Computing tails of compound distributions using direct numerical integration

Abstract

An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional Value at Risk (CVaR) of compound distributions using characteristic functions. A key innovation of the numerical scheme is an effective tail integration approximation that reduces the truncation errors significantly with little extra effort. High precision results of the 0.999 quantile and CVaR were obtained for compound losses with heavy tails and a very wide range of loss frequencies using the DNI, Fast Fourier Transform (FFT) and Monte Carlo (MC) methods. These results, particularly relevant to operational risk modelling, can serve as benchmarks for comparing different numerical methods. We found that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than MC and competitive with FFT in computing high quantiles and CVaR of compound distributions in the case of moderate to high frequencies and heavy tails.

An Introduction to Stochastic Particle Integration Methods: With Applications to Risk and Insurance

“…It is in these four key elements that we argue stochastic particle based numerical solutions to such inference on risk measures and tail functionals can be of direct utility to complement such asymptotic results. However, as all practitioners will know, the naive implementation of standard Monte Carlo and stochastic integration approaches to such problems will produce often poor results even for a considerable computational budget, see discussions in [63]. There is therefore a computational challenge for estimation of risk measures for such heavy-tailed annual loss distributions that we argue can be addressed by stochastic particle methods.…”

“…In this section we detail a special sub-set of algorithms from within the stochastic particle integration methods that was specifically developed to solve problems for risk and insurance in [36] and discussed in comparison to specific FFT methods in [63]. The class of recursive solutions developed is very general and applicable to a wide range of insurance and risk settings.…”

Calculation of aggregate loss distributions

“…is defined for a given quantile q α , that is, the quantile H −1 (α) has to be computed first. It is easy to show (see formulas (40)(41)(42)(43) in Luo and Shevchenko (2009)) that in the case of nonnegative severities, the above integral can be calculated via characteristic function as…”

Implementing loss distribution approach for operational risk

“…Typically, both methods are faster than Monte Carlo by a factor of a few orders. Other methods to calculate the compound distribution include direct integration of the CF [33] and a hybrid method combining Panjer recursion, importance sampling, and trans-dimensional Markov chain Monte Carlo (MCMC) considered in [34].…”

“…The prior distribution (h) can be estimated using appropriate expert opinions or using external data. Thus, the posterior distribution (h|y) combines the prior knowledge (expert opinions or external data) with the observed data using formula (33). In practice, we start with the prior (h) identified by expert opinions or external data.…”