Application of the convolution operator for scenario integration with loss data in operational risk modeling

ABSTRACT When using the advanced measurement approach to determine required regulatory capital for operational risk, expert opinion is applied via scenario analysis to help quantify exposure to high-severity events. A methodology is presented that makes use of the convolution operator to integrate scenarios into a baseline model. Using a baseline loss distribution model calibrated on historical losses and a scenario-derived loss distribution calibrated on scenario data points, the addition of both random processes equates to the convolution of the corresponding densities. Using an analogy from digital signal processing, the commutative property of convolution allows one function to smooth and average the other. The inherent uncertainty in scenario analysis has caused concern amongst practitioners when too much emphasis has been placed on absolutes in terms of quantified frequency/severity estimates. This method addresses this uncertainty and produces a combined loss distribution that takes information from the entire domain of the calibrated scenario distribution. The necessary theory is provided within and an example is shown to provide context.