  Numerical evaluation of performance measures in heavy-tailed risk models is
an important and challenging problem. In this paper, we construct very accurate
approximations of such performance measures that provide small absolute and
relative errors. Motivated by statistical analysis, we assume that the claim
sizes are a mixture of a phase-type and a heavy-tailed distribution and with
the aid of perturbation analysis we derive a series expansion for the
performance measure under consideration. Our proposed approximations consist of
the first two terms of this series expansion, where the first term is a
phase-type approximation of our measure. We refer to our approximations
collectively as corrected phase-type approximations. We show that the corrected
phase-type approximations exhibit a nice behavior both in finite and infinite
time horizon, and we check their accuracy through numerical experiments.
