Risk Measurement with a Safety Belt: Pareto Meets Chebyshev

Abstract

Risk measures based on the Gaussian distribution are prone to understate the probability of extreme events. To capture fat tails and extreme events, we combine the Pareto law with finite variance bounds of Chebyshev. This density encompasses the tail behaviour of a wide range of random variables with unknown distribution. It provides a well-defined conservative measure of risks. Applications to measurement of forecast uncertainty and to value at risk and expected shortfall illustrate the approach empirically.