  In this paper we establish the error rate of first order asymptotic
approximation for the tail probability of sums of log-elliptical risks. Our
approach is motivated by extreme value theory which allows us to impose only
some weak asymptotic conditions satisfied in particular by log-normal risks.
Given the wide range of applications of the log-normal model in finance and
insurance our result is of interest for both rare-event simulations and
numerical calculations. We present numerical examples which illustrate that the
second order approximation derived in this paper significantly improves over
the first order approximation.
