Distributions without finite moments

A statistician colleague of mine posed a question to me regarding certain distributions used in loss models. Naturally occurring distributions, such as inverse Pareto, do not have finite moments. But naturally occurring questions within actuarial science, such as those involving expected shortfall, involve moments or moments of right tails. Of course, this can be infinite.

When one is modeling with a distribution that has infinite first or second moment, and one has practical questions where infinite first or second moment provides some obstruction, is there some remediation? Is there some common practice for reaching meaningful answers while circumventing the lack of moments?

Solution 1:

Speaking as a non-life actuary, while we often use distributions without finite moments (such as a Pareto with $\alpha \leq 1$), we never do so on an unlimited basis. Agreed, that way lies madness. What we are often tasked to do is find the loss in a layer, or the loss subject to some cap. The cap may be the entire surplus of the company, although it is more often a percentage when dealing with company-wide ERM issues. Regardless, all distributions have finite moments if you truncate or even censor them from above. That makes the math and simulations work.