Keep It Real: Tail Probabilities of Compound Heavy-Tailed Distributions

TL;DR

This paper introduces a contour integration method for efficiently computing tail probabilities of heavy-tailed compound distributions, providing an alternative to Monte Carlo and FFT methods, with applications in financial risk assessment.

Contribution

The paper develops a novel analytical approach using contour integration to evaluate tail probabilities of heavy-tailed compound distributions, especially at high percentiles.

Findings

Efficient computation of tail probabilities via a rapidly convergent integral.

Application to Value at Risk (VaR) calculations for financial losses.

Extension potential for compound distributions of compound distributions.

Abstract

We propose an analytical approach to the computation of tail probabilities of compound distributions whose individual components have heavy tails. Our approach is based on the contour integration method, and gives rise to a representation of the tail probability of a compound distribution in the form of a rapidly convergent one-dimensional integral involving a discontinuity of the imaginary part of its moment generating function across a branch cut. The latter integral can be evaluated in quadratures, or alternatively represented as an asymptotic expansion. Our approach thus offers a viable (especially at high percentile levels) alternative to more standard methods such as Monte Carlo or the Fast Fourier Transform, traditionally used for such problems. As a practical application, we use our method to compute the operational Value at Risk (VAR) of a financial institution, where individual losses are modeled as spliced distributions whose large loss components are given by power-law or lognormal distributions. Finally, we briefly discuss extensions of the present formalism for calculation of tail probabilities of compound distributions made of compound distributions with heavy tails.