I'm new to this risk thing. I am trying to obtain the mean excess loss function evaluated at a point for the following Pareto distribution:
$F(x)=1-(\frac{\theta}{x})^{\alpha}$
The excess loss function is:
$e_{x}(v)=\frac{\int_{v}^{\infty}S(x)dx }{S(v)}$
where $S(x)=1-F(x)$. But that just explodes whenever $\alpha<1$, which isn't supposed to be a requirement for the distribution. Am I doing something wrong?
No, you are not doing anything wrong. The Pareto distribution does not have a finite expectation when $\alpha \le 1$; therefore, mean excess loss will also not be finite for such $\alpha$. Why do you think that it must be finite for all possible values of the shape parameter? There are many heavy-tailed distributions that fail to have a finite expectation.