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Computing tails of compound distributions using direct numerical integration. (English) Zbl 1183.91185

Summary: An efficient adaptive direct numerical integration (DNI) algorithm is developed for computing high quantiles and conditional value at risk (VaR) 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 conditional VaR are 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 methods. These results, particularly relevant to operational risk modeling, can serve as benchmarks for comparing different numerical methods. We find that the adaptive DNI can achieve high accuracy with relatively coarse grids. It is much faster than Monte Carlo and competitive with FFT in computing high quantiles and conditional VaR of compound distributions in the case of moderate to high frequencies and heavy tails.

MSC:

91G60 Numerical methods (including Monte Carlo methods)
65D30 Numerical integration
65C05 Monte Carlo methods
65T50 Numerical methods for discrete and fast Fourier transforms
91B30 Risk theory, insurance (MSC2010)

Software:

QUADPACK; UNCMND; pchip
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