# Closed-form approximations for operational value-at-risk

## Lorenzo Hernández, Jorge Tejero, Alberto Suárez and Santiago Carrillo-Menéndez

#### Abstract

ABSTRACT

In the loss distribution approach, operational risk is modeled in terms of the distribution of sums of independent random losses. The frequency count in the period of aggregation and the severities of the individual loss events are assumed to be independent of each other. Operational value-at-risk is then computed as a high percentile of the aggregate loss distribution. In this work we present a sequence of closed-form approximations to this measure of operational risk. These approximations are obtained by the truncation of a perturbative expansion of the percentile of the aggregate loss distribution at different orders. This expansion is valid when the aggregate loss is dominated by the maximum individual loss. This is the case in practice, because the loss severities are typically very heavy-tailed and can be modeled with subexponential distributions, such as the lognormal or the generalized Pareto distribution. The two lowest-order terms in the perturbative series are similar to the single-loss approximation and to the correction by the mean, respectively. Including higher-order terms leads to significant improvements in the quality of the approximation. Besides their accuracy and low computational cost, these closed-form expressions do not require that the moments of the severity distribution, including the mean, be finite.