Fast, Accurate, Straightforward Extreme Quantiles of Compound Loss Distributions

Abstract

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).