Using truncated Lévy flight to estimate downside risk

Abstract

It is well known that the normal distribution model fails to describe the fat tails of markets. The Lévy stable distribution model, meanwhile, has fat tails but leads to an infinite variance, thus complicating risk estimation. This study introduces truncated Lévy flight (TLF) — a better distribution model that has fat tails, finite variance, and more importantly, scaling properties. The paper uses TLF to estimate the downside risk of a variety of asset classes. The downside risk measure used is the conditional value-at-risk (CVaR). The study shows that the lognormal model can underestimate the monthly CVaR by 2.27 per cent for the S&P 500 Index, 0.48 per cent for the US Long-Term Government Bond Index, and 1.16 per cent for the MSCI UK Equity Index. Moreover, the paper extends a univariate TLF model to a multivariate TLF model to study the impact of fat tails on portfolios' downside risk and wealth accumulation.