  We determine the rate of decrease of the right tail distribution of the
exponential functional of a Levy process with a convolution equivalent Levy
measure. Our main result establishes that it decreases as the right tail of the
image under the exponential function of the Levy measure of the underlying Levy
process. The method of proof relies on fluctuation theory of Levy processes and
an explicit path-wise representation of the exponential functional as the
exponential functional of a bivariate subordinator. Our techniques allow us to
establish rather general estimates of the measure of the excursions out from
zero for the underlying Levy process reflected in its past infimum, whose area
under the exponential of the excursion path exceed a given value.
