameters of the distributions of interest will be considered to be random variables to describe their uncertainty, so that the premium (from now on we use the generic symbol \(P\)) is a random variable as well. To gauge the uncertainty on the premium, we will compute its variance. As a simplifying assumption, we consider the estimates of the two parameters as independent of each other, though they will be typically estimated on the same sample of observations. The general expression for the variance of \(P\) is then  
\[ \begin{split}\mathbb{V}[P]\simeq&\mathbb{V}[P(x_{0},y_{0})+P_{x}(x_{0},y_{0})(x-x_{0})+P_{y}(x_{0},y_{0})(y-y_{0})]\\ &=\mathbb{V}[P(x_{0},y_{0})]+\mathbb{V}[P_{x}(x_{0},y_{0})(x-x_{0 })+P_{y}(x_{0},y_{0})(y-y_{0})]\\ &=P_{x}^{2}(x_{0},y_{0})\mathbb{V}[x]+P_{y}^{2}(x_{0},y_{0}) \mathbb{V}[y]+P_{x}(x_{0},y_{0})P_{y}(x_{0},y_{0})\mathbb{C}ov[x,y]\\ &=P_{x}^{2}(x_{0},y_{0})\mathbb{V}[x]+P_{y}^{2}(x_{0},y_{0}) \mathbb{V}[y]\end{split}