Estimation of the capital under the Loss Distribution Approach (LDA) requires calculation of the distribution for the aggregate (compound) loss
where the frequency N is a discrete random variable. Closed-form solutions are not available for the distributions typically used in OpRisk and numerical evaluation is required. This is one of the classical problems in risk theory. Before the era of personal computers, it was calculated using approximations such as that based on the asymptotic central limit theory or on ad hoc reasoning using, for example, shifted Gamma approximation. With modern computer processing power, these distributions can be calculated virtually exactly using numerical algorithms. The easiest to implement is the Monte Carlo method. However, because it is typically slow, Panjer recursion and Fourier inversion techniques are also widely used. Both have a long history, but their applications to computing very high quantiles of the compound distribution functions with high frequencies and heavy tails are only recent developments and various pitfalls exist. This chapter describes numerical algorithms that can be successfully used for this problem. In particular, Monte Carlo, Panjer recursion, and Fourier transformation methods are presented. Several closed-form approximations are also reviewed.
13.1 Analytic Solution
In general, ...