Monte Carlo Simulation with Mathematical Convolution of Frequency and Severity Distributions in Operational Risk Capital Model in the Basel Accords
Authors: Mun Johnathan 
Published: March 2022

In October 2014, the Basel Committee on Banking Supervision released a Basel Consultative Document entitled, “Operational Risk: Revisions to the Simpler Approaches,” and in it describes operational risk as “the sum product of frequency and severity of risk events within a one-year time frame and defines the Operational Capital at Risk (OPCAR) as the tail-end 99.9% Value at Risk (VaR)” [1]. The Basel Consultative Document describes a Single Loss Approximation (SLA) model defined as \(F_{S}^{-1}(p)=F_{X}^{-1}\left[1-\frac{1-p}{\lambda}\right]+(\lambda-1) E[X]\), where the inverse of the compound distribution \(F_{S}^{-1}\) is the summation of the unexpected losses \(U L=F_{X}^{-1}\left[1-\frac{1-p}{\lambda}\right]\) and expected losses \(E L=(\lambda-1) E[X] ; \lambda\) is the Poisson distribution’s input parameter (average frequency per period; in this case, 12 months); and \(X\) represents one of several types of continuous probability distributions representing the severity of the losses (e.g., Pareto, Log Logistic, etc.). The Document further states that this is an approximation model limited to subexponential-type distributions only and is fairly difficult to compute. The distribution’s cumulative distribution function (CDF) will need to be inverted using Fourier transform methods, and the results are only approximations based on a limited set of inputs and their requisite constraints. Also, as discussed, the SLA model proposed in the Basel Consultative Document significantly underestimates OPCAR. The OPCAR methodology estimates a bank’s operational risk capital through the convolution of a single severity distribution and a single frequency distribution. Each bank’s OPCAR estimate was assumed to refer to a unique operational risk category, having a specific aggregated frequency and severity of losses [1].
The concept of significant loss events attributed to operational risk was introduced in the Basel II-IV accords by the Bank of International Settlements. Loss processes that contribute most to capital risk, the so-called high-consequence, low-frequency loss processes, with heavy-tailed loss process modeling where implications of such tail assumptions for the severity risk model is important in operational risk [2].
This paper provides a new and alternative convolution methodology to compute OPCAR that is applicable across a large variety of continuous probability distributions for risk severity and includes a comparison of their results with Monte Carlo risk simulation methods. As will be shown, both the new algorithm using numerical methods to model OPCAR and the Monte Carlo risk simulation approach tends to the same results and seeing that simulation can be readily and easily applied in the CMOL software and Risk Simulator software, simulation methodologies should be used for the sake of simplicity. While the Basel Committee has, throughout its Basel II-IV requirements and recommendations, sought after simplicity so as not to burden banks with added complexity, it still requires sufficient rigor and substantiated theory. Monte Carlo risk simulation methods pass the test on both fronts and are, hence, the recommended path when modeling OPCAR.