Let X i (i=1,2, ...) be a sequence of subexponential positive independent and identically distributed random variables. In this paper, we offer two alternative approaches to obtain higher-order expansions of the tail of [image omitted] and subsequently for ruin probabilities in renewal risk models with claim sizes X i . In particular, these emphasize the importance of the term [image omitted] for the accuracy of the resulting asymptotic expansion of [image omitted] . Furthermore, we present a more rigorous approach to the often suggested technique of using approximations with shifted arguments. The cases of a Pareto type, Weibull and Lognormal distribution for X i are discussed in more detail and numerical investigations of the increase in accuracy by including higher-order terms in the approximation of ruin probabilities for finite realistic ranges of s are given.