Using SAS ® PROC SEVERITY to evaluate Quantile Approximation Techniques for Compound Distributions popular in Operational Risk

The objective of this paper is to use a simulation comparison to evaluate quantile approximation methods in terms of their practical usefulness and potential applicability in an operational risk context. A popular method in modelling the aggregate loss distribution in risk and insurance is the Loss Distribution Approach (LDA). Many banks currently use the LDA for estimating regulatory capital for operational risk. 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 regulatory capital, an extreme quantile of the aggregate loss distribution has to be estimated. A number of numerical approximation techniques have been proposed to approximate the extreme quantiles of the aggregate loss distribution. We use PROC SEVERITY to fit various severity distributions to simulated samples of individual losses from a preselected severity distribution. The accuracy of the approximations obtained, is then evaluated against a Monte Carlo approximation of the extreme quantiles of the compound distribution resulting from the preselected severity distribution. 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 it is very easy to implement. INTRODUCTION Compound frequency distributions have many applications, particularly in insurance and risk management. For example, in insurance, they are used to model the distribution of aggregate claims on an insurance policy over a fixed period or, in operational risk, to model the annual aggregate loss distribution. An aggregate loss (or claims) distribution is a compound distribution resulting from a random sum of losses (or claims), where the individual losses (or claims) are independently and identically distributed according to some distribution (the severity distribution) and the number of losses (or claims) is independently distributed according to some frequency distribution. In operational risk the aggregate annual loss distribution is constructed in order to determine regulatory capital, which is equivalent to determining the 0.999 quantile of this distribution. In practice this quantity is estimated by using the so-called loss distribution approach (LDA). This method is usually implemented by fitting a distribution to the individual losses and then the aggregate loss distribution is constructed through brute force Monte Carlo simulation of random sums of losses, where the number of losses are generated from a (fitted) frequency distribution (e.g. Poisson) and the individual losses generated from the (fitted) severity distribution. This approach has been studied extensively and it has been shown that the resulting capital estimates are not trustworthy and depend on many uncontrollable factors (see e.g. Cope et al. 2009). In practice the LDA boils down to modelling the loss frequency by the Poisson or negative binomial distributions and the severity distribution by a wide class of sub-exponential distributions (e.g. the Burr, gand-h, lognig, lognormal and combinations of these with the generalised Pareto). The quantiles of a compound distribution can typically not be calculated exactly and have to be approximated in some way. In practice, brute force Monte Carlo (MC) simulation methods are mostly used to approximate these extreme quantiles. Depending on the accuracy required, these MC simulations are computer intensive and even utilising today’s computer power can become impractical when implemented. A number of numerical approximation techniques have been proposed to overcome this difficulty. Numerical approximation recursive techniques that can be used to approximate the quantile of the compound distribution include the Panjer recursion (Panjer, 1981) and techniques using Fourier inversion and the fast Fourier transform (see e.g. Grübel and Hermesmeier, 1999). These techniques require several input parameters that have to be selected carefully to ensure convergence and are computationally intensive.