  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.
