Maxentropic and Quantitative Methods in Operational Risk Modeling

Abstract

In risk management the estimation of the distribution of random sums or collective models from historical data is not a trivial problem. This is due to problems related with scarcity of the data, asymmetries and heavy tails that makes difficult a good fit of the data to the most frequent distributions and existing methods.In this work we prove that the maximum entropy approach has important applications in risk management and Insurance Mathematics for the calculation of the density of aggregated risk events, and even for the calculation of the individual losses that come from the aggregated data, when the available information consists of an observed sample, which we usually do not have any information about the underlying process.From the knowledge of a few fractional moments, the Maxentropic methodologies provide an efficient methodology to determine densities when the data is scarce, or when the data presents correlation, large tails or multimodal characteristics. For this procedure, the input would be the sample moments E[e −αS ] = µ(α) or some interval that encloses the difference between the true value of µ(α) and the sample moments (for eight values of the Laplace transform), this interval would be related to the uncertainty (error) in the data, where the width of the interval may be adjusted by convenience. Through a simulation study we analyze the quality of the results, considering the differences with respect to the true density and in some cases the study of the size of the gradient and the time of convergence. We compare four different extensions of Maxentropic methodologies, the Standard Method of Maximum Entropy (SME), an extension of this methodology allows to incorporate additional information through a reference measure, called Method of Entropy in the Mean (MEM) and two extensions of the SME that allow introduce errors, called SME with errors or SMEE.Although our motivating example come from the field of Operational Risk analysis, the developed methodology may be applied to any branch of applied sciences.