Simple approximations of ruin probabilities

A “simple approximation” of a ruin probability is an approximation using only some moments of the claim distribution and not the detailed tail behaviour of that distribution. Such approximations may be based on limit theorems or on more or less ad hoc arguments. The most successful simple approximation is certainly the De Vylder approximation, which is based on the idea to replace the risk process with a risk process with exponentially distributed claims such that the three first moments coincide. That approximation is known to work extremely well for “kind” claim distributions. The main purpose of this paper is to analyse the De Vylder approximation and other simple approximations from a more mathematical point of view and to give a possible explanation why the De Vylder approximation is so good.