Operational loss with correlated frequency and severity: an analytical approach

Daniel H. Stahl

Abstract

ABSTRACT

Operational risk modeling presents a number of difficulties. The severity distribution is often very heavy tailed (moments of second order and higher are infinite) making Monte Carlo simulations ineffective. Analytical solutions like the loss distribution approach (LDA) model are not flexible enough to model the empirical correlations often found between frequency and severity distributions. This paper proposes a significant generalization of the LDA model. This generalization involves treating operational risk as a Lévy jump-diffusion, which enables autocorrelation in the frequency distribution. By using a change of measure in the complex domain, this specification also allows for correlation between the severity and frequency distributions. The resulting characteristic function can be numerically approximated by solving a system of ordinary differential equations. Using the Runge-Kutta method, the number of steps to retain accuracy throughout the loss distribution is small: in computation tests even thirty-two steps retained excellent accuracy. This method is tested using three separate severity distributions. The impact of the resulting correlations on the capital required for operational risk can be large: in a computational experiment the required capital at the 99.9% level is over 55% larger than in a zero-correlation model.