A multiplier approach for nonparametric estimation of the extreme quantiles of compound frequency distributions

Helgard Raubenheimer, Tertius de Wet, Charl Pretorius and P. J. de Jongh

Abstract

Operational risk reserves are still widely estimated using the loss distribution approach. The accuracy of the estimation depends heavily on the accuracy with which the extreme quantiles of the aggregate loss distributions are estimated. Various approaches have been proposed to estimate the extreme quantiles of this compound distribution, including estimators based on the single-loss and perturbative approximations, which rely on estimating an even more extreme quantile of the underlying severity distribution. However, estimation of these extreme quantiles may be inaccurate due to fitting a parametric severity distribution that fits the bulk of the data well but struggles to capture the tail behavioral characteristics of the distribution that generated the loss data. To circumvent this problem, we propose estimating nonparametrically a less extreme or lower quantile of the severity distribution, hopefully with better accuracy, and then multiplying this lower quantile by a certain factor to obtain an estimate of the required extreme quantile of the compound distribution. The factor or multiplier is derived by using extreme value theory and the single-loss and perturbative approximations, after which these quantities are estimated nonparametrically. This approach is evaluated by means of a simulation study that suggests the second-order multiplier estimator based on the second-order perturbative approximation is a good choice for practical applications.