  We present an easily implemented, fast, and accurate method for approximating
extreme quantiles of compound loss distributions (frequency+severity) as are
commonly used in insurance and operational risk capital models. The
Interpolated Single Loss Approximation (ISLA) of Opdyke (2014) is based on the
widely used Single Loss Approximation (SLA) of Degen (2010) and maintains two
important advantages over its competitors: first, ISLA correctly accounts for a
discontinuity in SLA that otherwise can systematically and notably bias the
quantile (capital) approximation under conditions of both finite and infinite
mean. Secondly, because it is based on a closed-form approximation, ISLA
maintains the notable speed advantages of SLA over other methods requiring
algorithmic looping (e.g. fast Fourier transform or Panjer recursion). Speed is
important when simulating many quantile (capital) estimates, as is so often
required in practice, and essential when simulations of simulations are needed
(e.g. some power studies). The modified ISLA (MISLA) presented herein increases
the range of application across the severity distributions most commonly used
in these settings, and it is tested against extensive Monte Carlo simulation
(one billion years' worth of losses) and the best competing method (the
perturbative expansion (PE2) of Hernandez et al., 2014) using twelve
heavy-tailed severity distributions, some of which are truncated. MISLA is
shown to be comparable to PE2 in terms of both speed and accuracy, and it is
arguably more straightforward to implement for the majority of Advanced
Measurement Approaches (AMA) banks that are already using SLA (and failing to
take into account its biasing discontinuity).
