How to derive formula (5.5) with Taylor's expansion in the following link?
The difficult part of my question is that i had never seen Var(S) appear in Taylor's expansion
https://i.stack.imgur.com/j0dyh.png
EDIT: Let $\kappa_S(R)=\log(\mathrm{E}[e^{RS}])$, where $S$ is an aggregate loss random variable, and $R$ is a risk parameter (constant).
(5.5) $\pi(S)=\frac{1}{R}\kappa_S(R)=\frac{1}{R}[\mathrm{E}(S)R+\mathrm{Var}(S)\frac{R^2}{2}+...]\approx\mathrm{E}(S)+\mathrm{Var}(S)\frac{R}{2}$
So the OP's question is why $\mathrm{Var}(S)$ shows up in the Taylor series of $\kappa_S(R)=\log(\mathrm{E}[e^{RS}])$.