Journal of Operational Risk

Fast, accurate and straightforward extreme quantiles of compound loss distributions

J.D. Opdyke

  • The most widely used extreme quantile approximation for compound loss distributions (the Single Loss Approximation, “SLA,” of Degen, 2010) contains a discontinuity that systematically and materially biases capital estimates when the tail index approaches one.
  • The modified, interpolated SLA (MISLA) developed herein avoids that bias and is compared to all relevant competitors. 
  • Based on speed, accuracy, and ease-of-implementation, MISLA and one other method comparably outperform all others, but based on consistency with widespread practice, MISLA arguably is the preferred method of approximation.

In this paper, we present an easy-to-implement, fast and accurate method for approximating extreme quantiles of compound loss distributions (frequency + severity), which are commonly used in insurance and operational risk capital models. The interpolated single-loss approximation (ISLA) of J. D. Opdyke is based on the widely used single-loss approximation (SLA) of M. Degen. It maintains two important advantages over its competitors. First, ISLA correctly accounts for a discontinuity in SLA that can otherwise systematically and notably bias the quantile (capital) approximation under conditions of both finite and infinite mean. Second, because it is based on a closed-form approximation, ISLA maintains the notable speed advantages of SLA over other methods requiring algorithmic looping (eg, 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 (eg, in sme power studies). The modified ISLA (MISLA) presented herein increases the range of application across the severity distributions most commonly used in these settings; it is tested against extensive Monte Carlo simulation (one billion years’ worth of losses) and the best competing method (the perturbative expansion (PE2) of L. Hernández, J. Tejero, Al. Suárez and S. Carrillo-Menéndez) 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 approach banks that are already using SLA (and failing to take into account its biasing discontinuity).

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