The Asmussen–Kroese Monte Carlo estimators of P(S n > u) and P(S N > u) are known to work well in rare event settings, where S N is the sum of independent, identically distributed heavy-tailed random variables X 1,…,X N and N is a nonnegative, integer-valued random variable independent of the X i . In this paper we show how to improve the Asmussen–Kroese estimators of both probabilities when the X i are nonnegative. We also apply our ideas to estimate the quantity E[(S N -u)+].