In a wide class of problems in insurance and financial mathematics, it is of interest to study the extremal events of a perpetuity sequence.
This paper addresses the problem of numerically evaluating these rare event probabilities. Specifically, an importance sampling algorithm is described which
is efficient in the sense that it exhibits bounded relative error, and which is optimal in an appropriate asymptotic sense. The
main idea of the algorithm is to use a ``dual"
change of measure, which is employed to an associated Markov chain over a randomly-stopped time interval.
The algorithm also makes use of the so-called forward sequences generated to the given stochastic recursion,
together with elements of Markov chain theory.