J. D. Opdyke | Searcharxiv

arXiv2017

Fast, Accurate, Straightforward Extreme Quantiles of Compound Loss Distributions

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).

q-fin.RM↗

arXiv2014

Estimating Operational Risk Capital with Greater Accuracy, Precision, and Robustness

The largest US banks are required by regulatory mandate to estimate the operational risk capital they must hold using an Advanced Measurement Approach (AMA) as defined by the Basel II/III Accords. Most use the Loss Distribution Approach (LDA) which defines the aggregate loss distribution as the convolution of a frequency and a severity distribution representing the number and magnitude of losses, respectively. Estimated capital is a Value-at-Risk (99.9th percentile) estimate of this annual loss distribution. In practice, the severity distribution drives the capital estimate, which is essentially a very high quantile of the estimated severity distribution. Unfortunately, because the relevant severities are heavy-tailed AND the quantiles being estimated are so high, VaR always appears to be a convex function of the severity parameters, causing all widely-used estimators to generate biased capital estimates (apparently) due to Jensen's Inequality. The observed capital inflation is sometimes enormous, even at the unit-of-measure (UoM) level (even billions USD). Herein I present an estimator of capital that essentially eliminates this upward bias. The Reduced-bias Capital Estimator (RCE) is more consistent with the regulatory intent of the LDA framework than implementations that fail to mitigate this bias. RCE also notably increases the precision of the capital estimate and consistently increases its robustness to violations of the i.i.d. data presumption (which are endemic to operational risk loss event data). So with greater capital accuracy, precision, and robustness, RCE lowers capital requirements at both the UoM and enterprise levels, increases capital stability from quarter to quarter, ceteris paribus, and does both while more accurately and precisely reflecting regulatory intent. RCE is straightforward to implement using any major statistical software package.

q-fin.RM↗