A simulation comparison of quantile approximation techniques for compound distributions popular in operational risk

Abstract

Many banks currently use the Loss Distribution Approach (LDA) for estimating economic and regulatory capital for operational risk under Basel's Advanced Measurement Approach. The LDA requires, amongst others, the modelling of the aggregate loss distribution in each operational risk category (ORC). The aggregate loss distribution is a compound distribution resulting from a random sum of losses, where the losses are distributed according to some severity distribution and the number (of losses) distributed according to some frequency distribution. In order to estimate the economic or regulatory capital in a particular ORC, an extreme quantile of the aggregate loss distribution has to be estimated from the fitted severity and frequency distributions. Since a closed-form expression for the quantiles of the resulting estimated compound distribution does not exist, the quantile is usually approximated by using brute force Monte Carlo simulation which is computationally intensive. However, a number of numerical approximation techniques have been proposed to lessen the computational burden. Such techniques include Panjer recursion, the fast Fourier transform, and different orders of both the single-loss approximation and perturbative approximation. The objective of this paper is to compare these methods in terms of their practical usefulness and potential applicability in an operational risk context. We find that the second order perturbative approximation, a closed-form approximation, performs very well at the extreme quantiles and over a wide range of distributions and is very easy to implement. This approximation can then be used as an input to the recursive fast Fourier algorithm to gain further improvements at the less extreme quantiles.

Construction of Forward-Looking Distributions Using Limited Historical Data and Scenario Assessments

“…The large number of simulation repetitions involved in the MC approaches above motivates the use of other numerical methods such as Panjer recursion, methods based on fast Fourier transforms [5] and the single loss approximation (SLA) method (see e.g. [6] methods, the interested reader is referred to [7]. The SLA has become very popular in the financial industry due to its simplicity and can be stated as follows: If T is the true underlying severity distribution function of the individual losses and λ the true annual frequency then the 100 1 À γ ðÞ % VaR of the compound loss distribution may be approximated by T À1 1 À γ=λ ðÞ or, as modified by [8] for large λ,by T À1 1 À γ=λ ðÞ þ λμ, where μ is the finite mean of the true underlying severity distribution.…”

“…Next b and the threshold q ¼ q b must be specified. One possibility is to take b as the smallest of the scenario c-year multiples and to estimate q b as the corresponding smallest of the scenario assessmentsq b provided by the experts, in this caseq 7 . T e x ðÞcan be estimated by fitting a parametric family F e x, θ ðÞ (such as the Burr) to the data x 1 , x 2 , … , x K or by calculating the empirical distribution and then conditioning it to the interval 0,q b ÀÃ .…”