Evaluation of Value-at-Risk (VaR) using the Gaussian Mixture Models

Abstract

The normality of the distribution of stock returns is one of the basic assumptions in financial mathematics. Empirical studies, however, undermine the validity of this assumption. In order to flexibly fit complex non-normal distributions, this article applies a Gaussian Mixture Model (GMM) in the context of Value-at-Risk (VaR) estimation. The study compares the forecasting ability of GMM with other widespread VaR approaches, scrutinizing the data on the daily log-returns for a wide range of “S&P 500” stocks in two periods: from 2006 to 2010 and from 2016 to 2021. The statistical and graphical analysis revealed that GMM quickly and adequately adjusts to significant and rapid stock market changes, although the remaining methods delay. The study also found that the ratio of short-term and long-term standard deviations significantly improves the GMM and other methods’ ability to predict VaR, reflecting the observed features of analyzed stock log-returns.

Keywords

• Gaussian Mixture Model
• normal distribution
• heavy tails
• Value-at-Risk
• Monte-Carlo simulations
• backtesting