On the Basel Liquidity Formula for Elliptical Distributions

TL;DR

This paper justifies and generalizes the Basel liquidity formula for risk capital, extending it from Gaussian to elliptical distributions, and introduces a Fourier method for expected shortfall calculation.

Contribution

It provides a theoretical extension of the Basel formula to elliptical distributions and develops a Fourier-based approach for expected shortfall estimation.

Findings

Basel formula is conservative for heavy-tailed elliptical distributions.

Generalization to multivariate elliptical distributions broadens applicability.

Fourier method enables efficient expected shortfall calculation.

Abstract

A justification of the Basel liquidity formula for risk capital in the trading book is given under the assumption that market risk-factor changes form a Gaussian white noise process over 10-day time steps and changes to P&L are linear in the risk-factor changes. A generalization of the formula is derived under the more general assumption that risk-factor changes are multivariate elliptical. It is shown that the Basel formula tends to be conservative when the elliptical distributions are from the heavier-tailed generalized hyperbolic family. As a by-product of the analysis a Fourier approach to calculating expected shortfall for general symmetric loss distributions is developed.