Fitting Operational Risk Data Using Limited Information Below The Threshold

This paper presents a methodology to calibrate the distribution of losses observed in operational risk events. The method is specifically designed to handle the situation where individual event information is only available above an approved threshold and a limited set of below threshold information (BTI). The method requires the use of the number of events and the sum of the total losses below the threshold. Studies are performed demonstrating the improved stability of the fitted distribution parameters as the number of events above the threshold are reduced. This is compared to the conventional truncated maximum likelihood estimator (MLE). To demonstrate the improved stability of the parameters estimation using the BTI, a series of fits are performed to $m$ samples of $N$ events drawn from a Log-normal distribution, this is compared to estimates of the truncated MLE. The BTI fitting methodology produces a better estimator of the population distribution parameters with reduced bias and dispersion. An estimation is made of the $99.9^{th}$ percentile of the severity distribution. It is seen that the BTI provides a better estimator of the percentile with reduced uncertainty