Estimation of Truncated Data Samples in Operational Risk Modeling

Abstract

This article addresses challenges of estimating operational risk regulatory capital when a loss sample is truncated from below at a data collection threshold. Recent operational risk literature reports that the attempts to estimate loss distributions by the maximum likelihood method are not always successful under the truncation approach that accounts for the existence of censored losses—the likelihood surface is sometimes ascending with no global solution. The literature offers an alternative called the shifting approach, which estimates the loss distribution without taking into account censored losses. We present a necessary and sufficient condition for the existence of the global solution to the likelihood maximization problem under the truncation approach when the true loss distribution is lognormal, and derive a practically explicit expression for the global solution. We show by a simulation study that, as the sample size increases, the capital bias by the truncation approach declines while the bias by the shifting approach does not.

Estimating operational risk capital with greater accuracy, precision and robustness

“…This speeds computer runtime by almost an order of magnitude, so this approximation is used for this severity for both MLE and RCE-based capital estimates. Also derived in Appendix D is a closed-form analytic expression for the mean of 61 As mentioned previously, these findings are exactly consistent with other empirical results in the literature (for example, see Ergashev et al, 2014, andCavallo, 2012a). 62 See for example, Opdyke, 2013, Opdyke and Cavallo, 2012a, 2012b, Zhou et al, 2013, and Joris, 2013 These are three of the four parametric severity distributions listed in the most recent Interagency Guidance on Operational Risk AMA severity estimation (see OCC, 2014 the Truncated LogGamma severity, 64 which also decreases runtimes over the alternative requiring numeric integration.…”

“…So from an empirical perspective, we are squarely in the bias-zone: bias is material for many, if not most estimations of capital at the UoM level. 38 In fact, this is exactly what Ergashev et al (2014) found in their study comparing capital based on shifted vs. truncated lognormal severity distributions. The latter exhibited notable bias that disappeared as sample sizes increased up to n = 1,000, exactly as in the simulation study in this paper.…”

“…Sample sizes tested, for a ten year period, included average number of loss events = λ = 15, 25, 50, 75, and 100 for samples of approximately n ≈ 150, 250, 500, 750, and 1,000 loss events. 51 This is a very wide range of sample sizes compared to those examined in the relevant literature (see Ergashev et al, 2014, Opdyke and Cavallo, 2012aand 2012b, and Joris, 2013, and it arguably covers the lion's share of sample sizes in practice, unfortunately with the exception of the very small UoM's. For all sample 50 Again, "better" here means with greater capital accuracy, greater capital precision, and greater capital robustness.…”

“…Operational risk losses are simulated based on a Poisson frequency distribution, and six of the most commonly used severity distributions: 62 the LogNormal, LogGamma, and the Generalized Pareto (GPD) distributions, 63 as well as the truncated versions of each with a truncation threshold of H = $10,000, arguably the most widely used data collection threshold. As described above, the use of truncated distributions is the most widely accepted method for addressing data collection thresholds, unlike some alternatives that have been pillared in regulatory review processes and in the literature (for example, the use of so-called "shifted" distributions -see Schevchenko, 2009, andErgashev et al, 2014; but for a counter-argument supporting shifted distributions, see Cavallo et al, 2012).…”

Model Uncertainty in Operational Risk Modeling Due to Data Truncation: A Single Risk Case

“…This is the part of the aggregate model where initial assumptions about the data collection threshold are most influential. A number of authors have examined some aspects of this topic in the past (e.g., Cavallo et al 2012;Chernobai et al 2007;Ergashev et al 2016;Luo et al 2007;Moscadelli et al 2005). The modeling approaches they (collectively) considered include: the empirical approach, the "naive" approach, the shifted approach, and the truncated approach.…”