Determination of the fraction of losses and their probabilities by type of risk and business line from aggregate loss data

Henryk Gzyl and Argimiro Arratia

Abstract

Basel III rules require financial institutions to report operational losses and detail them in a database. Operational losses should be classified by the Basel II-determined risk types and also by line of business. While the operational risk capital requirement is now calculated using the standardized measurement approach (SMA) based on gross income, the collected data is nevertheless still useful for risk management purposes. However, the frequency of losses and the incurred losses per line of business and risk type may only be available in aggregate form, and this data needs to be disaggregated for better risk management. We propose a disaggregation method to derive the individual loss severities and the frequency of these losses per business line and risk type. This information is useful to the risk manager because it highlights where their attention should be focused. The mathematical problem in each case is similar to that of reconstructing a joint probability from its marginals. We solve this problem by minimizing a convex objective function (which happens to be an entropy of the Fermi–Dirac type) subject to the appropriate constraints. The method is nonparametric and model-free, and therefore it does not require parameters to be fitted to the data. If we think of the joint losses as the values of a random variable, and of the joint probabilities as the probabilities of obtaining these values, we can then use the tabulated data to compute expected losses as well as risk measures such as value-at-risk and expected shortfall.