Description: We propose asymptotically correct two-sided bounds for random sums where the number of summands has an arbitrary distribution which can be viewed as ruin probabilities or accumulated claim sizes in various risk processes. The bounds are fairly new and they reveal the real accuracy of some well-known and widely used asymptotic formulas. It turns out that this accuracy can be poor and the range where these formulas give appropriate approximations depends on the shape of the tail distribution of the summands. This dependence is examined in the research.