Journal of Operational Risk

Estimating operational risk capital with greater accuracy, precision and robustness

J.D. Opdyke


The largest US banks and systemically important financial institutions are required by regulatory mandate to estimate the operational risk capital they must hold using an advanced measurement approach as defined by the Basel II/III Accords. Most of these institutions use the loss distribution approach (LDA),which defines the aggregate loss distribution as the convolution of a frequency distribution and a severity distribution representing the number and magnitude of losses, respectively. Capital is a value-at-risk (VaR) estimate of this annual loss distribution (ie, the quantile corresponding to the 99.9th percentile, representing a one-in-a-thousand-year loss, on average). In practice, the severity distribution drives the capital estimate, which is essentially a very large quantile of the estimated severity distribution. Unfortunately, when using LDA with any of the widely used severity distributions (ie, heavy-tailed, skewed distributions), all unbiased estimators of severity distribution parameters generate biased capital estimates apparently due to Jensen's inequality: VaR always appears to be a convex function of these severities' parameter estimates because the (severity) quantile being estimated is so large and the severities are heavy-tailed. The resulting bias means that capital requirements always will be overstated, and this inflation is sometimes enormous (sometimes even billions of US dollars at the unit-of-measure level). In this paper an estimator of capital is presented that essentially eliminates this upward bias when used with any commonly used severity parameter estimator. The reduced-bias capital estimator (RCE), consequently, is more consistent with regulatory intent regarding the responsible implementation of the LDA framework than other implementations that fail to mitigate, if not eliminate this bias. The RCE also notably increases the precision of the capital estimate and consistently increases its robustness to violations of the independent and identically distributed data presumption (which are endemic to operational risk loss event data). So with greater capital accuracy, precision and robustness, the RCE lowers capital requirements at both the unit-of-measure and enterprise levels, increases capital stability from quarter to quarter, ceteris paribus, and does both while more accurately and precisely reflecting regulatory intent. The RCE is straightforward to explain, understand and implement using any major statistical software package.

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