Estimating Flexible, Fat-Tailed Asset Return Distributions

Abstract

We introduce new robust numerical methods, based on the minimum relative U−entropy (MRUE) principle, to estimate univariate probability density functions for power-law (fat-tailed) random variables. The semi-parametric models that we estimate via convex programming are flexible enough to conform well to potentially plentiful data for not-too-extreme values, while allowing for power-law tails (which need not be symmetric). We observe that a number of well-known power-law models, including the exponential, Pareto, Student-t, and skewed generalized-t (SGT) distributions, are special cases of the family of power-law probability densities that we consider. We benchmark our method against state-of-the-art asset return models on S&P500 index returns, individual stock returns, and power price returns and find that our models outperform the benchmarks out-of-sample. We attribute this out-performance to simultaneously conforming to data where it is plentiful, while building in reasonably conservative tails.

Keywords: Minimum Relative U−Entropy, Probability Distribution, Fattailed, Power-Law Distribution, Financial Data, Asset Returns